The talk will survey various models of traffic flow, with particular focus on new mathematical problems arising from some of these models. A large literature is currently available on particle models, describing the position of every car in terms of a large number of ODEs, and macroscopic models, where the traffic density is determined by a PDE.
Vehicular traffic can also be analyzed from the point of view of decision theory. Indeed, daily traffic patterns arise as the outcome of the decisions of a large number of drivers, who choose their departure time and route to destination in an ``optimal" way. Some recent work and open problems will be discussed.

This question has been much discussed in physics and one suggestion is that the
long time persistence of classical/quantum correspondence is due to interaction of a
small, observed system with a larger environment. Lindblad or GKSL evolution is one of the
standard models for describing such interactions. In that context the question of the
length of time of classical/quantum agreement was recently revisited in physics by
Hernández-Ranard-Riedel.
In my talk I will introduce the concept of Lindblad evolution and present results showing
that the evolution of a quantum observable remains close to the classical Fokker-Planck
evolution in the Hilbert-Schmidt norm for times vastly exceeding the Ehrenfest time (the
limit of such an agreement when there is no interaction with a larger system). The time
scale is the same as in two recent papers by Hernández-Ranard-Riedel but the statement
andmethods are different. The talk is based on joint work with J Galkowski and numerical
results obtained jointly with Z Huang. I will also comment on recent progress on trace
class estimates by Z Li and on the hypoelliptic case by H Smith.

Free boundary problems in partial differential equations (PDEs) are a class of mathematical problems in which both the solution to the PDE and the domain on which it is defined must be determined simultaneously, as the region is not known in advance. These problems arise in many physical and engineering contexts, such as fluid dynamics, solid mechanics, and heat conduction. In this talk, we will focus on certain free boundary problems related to thermal insulation, where either the boundary itself must be found, or the presence of an insulating material requires determining its optimal placement in order to minimize heat loss or maximize energy efficiency.

The irreversible behaviour of macroscopic processes governed by reversible laws of classical or quantum physics has been a captivating subject in statistical mechanics going back at least to the pioneering work of Boltzmann. The general consensus reached by the middle of the last century was that the second law of thermodynamics, which states that entropy increases with time, is empirical in nature, and that the probability of a negative fluctuation of entropy is so small that it cannot be observed in practice. A natural question is the quantitative description of that claim. A breakthrough on this subject came in the middle of nineties due to the work of Evans-Searles and Gallavotti-Cohen.
In this talk, I shall illustrate their discovery on the simplest example of finite state Markov chains. It will be shown that, under some natural hypotheses, one can define an entropy production observable whose time averages satisfy the large deviation principle. The resulting rate function possesses a symmetry property which implies that the probability of a negative value for the mean entropy production is exponentially suppressed by that of the opposite positive value. I shall also discuss the realisation of this programme in the context of fluid flows.

Currents mod p are a suitable generalization of classical chains mod p, i.e. of finite combinations of smooth submanifolds with coefficients in the cyclic group $\mathbb Z_p$. By the pioneering work of Federer and Fleming it is possible to minimize the area in this context and, for instance, represent mod $p$ homology classes with area minimizers. For $p>2$ typically (i.e. away froma small set of exceptional points) one would expect such minimizers to be a union of smooth minimal surfaces joining together (``in multiples of $p$'s'') at some common boundary. This is however surprisingly challenging to prove, especially for even $p$'s, and up until recently only known for $p=3$ and $4$ in codimension $1$. In this talk I will explain the outcome of a series of more recent works (some joint of the speaker with Hirsch, Marchese, Stuvard and Spolaor, some by Wickramasekera and Minter-Wickramasekera, and some joint of the speaker with Minter and Skorobogatova) which confirms this picture, with varying degrees of precision in
a variety of situations.

After recalling the basic notions coming from differential geometry, the talk will be focused on spaces satisfying Ricci curvature lower bounds. The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the 80s and was pushed by Cheeger and Colding in the 90s who investigated the fine structure of possibly non-smooth limit spaces. A completely new approach via optimal transportation was proposed by Sturm and Lott-Villani around twenty years ago. Via such an approach one can give a precise definition of what means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds. The talk is meant to be an introduction to the topic, accessible to non-specialists and as self-contained as possible.

At a large scale, minimal solutions of long-range phase coexistence equations tend to separate pure phases through an interface minimizing a nonlocal perimeter functional. These new geometric objects have something in common with classical minimal surfaces, but also exhibit several new and very specific patterns, which we will try to investigate and present.

In this lecture, we will discuss periodic and nonperiodic averaging principles for functional differential equations, as well as stability results.

The talk will be about a number-the quasicentral modulus-associated with an n-tuple of bounded Hilbert space operators and a rearrangement invariant norm. It underlies the multivariable generalizations
of the classical theorems of Weyl-von Neumann-Kuroda and Kato-Rosenblum in perturbation theory. On the other hand it provides an operator theory approach to Lebesgue measure and to certain other self-similar measures on fractals, as well as to the Kolmogorov-Sinai dynamical entropy. Recently a non-commutative analogy with condenser capacity in nonlinear potential theory is emerging.

We shall introduce the notion of m-twist condition for the optimal transportation problems. This notion together with Kantorovich duality provide an effective tool to study and describe the support of
optimal plans for the mass transport problem involving general cost functions. We also establish a criterion for the uniqueness of optimal plans not necessarily induced by a single map.

I will first give a brief overview of the notion of mean-field control and games. In particular, I will focus on the Eulerian formulation, through the notion of master equation, which is a nonlinear PDE on the space of probability measures. This PDE is notoriously difficult to solve, at least in a classical sense, except in cases presenting a form of convexity or monotonicity in the argument of the measure. I will then present two problems outside the convex/monotonic framework: (i) non-convex mean-field control problems and their particle approximation, (ii) mean-field games subject to certain non-standard forms of common noise.

Many deterministic systems display chaotic features: the stronger the chaotic features, the better the system can be by approximated by a probabilistic model, an idea that can be traced back to Boltzmann and explains the success of the branch of dynamical systems known as ergodic theory. In this talk we will discuss systems which display only 'mild’ chaotic features, such as the geodesic flow on surfaces which a flat geometry or the Ehrenfest model in mathematical physics.
Recent breakthroughs on our understanding of the latter model, introduced more than a century ago, were made possible by the powerful tools exploiting moduli spaces of surfaces and Teichmueller dynamics, an area which has attracted the work of several Fields medallists. We will in particular highlight some results of probabilistic flavor that can still be proven for these deterministic systems, hidden in the fractal structure of trajectories and the 'deterministic random walks' that describe them.

Let X be a smooth projective 3-fold over the complex
numbers. Following work of Thomas, Behrend-Fantechi, and others, one
has a virtual fundamental class in the Chow group of 0-cycles on the
Hilbert scheme of dimension 0, length n subschemes of X, the degree of
which is the nth Donaldson-Thomas invariant of X
Now take X over an arbitrary field k. We have developed a construction
of virtual fundamental classes with values in an arbitrary motivic
cohomology theory. An example of such, a ``quadratic'' analog of the
Chow groups, is the cohomology of the sheaf of Witt rings, which leads
to a refinement of the classical DT-invariants to quadratic
DT-invariants with values in the Witt ring of quadratic forms over k.
We will discuss some developments and conjectures for these refined DT
invariants, including some computations of the signature of these
invariants due to Anneloes Viergever.

A resolution of singularities of a singular algebraic variety is a proper morphism from a smooth algebraic variety which induces an isomorphism of dense open subsets. An old but extremely important theorem of Hironaka proves that, over a field of characteristic zero, any algebraic variety admits a resolution.
Similarly, an absorption of singularities of a singular algebraic variety is a proper morphism to a smooth algebraic variety which induces an isomorphism of dense open subsets. In contrast to resolution, existence of absorption is a very rare phenomenon.
I will talk about a categorical version of resolution and absorption of singularities, give some examples, and try to demonstrate how useful these notions are.

The interaction between learning theory and harmonic analysis was emphasized by mathematics of quantum computing. One of the outstanding open problems in this area concerns the Bohnenblust--Hille inequality
that generalizes a celebrated Littlewood’s 4/3 lemma. How to learn (approximately and with large probability) a very large matrix in a relatively small number of random quantum quarries? Motivated by this question, a non-commutative counterpart of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in Cambyse Rouz, Melchior Wirth, and Haonan Zhang. By waving the hands I will explain the proof of non-commutative Bohnenblust--Hille inequalities with constants that are dimension-free. As applications, we study learning problems of quantum observables. Using Heisenberg—Weyl basis one
reduces the quantum problem to commutative problem: the Bohnenblust—Hille inequality for cyclic groups (joint with Haonan Zhang, Joe Slote). To prove the Bohnenblust—Hille inequality for cyclic groups turned out to be a challenging problem. I will explain the progress in this area.

We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. A simpler model is then derived, describing the controlled evolution of a contaminated set.
The first part of the talk will focus on the optimal control of 1-dimensional traveling wave profiles.
Namely, we seek a control with minimum norm which produces a traveling profile with a given speed.
In turn, this leads to a family of optimization problems for a moving set, related to the original reaction-diffusion equation via a sharp interface limit. In connection with moving sets, the second part of the talk will present some results on controllability, existence of optimal strategies, and optimality conditions. Some open questions will be discussed.

The seminar will address general, sharp estimates of the singular values of quantum differentials of Connes spectral triples, on which Noncommutative Geometry is based. Special attention will concern the natural spectral triple of a Dirichlet form and those arising from negative definite functions on countable discrete groups.

The development of a small star on earth will lead to a sustainable, virtually limitless, safe source of base-load electricity.
The challenges, the successes and the remaining open questions along the road to a fusion reactor, in physics and technology, will be discussed, along with the European strategy.

We investigate the existence of non trivial doubly periodic solutions of the Benjamin-Ono equation, depending on the space period, the time period, the space average, and the space regularity of the solution. In particular, for doubly periodic solutions with space average 0 and with time period and space period equal to 2\pi, we prove that L^2 regularity implies analyticity, while there exists more such solutions which belong to the Sobolev space H^s for every s<0. The crucial tool is the Birkhoff map for the Benjamin-Ono equation recently obtained in collaboration with T. Kappeler and P. Topalov.

In this talk we will discuss a geometrical approach to the problem of the KPZ Universality.
Instead of looking at the height (interface) function and Airy processes, we will focus on the statistics
of shocks and points of concentration of mass. We will also discuss the connection with the problem
of the coalescing Brownian motions and coalescing fractional Brownian motions.

In order to define a differential operator on a domain with boundary one often needs
to specify boundary conditions. In the presence of singular terms this can be quite tricky. It may lead to interesting and surprising phase diagrams.
I will illustrate this with the so-called perturbed Bessel operators,
that is one-dimensional Schrodinger operators on the half-line of the form
$$-\partial_x^2+(m^2-1/4) 1/x^2+ Q(x).$$
The talk will be based on my joint work with Jeremy Faupin.

Il seminario prende in considerazione il ragionamento diagrammatico in geometria elementare e le sue origini nella pratica matematica greca. Si discutono alcuni limiti ed alcune potenzialità di questo approccio ala geometria. Si mostra che il ruolo delle inferenze diagrammatiche diminuì nel corso della storia della geometria moderna anche prima di arrivare alla matematica astratta del diciannovesimo secolo. Le ragioni dell'abbandono delle pratiche diagrammatiche sembra legato alla trasformazione della geometria da una scienza delle figure in una scienza dello spazio. Si mostra allora che tale trasformazione ha preceduto e reso possibile la nascita della matematica astratta e di un ideale dimostrativo non-diagrammatico.

The classic Liouville theorem states that any bounded harmonic function on Euclidean space is constant. The same holds for discretely harmonic functions on a lattice. But what happens when the Euclidean lattice is replaced by other graphs, in particular Cayley graphs of groups? We will survey old and new results on relations between geometric and algebraic properties of groups, harmonic functions and probability. Based on joint work with various subsets of Itai Benjamini, Ariel Yadin, Hugo Duminil-Copin, Gidi Amir and Maria Gerasimova.
Light refreshments will be served after the seminar.

By the early 1970's two important conjectures, the Wigner-Yanasee-Dyson conjecture and the Strong Subadditivity of Quantum Entropy conjecture, had attracted the attention on many mathematicians and physicists. These conjectures were both proved in 1973 by Leib and by Lieb and Ruskai respectively. The methods introduced for their solution were powerful, and immediately found other applications, for example in Lindblad's 1975 proof of the Data Processing Inequality, now one of the cornerstones of quantum information theory. In this talk I will briefly explain the history before turning to modern developments and a modern perspective that has led to new inequalities and stronger versions of known inequalities. In this latter part, I will focus on recent work of myself, and myself in collaboration with Alexander Mueller-Hermes and with Haonan Zhang. I will also briefly discuss some recent applications.

To participate in presence, please register using the link
https://forms.office.com/r/8HQCMBnec5
Maryna Viazovska was born in Kyiv, Ukraine. She did her bachelor studies at the Kyiv National Taras Shevchenko University and completed her MSc at the Technical University Kaiserslautern.
She obtained her PhD in 2013 in Bonn.
Viazovska was a visitor at the Institut des Hautes Etudes Scientifiques in 2013, Dirichlet postdoctoral fellow at the Berlin Mathematical School and the Humboldt University of Berlin in 2014–2016 and a Minerva Distinguished Visitor] at Princeton University in 2017. She joined EPFL in 2017 as Tenure-Track Assistant Professor and was promoted to Full Professor in 2018.
In 2016, Viazovska received the Salem Prize and, in 2017, the Clay Research Award and the SASTRA Ramanujan Prize for her work on sphere packing and modular forms. In December 2017, she was awarded a 2018 New Horizons Prize in Mathematics. She was an invited speaker at the 2018 International Congress of Mathematicians. For 2019 she was awarded the Ruth Lyttle Satter Prize in Mathematics and the Fermat Prize. She is one of the 2020 winners of the EMS Prize. In 2020, she also received the National Latsis Prize awarded by the Latsis Foundation. She was elected to the Academia Europaea in 2021. She was appointed Senior Scholar at the Clay Mathematics Institute in July 2022.
She was awarded the Fields Medal in July 2022.

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.
This is a joint work with C. Tauber and C.-A. Pillet

The cup product on topological manifolds or the intersection product
on algebraic varieties induce quadratic forms which turn out to be a
fine invariant of these geometric objects. We will discuss some old
theorems on the signature of these quadratic forms and some
applications both of geometric and arithmetic origins. Finally we will
study an old conjecture on Grothendieck about those signatures and
explain some new evidences.

We consider classical the question of prescribing the scalar curvature
of a manifold via conformal deformations of the metric, dating back to
works by Kazdan and Warner, and related to the well-known Yamabe problem.
It is challenging as it involves equations of critical type, leading to possible
blow-ups of solutions. The question is mainly understood in low dimensions,
where blow-ups of solutions are proven to be "isolated simple".
We will present some recent progress in arbitrary dimensions and
for the case of manifolds with boundary, where blow-ups of new type occur.

The Minimal Model Program (=MMP) aims to explain how
algebraic varieties are composed out of simpler pieces. The talk gives
a broad overview of the program and its applications. It surveys
recent progress in our understanding of holomorphic differential forms
on singular spaces, and explains the connection to the MMP.
Applications pertaining to classification and characterisation of
algebraic varieties will be mentioned in brief.

This talk is devoted to an overview of recent results on the optimal control of dynamical systems on probability measures modelizing the evolution of a large number of agents.
The system is composed by a number of agents so huge, that at each time only a statistical description of the state is available. A common way to model such kind of system is to consider a macroscopic point of view, where the state of the system is
described by a (time-evolving) probability measure on $R^d$ (which the underlying space where the agents move). So we are facing to a two-level system where the mascroscopic dynamic concerns probability measure while the microscopic dynamic - which describes the evolution of an individual agent - is a controlled differential equation on $ R^d$.
Associated to this dynamics on the Wasserstein space, one can associate a cost which allows to define a value function. We discuss the characterization of this value function through a Hamilton Jacobi Bellman equation stated on the Wasserstein space. We also discuss the problem of compatibility of state constraints with a multiagent control system. Since the Wasserstein space can be also viewed as the set of the laws of random variables in a suitable $L^2$ space, one can hope to reduce our problems to $L^2$ analysis. We discuss when this is possible.
This overview talk is based on several works in collaboration with I. Averboukh, P. Cardaliaguet, G. Cavagnari, C. Jimenez and A. Marigonda.

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

Purity in arithmetic algebraic geometry is the phenomenon of certain invariants of algebraic varieties being insensitive to removing closed subvarieties of sufficiently large codimension, the Hartogs' extension principle being the basic example. The most delicate case of such results is in mixed characteristic (0, p), in which a new approach based on perfectoid rings has recently been introduced. I will overview it and discuss the results that it has yielded so far.

In this talk, we are going to discuss the dynamics of a polaron, at large coupling. For initial data of Pekar product form, with a coherent phonon field and with the electron minimising the corresponding field energy, we provide a norm approximation of the evolution, valid up to times quadratic in the coupling constant. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations and corrected by a Bogoliubov dynamics describing quantum fluctuations. I will explain the similarities with the study of the evolution of interacting bosons. This talk is based on joint work with Nikolai Leopold, David Mitrouskas, Simone Rademacher and Robert Seiringer.

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various areas of Science. Research interests have evolved over the 80 years: while mathematicians were originally focussed on abstract properties (like notions of weak solutions and Cauchy problem in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. Entropy methods for nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as the optimal rate of decay of an entropy for an associated evolution equation. Much more can be learned on the way.
This lecture is intended to give an overview of various results on some Gagliardo-Nirenberg-Sobolev and Caffarelli-Kohn-Nirenberg inequalities obtained during the last decade. It will not be a global picture of an active area of research but more a series of snapshots aiming at the illustration of some emerging tools and new directions of research.

Convex billiards are a classical topic in conservative dynamics. Typically, their dynamics is qualitatively very intricate, since it showcases a coexistence of hyperbolic dynamics and KAM phenomena. Understanding long-term statistical properties of the dynamics with the current technology is essentially an intractable problem.
Here I venture in the opposite direction and I will discuss dynamical inverse problems: how much geometrical information can be extracted from the dynamics?
More precisely: what can be deduced about the billiard table if one knows the lengths of all periodic orbits? The quantum version of this question has been famously stated as "Can one hear the shape of a drum?"
In this talk I will review the latest results and describe the next steps in this direction. This is a joint project with Vadim Kaloshin.

We present a progress report on the development of Discontinuous Petrov-Galerkin methods for the convection-reaction problem in context of time-stepping and space-time discretizations of Boltzmann equations [1].
The work includes a complete analysis for both conforming (DPGc) and non-nonconforming (DPGd) versions of the DPG method employing either globally continuous or discontinuous piece-wise polynomials to discretize the traces.
The results include construction of a local Fortin operator for the case of constant convection and a global discrete stability analysis for both DPGc and DPGd methods.
The theoretical findings are illustrated with numerous numerical experiments in two space dimensions.
This is a joint work with Nathan Roberts from Sandia National Laboratories.
Slides (PDF)
[1] L. Demkowicz, N. Roberts, "The DPG Method for the Convection–Reaction Problem Revisited", submitted.

The talk is devoted to a mathematical introduction to Machine Learning
from the point of view of inverse problems. In the presentation, we focus on
supervised learning problems in the framework of kernel methods.

Prestrained plates are slender materials that develop internal
stresses at rest, deform out of plane even without external forces, and
exhibit nontrivial 3d shapes. Bilayer plates are slender structures made of
two materials that react differently to environmental (thermal,
electrical or chemical) actuation. In both cases the plates can exhibit large
bending deformations that are geometrically nonlinear. We present
reduced nonconvex models, develop variational formulations, and
design local discontinuous Galerkin methods (LDGs). Moreover, we prove
Gamma-convergence of the discrete energies and analyze discrete gradient
flows for the computation of minimizers that provide control of the
metric defect. We document the performance of the LDG methods with
several insightful simulations.

Plateau problem consists in finding a surface of minimal area among the ones spanning a given curve. It is among the oldest problem in the calculus of variations and its study lead to wonderful development in mathematics.
Federer and Fleming integral currents provide a suitably weak solution to the Plateau problem in arbitrary Riemannian manifolds, in any dimension and
co-dimension. Once this week solution has been found a natural question consists in understanding whether it is classical one. i.e. a smooth minimal surface. This is the topic of the regularity theory, which naturally splits into interior regularity and boundary regularity.
After the monumental work of Almgren, revised by De Lellis and Spadaro, interior regularity is by now well understood. Boundary regularity is instead less clear and some new phenomena appear.
Aim of the talk is to give an overview of the problem and to present some boundary regularity results we have obtained in the last years.

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects. But how much understanding can one really gain about an infinite group by examining its finite images? Sometimes little, sometimes a lot. In this colloquium talk, I will sketch the rich history of this problem and describe how input from geometry and low-dimensional topology, mingling with algebra and arithmetic, have transformed the subject in recent years. I shall also describe some open problems.

After presenting some of the recent progress on existence of minimal surfaces, I will talk about my recent work with Calegari and Marques where we introduce a quantity that counts some minimal surfaces in negatively curved manifolds and which is minimized by the hyperbolic metric

A third-order (in time) JMGT equation is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a non-linear propagation of high frequency acoustic waves. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences-including high intensity focused ultrasound [HIFU] technologies, lithotripsy, welding and others. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing a classical heat transfer by heat waves gives rise to the third order in time derivative scaled by a small parameter $\tau > 0$, the latter represents the thermal relaxation time parameter and is intrinsic to the properties of the medium where the dynamics occurs.
The aim of the present lecture is to provide a brief overview of recent results in the area which are pertinent to both linear and non-linear dynamics. From the mathematical point of view JMGT, can be seen as a nonlinear perturbation of a third order strictly hyperbolic system, which however has a characteristic boundary. This feature has, of course, strong implications on boundary behavior [both regularity and controllability] which can not be patterned after classical hyperbolic systems theory [as it is the case for the wave equation]. As a consequence, the analysis of regularity [both forward and inverse estimates] is particularly challenging-even in the linear case. Several recent results pertaining to boundary stabilization, optimal control and asymptotic analysis of the solutions with vanishing time relaxation parameter will be presented and discussed. In all these case, peculiar features associated with the third order dynamics leads to novel phenomenological behaviors.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no cure. Yet, a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages associated with various cognitive deficits and pathologies. How can we use mathematics to gain insight into this process and, doing so, gain understanding about how the brain works? In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions.

The description of all the solutions of the equation $x^2+y^2=z^2$ in integral numbers (a.k.a. Pythagorean triples) is a very ancient problem: a Babylonian clay tablet from about 1800BC may contain some solutions, Pythagoras (about 500BC) seems to have known one infinite family of solutions, and so did Plato... This gives a first example of a rational variety: the rational points on the circle with equation $x^2+y^2=1$ can be algebraically parametrized by one rational parameter. More generally, one says that a variety, defined by a system of polynomial equations, is rational if its points (the solutions of the system) can be algebraically parametrized, in a one-to-one fashion, by independent parameters. I will begin with easy standard examples, then explain and apply some (not-so-recent) techniques that can be used to prove that some varieties (such as the set of rational solutions of the equation $x^3+y^3+z^3+t^3=1$) are not rational.

L-functions are one of the central objects of study in
number theory. There are many beautiful theorems and many more open
conjectures linking their values to all kinds of arithmetic problems.
I will talk about the mysteries surrounding these L-values and
describe some of the progress that has recently been made towards
understanding them.

The study of the structural properties of the set of points at which the viscosity solution of a first order Hamilton-Jacobi equation fails to be differentiable - in short, the singular set - started with the paper [On the singularities of viscosity solutions to Hamilton-Jacobi-Bellman equations, Indiana Univ. Math. J. 36 (1987), pp.501-524] by Mete Soner and myself. These thirty years have registered enormous progress in the comprehension of the way how singularities propagate: a fine measure theoretical analysis of the singular set has been developed, it has been understood how singular dynamics is driven by generalised characteristics, and deep topological applications to the structure of the cut locus on a Riemannian manifold have been pointed out. In this talk, I will revisit the milestones of the theory and discuss possible developments and open problems.

Automorphism groups of discrete structures such as graphs carry a totally disconnected topology: both the groups and the structures they act on are thus zero-dimensional. The group topology is often non-discrete, and it has been found in recent years that there is a strong interplay between the algebraic and topological properties of these groups.
Totally disconnected locally compact groups are, it turns out, locally profinite and have a rich structure theory which has parallels with the theory of Lie groups, although it is more complicated.
This theory is still being developed and, in addition to Lie theory, draws on results about finite groups and from geometric group theory.
Summarising progress so far, an analysis of the locally profinite structure of the groups corresponds to the local theory of Lie groups, and a canonical form for group elements corresponds to eigendecomposition in the Lie algebra of a Lie group. A decomposition theory separates the cases of discrete and profinite groups, which are treated as negligible, from cases which are not negligible such as simple Lie groups over $p$-adic or function fields and automorphism groups of regular trees.
The techniques developed have been applied so far to answer questions about ergodic theory, random walks and arithmetic groups.

In this talk I wish to present some of the ideas at the heart of the theory of regularity structures (RS), introduced by Martin Hairer in 2014. RS are perhaps best described as a theory of Taylor expansions in a fractal (random) setting. I plan to show how this theory is based on a fascinating interplay between several different disciplines, as announced by my title.

The effect of non-degeneracy conditions on the applicability of variational gluing arguments for some variational problems possessing mountain pass structure will be discussed.

Looijenga—Lunts and Verbitsky (LLV) have shown that the cohomology of
a compact hyper-Kaehler manifold admits the action of a big Lie
algebra g, generalizing the usual sl(2) Hard Lefschetz action. We
compute the LLV decomposition of the cohomology for the known classes
of hyper-Kaehler manifolds (i.e. K3^n, Kim_n, OG6, and OG10). As an
application, we easily recover the Hodge numbers of the exceptional
example OG10. In a different direction, we establish the so-called
Nagai’s conjecture (on the nilpotency index for higher degree
monodromy operators) for the known cases. More interestingly, based
on the known examples, we conjecture a new restriction on the
cohomology of compact hyper-Kaehler manifolds, which in particular
implies the vanishing of the odd cohomology as soon as the second
Betti number is large enough relative to the dimension.
This is joint work with M. Green, Y. Kim, and C. Robles.

Wetting phenomena at small scales are an area where chemistry,
physics, mathematics, and engineering intersect. In recent years,
driven also by molecular dynamics, new concepts have been introduced
to describe the statics and dynamics of wetting, allowing new
insights into the old problems of surface forces. Among these
problems, two prominent ones are an appropriate mathematical modeling
of the moving contact line where liquid, solid, and surrounding vapor
meet, and how such models influence the macroscopic properties of the
flow. After a general framing -- the classical setting of droplets'
statics and dynamics, diffuse and sharp interface models, classical
and new descriptions of the contact line -- I shall review the PDE
theory for one of such models -- the so-called thin-film equation --, mainly focusing on the two aforementioned problems and on some of the most interesting current challenges.

In my talk, I will discuss a reformulation of quantum mechanics in which each quantum system at any time is described by two Hilbert space vectors rather than one. One of the vectors propagates from past boundary condition towards the present and the other propagates back to the present from a future boundary condition. I will show that this reformulation uncovers a host of fascinating new physical phenomena and new mathematics. Examples of the former, i.e. new physical phenomena, are Weak Measurements and Weak Values. Examples of the latter, i.e. new mathematics, are superoscillations. Both examples will be described in detail within this talk.

I will present a few results concerning spectral theory of
graphs and groups that are based on the use of multidimentional rational
maps and self-similar groups. It will be explained how these maps
appear, how they help to compute the spectrum, and why they are
``strange'' (or better to say, non typical). As part of the story there
will be a short survey on spectra of infinite graphs and groups with
some old and new results belonging to the speaker and his collaborators
(L.Bartholdi, A.Dudko, D.Lenz, T.Nagnibeda, B.Simanek, A.Zuk). The
relation to the random Schrodinger operator will be mentioned at the end
of the talk if time permits.

This talk offers an introductory look at how superoscillatory sequences can be utilized to approximate generalized functions. After an introduction to superoscillations, I will briefly discuss how such sequences can be used to approximate tempered distributions, and will then focus on their role in the context of the theory of hyperfunctions. The talk will be based on a series of papers jointly coauthored with F.Colombo, I.Sabadini, and A.Yger.

In one space dimension, it is well known that hyperbolic conservation
laws have unique entropy-admissible solutions, depending continuously on
the initial data. Moreover, these solutions can be obtained as limits of
vanishing viscosity approximations.
For many years it was expected that similar results would hold in
several space dimensions. However, fundamental work by De Lellis,
Szekelyhidi, and other authors, has shown that multidimensional
hyperbolic Cauchy problems usually have infinitely many weak solutions.
Moreover, all known entropy criteria fail to select a single admissible one.
In the first part of this talk I shall outline this approach based on a
Baire category argument, yielding the existence of infinitely many weak
solutions.
I then wish to discuss an alternative research program,
aimed at constructing multiple solutions to some specific Cauchy
problems. Starting with some numerical simulations, here the eventual
goal is to achieve rigorous, computer-aided proofs of the existence of
two distinct self-similar solutions with the same initial data.
While solutions obtained via Baire category have turbulent nature, these
self-similar solutions are smooth, with the exception of one or two
points of singularity. They are thus much easier to visualize and
understand.

Da qualche anno a questa parte nel mondo matematico sono in corso molte ricerche
per cercare di descrivere in modo soddisfacente e predittivo il comportamento
di aggregati cellulari a vari livelli di evoluzione e organizzazione.
Questi modelli possono aiutare biologi e medici a validare meglio la loro ricerca,
esplorando ipotesi ed alternative altrimenti impraticabili.
In questo seminario esporrò alcuni risultati che ho ottenuto negli ultimi anni su alcuni di questi problemi: crescita di biofilm, movimento di protisti su reti, morfogenesi.
I modelli matematici sono tutti di tipo differenziale e hanno in comune il movimento a velocità finita delle cellule, pur in contesti matematici abbastanza diversi.

Some aspects of the global regularity of solutions to boundary value problems for nonlinear elliptic equations and systems of p-Laplacian type will be discussed. In particular, second-order regularity properties of solutions, and the boundedness of their gradient will be focused. The results to be presented are optimal as far as the regularity of the right-hand sides of the equations and of the boundary of the ground domains are concerned. The talk is based on joint researches with V.Maz'ya.

I describe some conjectures on smoothing toric Fano 3-folds
motivated by mirror symmetry.

It is a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A theorem by Lanford shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. This raises the question whether and how Lanford’s theorem succeeds in deriving this remarkable emergence of irreversibility. Many authors (Cercignani, Illner & Pulvirenti, 1994; Lebowitz 1983, Spohn 1991) have expressed very different views on this question. In this talk, I will argue that the theorem actually does not imply irreversibility at all.

In "Esquisse d'un programme" Grothendieck argues that general topology, which was developed for the needs of analysis, should be replaced by a "tame topology" if one wants to study the topological properties of natural geometric forms.
Such a tame topology has been developed by model theorists under the name "o-minimal structures". The goal of this lecture will be to explain in simple topological terms the notion of o-minimal structure and its applications in algebraic geometry, in particular for studying periods of algebraic varieties.

Let $x_n$ be a complete and minimal system of vectors in a Hilbert space $H$. We say that this system is hereditarily complete or admits spectral synthesis if any vector in $H$ can be approximated by linear combinations of partial sums of the Fourier series with respect to $x_n$. It was a long-standing problem whether any complete and minimal system of exponentials in $L^2(-a,a)$ admits spectral synthesis. Several years ago we gave a negative answer to this question. At the same time we showed that any such system admits the synthesis up to a one- dimensional defect. In the talk we will also discuss related problems for systems of reproducing kernels in Hilbert spaces of entire functions (such as Paley-Wiener, de Branges, Fock).

Finding rational solutions of polynomial equations is one of the most difficult questions in arithmetic geometry. The Birch-Swinnerton-Dyer conjecture (one of the millennium problems) proposes an answer to this question in the case of elliptic curves. In the last years, using techniques like Euler systems in combination with methods involving p-adic families of modular forms, new insights and results concerning refinements of this conjecture were obtained.
In this talk we want to give an introduction to the Birch-Swinnerton-Dyer conjecture, avoiding all technicalities and review what is known about it. In the end we want to explain the ideas which lead to new results on a refinement of the Birch-Swinnerton-Dyer conjecture.

It is much easier to present nice rational linear analysis than it is to wade into the morass that is our understanding of turbulence dynamics. With the analysis, professor and students feel more comfortable; even the reputation of turbulence may be improved, since the students will find it not as bad as they had expected. A discussion of turbulence dynamics would create only anxiety and a perception that the field is put together out of folklore and arm waving.” John Lumley, 1987.
From the outset I have to confess that I find myself 99% in agreement with John Lumley’s concern on “theories of turbulence”. This includes the first premise – i.e. the absence of a theory based on first principles. The second aspect concerns the importance of experiments and observations (both physical and numerical), below referred to as evidence. This lecture is intended to be, first and foremost, a critical presentation and examination of some fundamentally important issues.
* What do we really mean by ‘conventionally defined inertial range’ (CDIR)? Are its properties really independent of (the nature of) dissipation and/or large-scale forcing? Thus, is the inertial range a well defined concept or is it a mis-conception? Who is the guilty party for dissipation anomaly in turbulent flows? And what about the role of the self-amplification processes of vorticity, strain and super-helicity? Also, how well-defined and meaningful is the so-called ‘decomposition’ of energy in inertial and dissipative ranges?
* Is the ‘anomalous scaling’ an attribute of the inertial range? And of passive turbulence?
* Is the ‘4/5 law’ a purely inertial relation?
* Why should one expect that in the CDIR at very high Reynolds numbers the Navier–Stokes equations (NS) are invariant under infinitely many scaling groups (like the Euler equations), in the statistical sense of K41 labeled by an arbitrary real scaling exponent h? And more generally, should one expect to restore in some sense all the symmetries of Euler equations in the CDIR? And why necessarily Euler?
* Are weak solutions of Euler equations going to describe adequately a turbulent flow? Is the inviscid limit of NS always independent of the nature of dissipation and viscosity? Is it possible that the Reynolds dependence differs, but the limit (in distributional sense) remains the same? What does it happen to the solenoidal part of the acceleration as viscosity goes to zero? Could the ‘real’ inertial range of turbulence be adequately described by a suitable singular solution of some sort of Euler-like equations?
* About the concept of ‘non-locality’ of turbulence: is ‘cascade’ a well defined concept and is there a cascade in physical space? Is ‘cascade’ Eulerian, Lagrangian or what? These and other related questions will be briefly touched upon depending on the discussion and interest.
References
TSINOBER, A. 2009 An Informal Conceptual Introduction to Turbulence, Springer-Verlag.
TSINOBER, A. 2018 The Essence of Turbulence as a Physical Phenomenon. II edition (in press), Springer-Verlag.

In 1950, de Bruijn studied the effect of evolving the Riemann zeta function (or more precisely, a closely related function known as the Riemann xi function) by the (backwards) heat equation. His analysis, together with later work by Newman, showed that there existed a finite constant Lambda, at most 1/2 in value, such that the Riemann hypothesis for this evolved function was true at times greater than or equal to Lambda, and false below that threshold. Thus the Riemann hypothesis for the zeta function is equivalent to Lambda being non-positive. Recently, in joint work with Brad Rodgers, I was able to establish the complementary estimate that Lambda is non-negative, confirming a conjecture of Newman; thus, the Riemann hypothesis for zeta, if true, is only "barely so". The proof relies on an analysis of the dynamics of zeroes of entire functions under heat flow; it turns out that as one evolves forward in time, the zeroes "freeze" into approximate arithmetic progressions, while if one evolves backwards, the zeroes "vaporize" to leave the critical line. In followup work in an online collaborative "Polymath" project, the upper bound on Lambda has also been improved. We describe these results and their proofs in this talk.

Compactness of the $\overline\partial$-Neumann operator is
closely connected to the compactness of Hankel operators on the Bergman
space. At first, for convex domains in $\mathbb{C}^n$, we use the
$\overline\partial$ methods to relate the compactness of a Hankel
operator to the boundary behavior of its symbol. In the absence of
compactness, we give the essential norm estimates of Hankel operators.
This in turn, led us to obtain the essential norm estimates for the
$\overline\partial$-Neumann operator on convex domains. (This is joint
work with Sonmez Sahutoglu)

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

After de
ning the spectral theory of orthogonal polynomials on the unit circle
(OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's
theorem as a sum rule for OPUC and the Killip-Simon sum rule for OPRL
and their spectral consequences. Next I'll explain the original proof of Killip-Simon using representation theorems for meromorphic Herglotz functions.
Finally I'll focus on recent work of Gamboa, Nagel and Rouault who obtain
the sum rules using large deviations for random matrices.

The Alexander-Briggs tabulation of knots in R^3 (started almost
a century ago, and considered as one of the most traditional ones
in classical Knot Theory) is based on the minimal number of crossings
for a knot diagram. From the point of view of Real Algebraic Geometry
it is more natural to consider knots in RP^3 rather than R^3, and use
a different number also serving as a measure of complexity of a knot:
the minimal degree of a real algebraic curve representing this knot.
As it was noticed by Oleg Viro about 20 years ago, the writhe of a knot
diagram becomes an invariant of a knot in the real algebraic set-up,
and corresponds to a Vassiliev invariant of degree 1. In the talk we’ll
survey these notions, and consider the knots with the maximal possible
writhe for its degree. Surprisingly, it turns out that there is a unique
maximally writhed knot in RP^3 for every degree d. Furthermore, this
real algebraic knot type has a number of characteristic properties, from
the minimal number of diagram crossing points (equal to d(d-3)/2) to
the minimal number of transverse intersections with a plane (equal to
d-2). Based on a series of joint works with Stepan Orevkov.

We say that an algebraic variety is unirational if it can be parametrized by rational functions, rational if moreover the parametrization can be chosen to be one-to-one. A very classical problem, called nowadays the Luroth problem, asks whether a unirational variety is necessarily rational. This holds for curves (Luroth, 1875) and for surfaces (Castelnuovo, 1894); after various unsuccessful attempts, it was shown in 1971 that the answer is quite negative in dimension 3: there are many examples of unirational varieties which are not rational. Up to 3 years ago the known examples in dimension >3 were quite particular, but a new idea of Claire Voisin has significantly improved the situation. I will survey the colorful history of the problem, then explain Voisin's idea, and how it leads to a number of new results.

I will survey the recent results about the Yamabe problem on stratified spaces. I will first introduce the scalar curvature and the Yamabe equation for the constant scalar curvature equation and its variational formulation and the results of Obata, Trudinger, Aubin and Schoen for smooth compact manifold. Then I will describe the geometry of stratified space with some 2D and 3D examples. Eventually I will formulate the Yamabe problem for stratified space and explained some of the recent results and will explain some perspectives.

In Hilbert or Banach spaces $X$ endowed with a good probability measure $\mu$ there are a few "natural" definitions of Sobolev spaces and of spaces of bounded variation functions. The available theory deals mainly with Gaussian measures and Sobolev and BV functions defined in the whole $X$, while the study and Sobolev and BV spaces in domains, and/or with respect to non Gaussian measures, is largely to be developed.
As in finite dimension, Sobolev and BV functions are tools for the study of different problems, in particular for PDEs with infinitely many variables, arising in mathematical physics in the modeling of systems with an
infinite number of degrees of freedom, and in stochastic PDEs through Kolmogorov equations.
In this talk I will describe some of the main features and open problems concerning such function spaces.

We consider PET (Positron-emission tomography) which a medical imaging technique. We deal with dynamical PET (only static one has a clinical use by now) . I will explain the physical process and set a variational model that takes into account the whole spatio-temporal infomation. Theoretical and numerical results are presented.

Se X e' una varieta' iperkaehler di tipo Kummer, il gruppo di coomologia H^3(X) ha dimensione 8, e quindi possiamo associare a X una Jacobiana intermedia J^3(X) (un toro complesso compatto di dimensione 4, proiettivo se X e' proiettiva). Faro' vedere come ricostruire esplicitamente J^3(X) a partire dalla struttura di Hodge su H^2(X). In particolare seguira' che, se X e' proiettiva, allora J^3(X) e' una varieta' abeliana di tipo Weil. Lo studio di J^3(X) suggerisce come (tentare di) costruire famiglie esplicite localmente complete di varieta' iperkaehler di tipo Kummer proiettive. of these results extend or not to non linear operators and to degenerate elliptic operators.

Global bifurcations along the family of radially symmetric vortices are analyzed for the Gross-Pitaevskii equation with a symmetric harmonic potential and a chemical potential under the steady rotation. The families are constructed in the small-amplitude limit when the chemical potential is close to an eigenvalue of the Schrodinger operator for a quantum harmonic oscillator. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross-Pitaevskii equation and the zeros of Hermite-Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex, the asymmetric vortex pair, and the vortex polygons.

In this talk, I show how (local) dynamical systems can be used in order to understand the geometry of real submanifolds in the complex euclidean space

The mixing time of a random walk is the time it needs to approach its stationary distribution. For random walks on graphs, the characterisation of the mixing time has been the subject of intensive study. One of the motivations is the fact that the mixing time gives information about the geometry of the graph. In the last few years, much attention has been devoted to the analysis of mixing times for random walks on \emph{random graphs}, which poses interesting challenges.
Many real-world networks are dynamic in nature. It is therefore natural to study random walks on \emph{dynamic random graphs}. In this talk we consider a random walk on the configuration model, i.e., a random graph with prescribed degrees. We investigate what happens when at each unit of time a fraction $\alpha_n$ of the edges is randomly relocated, where $n$ is the number of nodes.
We identify \emph{three regimes} for the mixing time in the limit as $n \to \infty$, depending on the choice of $\alpha_n$. These regimes exhibit surprising behaviour.
Joint work with Luca Avena (Leiden), Hakan Guldas (Leiden) and Remco van der Hofstad (Eindhoven)

ABOUT THE SPEAKER: Wendelin Werner is a world renowned scientist, whose exciting work intertwines probability theory with mathematical physics and complex analysis. Werner is often motivated by problems in statistical physics which involve critical phenomena in the large scale behavior of complex multi-particle systems; such as those that arise in phase transitions and percolation. His profound work on stochastic Loewner evolution, the fractal geometry of two dimensional Brownian motion and conformal field theory earned Werner both the Fields medal and the SIAM Pólya Prize (together with collaborators Greg Lawler and Oded Schramm) in 2006. In addition, Werner is the recipient of the European Mathematical Society Prize (2000), the Fermat Prize (2001), the Jacques Herbrand Prize (2003), the Loève Prize (2005) and was inducted into the French Academy of Sciences in 2008. Werner received his Ph.D in 1993 from the Universitè Paris VI under the direction of Jean-François Le Gall and has held positions at the Cambridge University, Universitè Paris-Sud and Ecole Normale Supèrieure before being named Professor of Mathematics at ETH Zürich in 2013.

The link between the principal eigenvalue and the maximum principle has been made very clear since the works of Protter - Weinberger, Pucci and Berestycki-Nirenberg-Varadhan. Its consequences are important and go from regularity of solutions to their very existence.
I will start by giving a survey of this rich field in the classical case of the second order uniformly elliptic operators. Then I will go on describing what and how some of these results extend or not to non linear operators and to degenerate elliptic operators.

In this talk, I will discuss some questions related to the nonlinear theory of electromagnetism formulated by Born and Infeld in 1934. I will focus on the set of PDEs arising from this theory and mainly on the static regime.
I will discuss also some links between this theory and the curvature operators in the Euclidean and in the Lorentz-Minkowski space. Finally I will address the solvability of the electrostatic Born-Infeld equation with sources and some related questions.
This seminar is organized within the PRIN 2012 Research project «Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration number 2012TC7588_003, funded by MIUR - Project local coordinator Prof. Filippo Gazzola

Nonlinear Potential Theory aims at reproducing, in the nonlinear setting, the classical results of potential theory concerning the fine and regularity properties of solutions to linear elliptic and parabolic equations. Potential estimates, integrability, differentiability and continuity properties of solutions are at the heart of the matter. I will give a brief survey of a few recent results.

I will review recent results and not so recent open problems concerning
constructions and ergodicity of dissipative dynamics for classical and noncommutative large interacting systems. In particular I will talk about dynamics defined by formally hypoelliptic generators and some generalised Dunkl type extensions, and how to control their long time behaviour (by use of coercive inequalities or generalised gradient type bounds).

Geometric and functional inequalities play a crucial role in several PDE problems.
Very recently there has been a growing interest in studying the stability for such inequalities. The basic question one wants to address is the following:
Suppose we are given a functional inequality for which minimizers are known. Can we prove, in some quantitative way, that if a function “almost attains the equality” then it is close to one of the minimizers?
Actually, in view of applications to PDEs, a even more general and natural question is the following: suppose that a function almost solve the Euler-Lagrange equation associated to some functional inequality. Is this function close to one one of the minimizers?
While in the first case the answer is usually positive, in the second case one has to face the presence of bubbling phenomena.
In this talk I’ll give a overview of these general questions using some concrete examples, and then present recent applications to some fast diffusion equation related to the Yamabe flow.

A number of important questions in arithmetic and algebraic geometry, like the Hodge conjecture, deal with construction methods for subvarieties of algebraic varieties. These conjectures deal with complex-analytic questions, number-theoretic ones and questions regarding transcendental number theory. We will describe various aspects of these conjectures, and explain the geometric ideas underlying the proof of some of these in specific cases.

We are going to discuss many applications and generalizations of Freiman's theorem in discrete mathematics and elsewhere. No previous knowledge is assumed.

The evolution of vortex filaments in three dimensions is an important problem in mathematical hydrodynamics. It appears in questions on solutions of the Euler equations as well as in the fine structure of vortex filamentation in a superfluid. It is also a setting in the analysis of partial differential equations with a compelling analogy to Hamiltonian dynamical systems. I will give an analysis of a system of model equations for the dynamics of near-parallel vortex filaments in a three dimensional fluid. These equations can be formulated as a Hamiltonian system of partial differential equations, and the talk will describe some aspects of a phase space analysis of solutions, including the construction of periodic and quasi-periodic orbits via a version of KAM theory for PDEs, and a topological principle to count multiplicity of solutions. This is ongoing joint work with L. Corsi (McMaster and Georgia Tech), C. Garcia (UNAM) and C.-R. Yang (McMaster and Shantou Univ.)

Arithmetic quotients of algebraic groups such as the space of unit volume lattices in R^n can be studied fruitfully from many directions and contain deep and subtle arithmetic information. Homogeneous dynamics studies these spaces by considering the action of a subgroup of the algebraic group on such a quotient. Of particular interest is the action of multiparameter diagonal groups: these display remarkable rigidity properties that are absent in the context of one parameter diagonalizable group actions.
One aspect of this rigidity is joining rigidity: under suitable conditions, knowing that an orbit of a multiparameter diagonalizable group in a product of two arithmetic quotients is equidistributed in each one of these quotients individually implies joint equidistribution.
I would explain this phenomena as well as some arithmetic consequences.

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others.
In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d-dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders.
By developing new isoperimetric methods for "locally minimal small" F_2- co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.
it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.
We will describe these developments and the general area of high dimensional expanders and some of its open problems.

Two-dimensional dimer models are popular models, which are used to describe either the liquid phase of dense anisotropic molecules or, thanks to a well-known mapping between dimer configurations and discrete height functions, the rough phase of fluctuating random surfaces. The last few years witnessed important progresses in the understanding of the critical phase of dimer systems, including the proof of existence and conformal covariance of the scaling limit, and the proof of convergence of the discrete height field to the massless Gaussian Free Field (GFF), due to Kenyon, Okounkov and Sheffield. In this talk I will review some aspects of the theory of critical dimer models, which is based, in large part, on the celebrated Kasteleyn solution for `non-interacting' dimers, combined with discrete holomorphicity methods. I will also discuss a novel approach to *interacting* dimer models, based on constructive Renormalization Group techniques, which recently allowed us to prove the convergence of the discrete height function to the GFF, in the presence of non-integrable perturbations of the dimers' Gibbs measure, as well as the validity of a universality relation between the renormalized variance of the GFF and the critical exponent of the dimer-dimer correlations (in collaboration with V. Mastropietro and F. Toninelli)

I will explain my proof of the Tate and Mumford-Tate conjecture for divisor classes on a variety with h^{2,0}=1 (e.g., surfaces with p_g=1), under a mild assumption on their moduli. The proof involves a combination of several new techniques, including a generalization of the Kuga-Satake construction and a new interpretation of 'half-twists' of CM-Hodge structures. I will start by explaining the method of Deligne-Andre for K3 surfaces, which is the main source of inspiration for the proof.

We review some Liouville type results from the classical Cauchy-Liouville theorem for harmonic functions up to some recent contributions related to quasilinear second order and semilinear higher order problems.
This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. A successful existence theory, that of integral currents, was developed by De Giorgi in the case
of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties.
When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es,70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).
In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2.
However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry.

Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.

As a wave propagates over a shock profile, there are new waves
reflecting, transmitting and moving along the profile. The rich wave
phenomenon results from the strong nonlinear nature of the shock waves,
as well as the coupling of waves for a general system of conservation
laws. For one spatial dimension, there are sub-scale waves as a result
of nonlinear coupling of distinct characteristic families. For more than
one spatial dimension, the propagation of dispersion waves over a shock
profile give rise to new wave phenomena, which we illustrate with a
simple model inspired by gas dynamics. Strongly quantitative methods
based on explicit construction of Green's function are introduced for
our analysis.

It is classically known that the general principally polarized abelian variety of dimension at most five is a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully
concrete and rich theory of algebraic curves.
I will discuss decisive recent progress on finding a structure theorem for
principally polarized abelian varieties of dimension six, and the implications this uniformization result has on the geometry of their
moduli space.

Living tissues, such as stems, leaves and flowers in plants
and bones in animals, grow into a great variety of shapes.
In some cases, Nature has found ways to control this growth with extremely high accuracy.
In this talk I plan to discuss a few related questions, from a mathematical perspective.
What is the simplest set of PDEs that can describe controlled growth in such a variety of forms?
How can one break away from radial symmetry?
Can one recover familiar shapes of leaves and flowers as stable solutions to a nonlinear eigenvalue problem?
A few results and many open problems in this direction will be presented.

The incompressible Euler equations were derived more than 250 years ago
by Euler to describe the motion of an inviscid incompressible fluid.
It is known since the pioneering works of Scheffer and Shnirelman that
there are nontrivial distributional solutions to these equations which
are compactly supported in space and time. If they were to model the
motion of a real fluid, we would see it suddenly start moving after
staying at rest for a while, without any action by an external force.
A celebrated theorem by Nash and Kuiper shows the existence of C1 isometric
embeddings of a fixed flat rectangle in arbitrarily small balls
of the threedimensional space. You should therefore be able to put a
fairly large piece of paper in a pocket of your jacket without folding
it or crumpling it.
In a first joint work with Laszlo Szekelyhidi we pointed out that these
two counterintuitive facts share many similarities. This has become even
more apparent in some recent results of ours, which prove the existence
of Hoelder continuous solutions that dissipate the kinetic energy. Our
theorem might be regarded as a first step towards a conjecture of Lars
Onsager, which in his 1949 paper about the theory of turbulence asserted
the existence of such solutions for any Hoelder exponent up to 1/3.
Recently, the threshold 1/5 has been reached by Philip Isett in his PhD
thesis and in a joint work with Tristan Buckmaster we show that the treshold
1/3 can be achieved at the price of giving up the time-regularity.

Svariati fenomeni (corde vibranti, segmentazione di segnali) coinvolgono energie che dipendono dalla curvatura. Il caso di curve in codimensione maggiore di uno presenta fenomeni interessanti, che vengono analizzati da un punto di vista geometrico/analitico; usando (come strumento) le correnti cartesiane e' possibile determinare una rappresentazione in BV dell'inviluppo semicontinuo di funzionali molto naturali, ottenendo un risultato altrettanto naturale. Al termine viene fatto un cenno al caso di crescite piu' che lineari.

We review some recent learning approaches in variational image regularisation — in particular for variational image denoising — based on PDE constrained (also called bilevel) optimisation. Optimal solutions are typically parameters determining the type of regularisation and data discrepancy. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV), as well as general data discrepancies encoding different noise distributions. We analyse those models for existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are employed to solve the models numerically. The computational verification of the developed techniques will be discussed, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.
This is joint work with Luca Calatroni, Chung Cao, Juan Carlos De los Reyes, and Tuomo Valkonen.

L'effetto Kondo si manifesta con la non divergenza a temperatura 0 della suscettivita` magnetica di una impurita` in un gas di fermioni: la non divergenza avviene solo in presenza di una interazione antiferromagnetica e non nel caso ferromagnetico.
E` un effetto quantistico, non deducibile da una teoria perturbativa: la sua teoria, in un modello unidimensionale, fu risolta da Wilson attraverso il gruppo di rinormalizzazione. Fu uno dei primi casi in cui si pote` capire un fenomeno multiscala controllato da un punto fisso non banale: in questo seminario illustrero` il problema in un modello che pur mantiene la caratteristica di essere un problema non perturbativo a molte scale e che sembra richiedere il metodo del gruppo di rinormalizzazione per la sua soluzione.

Richiamerò la definizione di superficie di Godeaux,
illustrando rapidamente lo "stato dell'arte".
Seguendo l'approccio di Kollàr, introdurrò le superfici algebriche
stabili e illustrerò la classificazione delle superfici di Godeaux
stabili Gorenstein (in coll. con Franciosi e Rollenske), descrivendo
in particolare un inaspettato collegamento tra una classe di tali
superfici e le curve bi-triellittiche di genere 2. Se il tempo lo
permette, accennerò alla classificazione delle curve (p,d)-ellittiche
di genere 2, dove p e' un primo e d e' un intero qualunque.

We consider evolution equations induced by an important class of quadratic differential operators which arise from the Weyl quantization of quadratic forms. This class of operators has a number of special features which makes their study quite involved. In general:
1) They are far from being selfadjoint,
2) Their eigenvectors form a minimal set with dense span, but not a Riesz basis,
3) The norm of their resolvent grows exponentially towards infinity within certain regions in the complex plane.
The aim of the talk is to present a model for such operators on Fock spaces in several complex variables, which offers a complex analysis perspective and can be used to address a number of questions about the solutions of these evolution equations.
The material is based on joint work with J. Viola.

Si discuteranno diversi aspetti (analitici, geometrici, topologici) dei problemi differenziali (di tipo ellittico) legati alla costruzione di configurazioni di tipo vortice (o solitone) nell’ambito della teoria di Chern-Simons.

Several variational problems related to image analysis have been
studied in recent years. This talk focuses on functionals coupling bulk energy depending on second order derivatives of the admissible
functions, and surface energy due to discontinuity and gradient discontinuity. The case of vector-valued admissible functions is considered too, in order to analyze RGB color images.
As an application to image analysis, we discuss the problem
of inpainting, say the reconstruction in the whole domain of a 2-D image which is damaged or even completely lost in a small subdomain.

For several decades the mathematical model of Quantum Probability (QP) has been considered as a generalization of classical probability.
However some discoveries of the past 15 years show that the whole quantum theory, including quantum fields, is not a generalization,
but rather a deeper level of classical probability.
In fact, combining classical probability with the theory of
orthogonal polynomials in 1 or several real variables, it is possible to
prove that the canonical commutation relations, both Fermi and Bose
(and in fact even their $q$-deformations), arise canonically from the Bernoulli and Gaussian random variables respectively.
More generally one can prove that there is a one-to-one correspondence between Heisenberg-type commutation relations
and equivalence classes of probability measures on R with all moments.
The equivalence relation being defined, in the one-dimensional case,
by the fact that all measures in a class share the same principal
Jacobi sequence.
To each of these equivalence classes it is canonically associated a
free evolution, generalizing the classical harmonic oscillator evolution.
The characterization of the equilibrium states with respect to any such evolution naturally leads to a generalization of the Planck factor.
Similar arguments, applied to the recently introduced local
equilibrium states, lead to non-linear extensions of the Planck factor and non-linear Gibbs states.
Being functorial, the above construction also provides a generalization
of the second quantization procedure both at Hilbert space (Fock)
and $*$-algebra level and in some special cases (e.g. probability measures
on $R^d$ with compact support) even at $C^*$-algebra level.
However in general the class of morphisms will be much narrower than
in usual second quantization.
This fact supports the intuition that the new quantizations have a physical meaning in terms of non-linear completely integrable classical systems.
The present talk is concentrated on the goal to illustrate
the classical roots of quantum theory, however
if time allows it will be also mentioned how these new ideas have allowed
to solve a multiplicity of long standing open problems both in
classical probability and in the theory of orthogonal polynomials.

Sottotitolo: Come l' angelo della topologia può convivere col diavolo della algebra astratta*.
Il legame tra algebra e topologia è duplice: la topologia algebrica, si pensi al teorema di Brouwer del punto fisso, od al teorema di Borsuk-Ulam, traduce la esistenza di certe applicazioni continue in esistenza di omomorfismi con certe proprietà algebriche .
Viceversa, la teoria dei fibrati e della omotopia dà una incarnazione topologica di un gruppo.
Tale teoria, detta degli spazi classificanti, traduce viceversa omomorfismi di gruppi in applicazioni continue. La questione della regolarità di tali applicazioni e della loro eventuale olomorficità, sviluppata negli ultimi 40 anni, ha potenti applicazioni nella teoria dei moduli.
Il caso più noto è la teoria delle varietà di Albanese associate a varietà algebriche, meno nota è la teoria delle loro varietà quozienti,
tra cui le cosiddette varietà di Bagnera De Franchis.
Dopo avere dato esempi di classi di varietà proiettive che sono spazi classificanti, ed avere illustrato come la topologia dia risultati molto forti in teoria dei moduli, mi dedicherò alla azione del gruppo di Galois assoluto Gal, ed alla sua azione su spazi di moduli.
In particolare sugli spazi di moduli di varietà proiettive classificanti. Si ottengono così azioni fedeli, realizzando un programma di Grothendieck, ed un importante fenomeno, cioè che ci sia, per ogni g
in G non equivalente alla coniugazione complessa, sempre una
superficie S tale che S e g(S) hanno gruppi fondamentali non isomorfi.
*Nota storica: In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.Hermann Weyl (1939), p.500 di "Invariants", Duke Mathematical Journal 5 (3): 489-502.

In this talk I will discuss shape optimization problems with Robin boundary conditions on the free part. For general shape functionals, the existence of a solution may not occur, but for a suitable class of energy type functionals one can prove the existence of a solution and some partial regularity results. The main tools are of free discontinuity type. As example, one could consider the minimization of the first Robin eigenvalue of the Laplacian, among arbitrary open sets of prescribed measure, contained in a given design region.

In this talk I will discuss shape optimization problems with Robin boundary conditions on the free part. For general shape functionals, the existence of a solution may not occur, but for a suitable class of energy type functionals one can prove the existence of a solution and some partial regularity results. The main tools are of free discontinuity type. As example, one could consider the minimization of the first Robin eigenvalue of the Laplacian, among arbitrary open sets of prescribed measure, contained in a given design region.

Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible patterns. I have explored an alternative viewpoint in a series of recent projects with Jacob Bedrossian, Peter Bella, Jeremy Brandman, and Hoai-Minh Nguyen. Our goal is to identify the scaling law of the minimum elastic energy (with respect to the sheet thickness, and in some cases with respect to other small parameters). Success entails proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain why we see particular patterns. My talk will give an overview of this activity, and details of some examples.

Nel seminario verrà discusso un modello variazionale di Ohta e Kawasaki usato per descrivere la separazione di fase nei composti polimerici e che consiste in un’energia risultante dalla somma di un termine di superficie e di un termine non locale di volume. Mostreremo che le configurazioni critiche con variazione seconda positiva sono minimi locali dell’energia. Da ciò dedurremo in particolare dei nuovi risultati relativi ai minimi locali periodici dell’area e la minimalità locale e globale di certe configurazioni osservate in molti di questi composti polimerici.

La teoria delle forme di Dirichlet (Fukushima 1980, Albeverio, Ma-Rockner 1992,…) permette di costruire processi di diffusione in spazi di misura molto generali.
Ad essi è possibile associare una geometria intrinseca grazie alla distanza di Biroli-Mosco (1991), un formalismo di calcolo differenziale (il cosiddetto Gamma-calcolo) e una nozione di curvatura-dimensione seguendo l’approccio introdotto da Bakry-Emery (1984), legata ad importanti disuguaglianze geometrico-funzionali.
Più recentemente, partendo da uno spazio metrico di misura invece che da una forma di energia, si è sviluppata una teoria metrica degli spazi di Sobolev (Koskela-MacManus 1998, Cheeger 1999, Shanmugalingam 2000) che porta alla costruzione di una diffusione (generalmente nonlineare) e di un calcolo non-smooth al primo ordine.
I lavori di Lott-Villani e Sturm (2006-2009) hanno poi introdotto il punto di vista del trasporto ottimo per definire una condizione di curvatura-dimensione stabile per convergenza di Gromov-Hausdorff e con importanti applicazioni geometriche.
Questi due punti di vista mettono in luce interessanti aspetti anche nel caso più semplice di una equazione di diffusione-trasporto in R^n con coefficienti sufficientemente regolari o, più in generale, in varietà Riemanniane.
Il seminario si propone di introdurre queste tematiche e di presentare alcuni recenti risultati, ottenuti in collaborazione con Ambrosio e Gigli, che mostrano come i due punti di vista sono essenzialmente equivalenti.

Optimal control theory is an active area of research in both mathematics and engineering sciences. Still, many mathematically challenging problems related to second order optimality conditions remain open, mostly because the structural properties of control systems do not fit well the framework of contemporary optimization literature. In this talk I will discuss second-order necessary optimality conditions in deterministic optimal control from the point of view of second order variational analysis. The new approach introduces second order jets of set valued maps and second order approximations of differential equations and, more generally, control systems and differential inclusions. Then a second order variational differential inclusion leads to a second order Fermat like rule of optimality. In particular, it allows to obtain a new pointwise second order condition verified by the adjoint state of the maximum principle and opens new questions to be investigated.

Let us consider a class of homogeneous quasilinear parabolic equations whose prototypes are the p-Laplacian and the Porous medium equation.
We assume suitable monotonicity and Lipschitz conditions that are suffcient to have a comparison principle, to preserve the positivity of solutions and to guarantee the existence of the solution of a Cauchy problem with L¹ initial
datum.
By using recent results obtained in collaboration with Bögelein, Calahorrano, Piro Vernier and Ragnedda we are able to give some pointwise estimates from above and from below starting from the value of the solution attained in a point. We apply these results to give sharp estimates to the fundamental solution of such class of equations.

In my talk I shall discuss consequences of the requirement of covariance (structure invariance under diffeomorphism-based changes in observers) of the mechanical dissipation inequality, written for classes of bodies admitting processes along which their material texture changes irreversibly – mutant bodies, in short.
Mine is a tentative to answer the following question: Do the ideas leading to Nöther’s theorem admit counterparts in fully dissipative setting?
I shall discuss the question in the special case of standard elastic-plastic materials in large-strain regime, showing not only that the second law is a source of a priori constitutive restriction and the expression of the dissipation (the standard use of the second law in continuum mechanics), but also that the requirement of its covariance allows us to determine in addition the existence of the stress tensor, and the pointwise balances of standard and configurational actions, all arising from a unique source.
The choice of considering elastic-plastic materials is dictated just by the will of having at disposal a classical setting that can be easily detected by a not particularly specialized audience.
However, the technique that I shall indicate can be used also for viscous and general complex bodies (and I shall sketch briefly the way). In the last case, it would furnish a safe tool for determining appropriately balance equations for wide classes of complex materials in dissipative setting in case we are interested in constructing new models.

For an automorphism of an algebraic surface, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Helene Esnault.

We will give an expository talk comparing two approaches to 2D lattice models of critical phenomena. Developed over two decades ago, Conformal Field Theory led to spectacular predictions for 2D lattice models. While the algebraic framework of CFT is rather solid, rigorous arguments relating it to lattice models were lacking. More recently, a geometric approach involving random SLE curves was proposed by Oded Schramm, and developed by him, Greg Lawler, Wendelin Werner, Steffen Rohde and others. Not only this approach is completely rigorous, it also constructs new objects of physical interest and gives results inaccessible by CFT means.

Optimal control theory provides a unified framework for studying the minimization of a performance index over a class of state trajectories satisfying a dynamic constraint. Minimizing state trajectories may be optimal flight paths in aeronautical engineering, a most profitable resource extraction policy in mathematical economics, a solution to a Hamiltonian system, or have other interpretations. Typically the dynamic constraint takes the form of a controlled differential equation. But in certain applications the differential equation involves time delays in state and control variables, which may arise from transportation delays in chemical processing, finite speed of signals in communications links, or by other mechanisms.
From a theoretical point of view, the optimal control of systems with time delay have many fascinating and unusual features. These systems are infinite dimensional, to the extent that the true state is an entire trajectory segment (an element in an infinite dimensional function space), yet necessary conditions of optimality may be studied by means of variational techniques developed for finite dimensional, delay-free systems. One the other hand, questions of existence of optimal controls and sufficient conditions via Hamilton Jacobi equations are, in some ways, much more complicated for time delay systems and, currently, only partly resolved.
This talk will provide an overview of the theory. It will include recent advances in the derivation of necessary conditions of optimality for time delay systems. Illustrations of their practicality will be provided by applications to problems in ecological control and other areas.

Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry.
Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry.
Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli of marked del Pezzo surfaces.
Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G2.

In this talk I will discuss the contribution of points and curves on a K3 surface to the Chow ring. The Chow ring of a K3 surface remains mysterious and reflects the interplay between
the arithmetic and geometry of the surface.
Classical results as well as open conjectures will be explained.

In questo seminario si studia il comportamento asintotico di equazioni di diffusione non lineari, in dimensione $n$, nell intervallo degli esponenti $p > (n - 2)/n$,
per dati iniziali di momento secondo momento limitato. In questo range di non linearità si determina la convergenza a profili di tipo Barenblatt in termini di entropia relativa di tipo Renyi, per soluzioni rinormalizzate ad ogni tempo rispetto al proprio mmento secondo.
L analisi mostra che la entropia di Renyi relativa presenta un decadimento migliore, per tempi intermedi, rispetto alla standard entropia di tipo Ralston-Newton usata di solito in questo contesto. Il risultato consegue dalla proprietà di concavità della Renyi entropy power, recentemente dimostrata in un lavoro congiunto con G. Savaré.

Estimation of the integrated volatility (or, quadratic variation of the continuous martingale
part) of Itò semimartingales which are discretely observed at n points is a central topic in
financial statistics. In this talk I will give an overview of the topic, and also indicate some
of the recent developments, concerning the case where the underlying process has jumps.
The focus will be on efficient estimation procedures.

In this talk we will connect moduli spaces of stability conditions of categories to dynamical systems.
We will show how classical ideas from category theory and from ergodic theory unify to benefit both subjects.

A classical approach to the problem of contact of an elastoplastic body with an elastoplastic obstacle consists in applying a variant of the
penalty method. Instead, we propose here to reformulate it equivalently as a PDE with hysteresis operators both in the constitutive law and in the contact boundary condition. Analytical properties of the hysteresis operators (Lipschitz continuity in suitable function spaces, monotonicity, energy inequalities) enable us to construct a regular solution by conventional Galerkin approximations and prove its uniqueness for each given data. This is a joint work with Adrien Petrov, INSA Lyon.

Lo scopo di questo seminario è di fornire una introduzione allo studio di curve razionali
su varietà proiettive, sia da un punto di vista algebrico che da un punto di vista analitico.

I will discuss the emergent Dirac quasiparticles in the lattice systems of electrons, exemplified by graphene, in the presence of topological defects of the allowed superconducting and insulating, order parameters. These orders appear as possible mass terms in the Dirac equation, and their topological defects have been known to carry non-trivial quantum numbers such as charge and spin since the work of Jackiw and Rebby in 1976.
In the talk I will discuss their additional internal degree of freedom: irrespectively of the nature of orders that support the defect, an extra
mass-order-parameter spontaneously emerges in the defect's core. The determination of the quantum state of the topological defect turns out to be an interesting problem in the representation theory of (real) Clifford algebras; with the Clifford algebra C(2,5) playing a fundamental role in graphene, for example. Ultimately, the particle-hole symmetry restricts the defects to always carry the quantum numbers of a single effective isospin-1/2, quite independently of the values of their electric charge or true spin. Examples of this new degree of freedom in graphene and on surfaces of topological insulators will be given.

In 1985 Moffatt proposed the method of magnetic relaxation to construct stationary Euler flows with non-trivial topology. The idea is to take an initial magnetic field and let it evolve under the dynamics of the MHD system with zero viscosity; the time asymptotic limit of the magnetic field should then yield a function that satisfies the stationary Euler equations.
Since the MHD equations are only used to produce the limiting field, the problem can be changed in a number of ways and still (at least heuristically) provide a stationary Euler flow.
In this talk I will discuss a simplification of the system in which the fluid evolution is replaced by an elliptic equation. Our aim is to show that the resulting system is locally well-posed. Despite the apparent simplicity of the equations it turns out that this requires results that are at the limit of what is available - elliptic regularity in $L^1$, the limiting case of the Aubin-Lions compactness theorem, and a strengthened form of the Ladyzehnskaya inequality derived using the theory of interpolation.

Come dedurre, in maniera matematicamente chiara, i principi della termodinamica (con le sue trasformazioni isoterme, adiabatiche, i cicli di Carnot ecc.) dalle dinamiche 'microscopiche' classiche hamiltoniane? I limiti idrodinamici permettono di ottenere la termodinamica classica come leggi 'macroscopiche', attraverso un riscalamento spazio-temporale della evoluzione delle quantità conservate (lente). Il problema è quindi puramente matematico, anche se di soluzione molto difficile. Il concetto di equilibrio locale e della ergodicità delle dinamiche infinite è centrale in questo approccio.

We review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian.

The theory of random matrices appears in many parts of mathematics such as probability, statistics, quantum chaos, number theory and the spectral theory of random Schrödinger operators. This lecture will give a brief introduction to the history and conjectures of this subject. We show how certain models of statistical mechanics provide a dual representation for spectral problems in random matrix theory. This representation enables one to obtain numerous identities arising from symmetry and to apply new
tools of analysis and phase transitions. Ordered and disordered phases correspond to different spectral types and time evolution of a random matrix Hamiltonian. A particular statistical mechanics model is equivalent to a history dependent random walk which prefers to jump to vertices it has visited in the past. The phase transition for this process is reflected in a change of the long time behavior of the walk.

I will start with a quote from the most famous scientist of the first half of the 20th century:Why were another seven years required for the construction of the general theory of relativity. The main reason is the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning.
The following quote is from a far less famous scientist:
The approach of this treatise is conceptual, geometric, and uncompromisingly coordinate-free. In some of the literature tensors are still defined in terms of coordinates and their transformations. To me, this is like looking at shadows dancing on the wall rather than at reality itself.
The first quote is, of course, by Albert Einstein, and is cited in Section 1.2, (entitled spacetime with and without coordinates) of the book Gravitation by Misner, Thorne, and Wheeler. The second quote is by a far less famous scientist, namely me (Walter Noll) in part F of the Introduction to the Book entitled Finite-Dimensional Spaces, Algebra, Geometry, and Analysis.
I will discuss specific examples of coordinate-free treatments of the following topics:
1. Continuum Mechanics
2. Geometry
3. Special Relativity
4. General Relativity
5 Lineons versus Matrices

We discuss sharp continuity and regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity properties beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions.
Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron.
The techniques developed recently allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. An appropriate
notion of polyharmonic capacity is introduced which allows one to describe the precise correlation between the smoothness of solutions and the geometry of the domain.

I will discuss the geometry of heterotic string compactifications with fluxes. The compactifications on 6 dimensional manifolds which preserve N=1 supersymmetry in 4 dimensions must be complex conformally balanced manifolds which admit a now-where vanishing holomorphic (3,0)-form, together with a holomorphic vector bundle on the manifold which must admit a
Hermitian Yang-Mills connection. The flux, which can be viewed as a torsion, is the obstruction to the manifold being Kahler. I will describe how these compactifications are connected to the more traditional compactifications on Calabi-Yau manifolds through geometric transitions like flops and conifold transitions. For instance, one can construct solutions by flopping rational curves in a Calabi-Yau manifold in such a way that the resulting manifold is no longer Kahler. Time permitting, I will discuss open problems, for example the understanding of the the moduli space of heterotic
compactifications and the related problem of determining the massless
spectrum in the effective 4 dimensional supersymmetric field theory. The study of these compactifications is interesting on its own right both in string theory, in order to understand more generally the degrees of freedom of these theories, and also in mathematics. For example, the connectedness between the solutions is related to problems in mathematics, for instance
Reid s fantasy, that complex manifolds with trivial canonical bundle are all connected through geometric transitions.

Gli oscillatori armonici non-commutativi sono quantizzazioni di tipo Weyl di forme quadratiche a valori matrici. Si può pensare ad essi come a deformazioni a valori matrici dell'usuale oscillatore armonico (quantistico). La non-commutatività a cui ci si riferisce è legata alla non-commutatività del prodotto tra matrici ed al principio di indeterminazione. In questo seminario darò una panoramica sui risultati di tipo spettrale di tali oggetti e problematiche correlate.

After introducing the notion of spectral minimal partitions we give some explicit examples for non-nodal spectral minimal partitions. In particular for sufficiently thin cylinders with Neumann boundary conditions and for the torus. Additional problems and observations are discussed.
This is joint work with Bernard Helffer.

I will present our recent results on the uniqueness in determining coefficients in 2-dimensional elliptic equations by all the set of
Cauchy data with Dirichlet data supported on arbitrary subboundary $\Gamma$ and Neumann data on $\Gamma$.
Classical Dirichlet-to-Neumann map corresponds to a special case where $\Gamma$ is the whole boundary, and our results are the best possible uniqueness results within some smoothness assumptions.

Nella prima parte discuterò alcune questioni che stanno alla base del teorema di supporto di B.C. Ngo. Nella seconda parte discuterò una estensione di questo teorema, dovuta indipendentemente a Maulik-Yun e a V.Shende in collaborazione con me, allo schema di Hilbert relativo di una famiglia di curve piane.
Tale teorema si puo' interpretare come estensione al caso singolare della classica formula di MacDonald per la coomologia del prodotto simmetrico.

In this talk I will give an overview of some geometric and analytic issues related to the regularized determinant of an elliptic operator.
I will begin with the work of Osgood-Phillips-Sarnak on the determinant of the laplacian for surfaces, which has its origins in a formula of Polyakov, and explain the connection to the uniformization theorem and the Ricci flow. In four dimensions, the starting point is a beautiful but daunting formula of Branson-Orsted for conformal variations of the determinant. I will explain the connection of this formula to mathematical physics and conformal geometry, then discuss some of it variational properties. I will end with a question posed by Connes about the determinant of the Paneitz operator, and some work in progress.

Da circa 20 anni, in geometria algebrica stanno avendo successo alcune idee, proposte inizialmente da Deligne, Drinfeld, Stasheff e Kontsevich, secondo le quali i problemi di deformazione su di un campo di caratteristica 0 sono governati da oggetti nella categoria omotopa
delle algebre di Lie differenziali graduate. Queste idee hanno permesso di studiare tali problemi utilizzando tecniche classiche di
topologia algebrica e omotopia razionale che si sono rivelate estremamente ricche di applicazioni e ispiratrici di nuovi sviluppi teorici.
Nel seminario ripercorreremo alcune di queste idee con applicazioni alla teoria delle variazioni di cicli algebrici su varietà' proiettive.

Viene studiata l'esistenza e molteplicità di
soluzioni per equazioni ellittiche semilineari con secondo membro misura, introducendo un approccio variazionale diretto.

We consider a general formulation of the domain evolution by means of gradient flows, obtained through the minimizing movements procedure, which
has its natural framework in the metric spaces setting. The functionals which govern the evolutions are of spectral type, involving the
eigenvalues of the Laplace operator with Dirichlet boundary conditions. Several dissipation distances are considered and the phenomenon of occurrence of relaxed capacitary measures during the evolution is discussed.

Critical point theory is used to obtain multiplicity results for the periodic solutions of systems of differential equations of the form
$$
\left(\nabla \Phi(u^\prime)\right)^\prime
=\nabla_u F(t,u) + h(t).
$$

In this talk, we analyze a new type of highly nonlinearly coupled phase field equations that was recently introduced by P. Podio-Guidugli as a model for diffusive phase segregation on an atomic lattice. The unknowns in this system of PDEs are an order parameter and the chemical potential. We derive results concerning well-posedness, asymptotic behavior and optimal control. This is joint work with P. Colli, G. Gilardi and P. Podio-Guidugli.

Various problems arising in games and sports will be analyzed mathematically. For example: How many shuffles to mix a deck of cards? Given the solution, can you find the problem? When is a team eliminated? How should teams be ranked? How should a runner vary his speed to achieve shortest time? How do world records vary with time? What is the probability of heads in a coin toss? Does it pay to exercise?

Our story begins more than a century ago with Kronecker Jugendtraum and Hilbert 12th problem, where complex multiplication appears as a way to understand Galois extensions of number fields - a problem still at the heart of number theory. Our lecture will have a strong historical flavour; we shall attempt to survey the development of the theory of complex multiplication and the philosophy behind it. On this background, we will present some exciting recent results, some of which build in an essential way on Borcherds theory. Finally, we shall sketch some of the key challenges of the theory of complex multiplication today and future directions.

This talk is a partial survey of some developments over recent years around the expansion theory ofthe Cayley graphs of linear groups and spectral gaps of the corresponding Hecke operators. A sample resultis the extension of Selberg's theorem for congruence subgroups of SL2(Z) to the setting of arbitrary non-elementary subgroups. Underlying these advances are techniques from arithmetic combinatorics that will be briefly indicated. The main emphasis of this (non-technical) presentation will be rather the applications however. They are quite diverse and relate to a variety of different issues in various fields: hyperbolic lattice point counting, number theory, geometry and questions from computer science.

This lecture will give an introduction to the topology and geometry of DNA. Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate and carefully control the geometry and topology of cellular DNA perform many important cellular
processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This talk is intended for a general scientific audience, and will discuss topological models for DNA strand passage and exchange, including the analysis of topoisomerase experiments on circular DNA using knot theory.

In the talk we will make a presentation of the theory of Nonlinear Diffusion centered on one of the popular models, the porous medium equation and its close relative, the fast diffusion equation. The existence of free boundaries is one of the most peculiar properties of the former equation. Related models and literature will be mentioned.
In the final section, I will present recent work that combines degenerate nonlinear diffusion with nonlocal operators of fractional Laplacian type. Apart from the unexpected existence of free boundaries, the model admits mass preserving self-similar solutions that are found by solving an elliptic fractional-Laplacian obstacle problem. We use entropy methods to show that the asymptotic behaviour is described after renormalization by these self-similar solutions.
This is intended to give an idea of the variety of topics and techniques in the area.

The theory of multiphase systems was developed at the beginning of the 19th century assuming that different phases are at local equilibrium and are separated by a sharp (i.e. with zero thickness) interface. This approach breaks down when the real interface thickness is comparable to the lengthscale of the phenomenon that is being studied, as it happens near a contact line or in the breakup or coalescence of liquid droplets. A different approach consists in treating the interface as a finite (although thin) region where the density, or the composition, of the mixture varies from one value (not necessarily of equilibrium) to the other. The drawback of this approach is that we have to add a mass conservation equation to the equation of conservation of momentum and of energy, as we need to determine the density (or concentration) profile of the mixture in the interface region. The advantage is that the position of the interface is automatically determined through the concentration profile and so no interface tracking is required. This approach, which is generally referred to as the diffuse interface method, is based on one of the many intuitions by Van der Waals and was later generalized by Ginzburg and Landau to formulate the mean field theory.
After deriving the basic equations of the model, results of several recent simulations are presented and commented. In particular, we will describe spinodal decomposition and nucleation of both liquid binary mixtures and single component, vapor-liquid systems.

Coupled nonlinear Schroedinger equations have found considerable interest for many years, being motivated by various problems in mathematical physics. We present existence results for vector solitary wave solutions achieved in the last years using variational methods.

La teoria degli integrali singolari si è sviluppata a partire dagli anni $'$50, ad opera di Calderon e Zygmund, come strumento per l$'$analisi della regolarità $L^p$ di soluzioni di equazioni ellittiche, e al tempo stesso incorporando metodi, nati con Hilbert, Riesz, Hardy e Littlewood, nell ambito dell$'$analisi complessa in una variabile. Nel corso degli anni, la teoria ha visto significativi ampliamenti, raffinamenti e sistematizzazioni, con un forte impatto in diversi ambiti dell$'$analisi. Verranno presentati alcuni recenti sviluppi della teoria, in un contesto motivato da problemi di analisi complessa in più variabili e teoria degli operatori ipoellittici.

In polymer physics and biomechanics the need for models based on micro-mechanics considerations is clear. In this talk we show that it is possible to study the behavior of disordered media constituted by considering at the microscale a bimodal distributions of elastic and breakable links with variable activation and fracture thresholds. Depending on the microscopic distribution properties, the material may be characterized by an unstable strain domain, which gives the possibilities of having homogeneous or localized damage. This simple idea delivers a theoretical scheme to describe many experimental effects observed at the microstructure and macroscopic scale in disordered materials like synthetic or biological polymers. Moreover, we discuss how these idea may fitting in the theory of continuum mechanics with multiple configurations.

La conferenza ha come scopo la presentazione di una formulazione dinamica di alcuni problemi di trasporto ottimo di massa, ottenuta utilizzando
l approccio alla Benamou-Brenier che consiste nella minimizzazione di un opportuno funzionale dipendente dalla densità di massa e dalla velocità
del flusso, accoppiata con la equazione di continuitaà.
Verranno considerati problemi di transporto in cui sono presenti effetti di congestione, come ad esempio in diversi problemi di traffico e nel
movimento di una folla in caso di panico, e di concentrazione, come ad esempio nel trasporto ramificato presente nelle radici di piante, reti di comunicazione, sitemi di circolazione sanguigna.

La Luna rivolge sempre la stessa faccia verso la Terra.
Questo fenomeno si chiama risonanza sincrona, e vuol dire che coincidono il periodo di rivoluzione della Luna attorno alla Terra ed
il periodo di rotazione della Luna attorno a se stessa. Ben lungi dall essere un caso isolato, le risonanze sincrone sono molto comuni
tra i corpi del sistema solare. Il modello matematico che descrive questa situazione rientra nella classe dei sistemi quasi-integrabili con dissipazione. In tale contesto, studiamo l esistenza di attrattori periodici e quasi-periodici attraverso metodi analitici
(teoria perturbativa, teorema KAM, teorema di Nekhoroshev) oppure numerici (bacini di attrazione, analisi in frequenza, metodo di Newton).
I risultati ottenuti forniscono un importante strumento per comprendere il ruolo degli effetti dissipativi nella selezione delle risonanze,
trovando ampie applicazioni sia nel caso della dinamica rotazionale dell esempio Terra-Luna, sia in contesti più generali come il problema dei tre corpi soggetto ad effetti dissipativi.

We study Chow groups of smooth projective varieties over algebraically closed fields and construct examples, where the torsion and cotorsion of these groups is very large.

Our aim in this talk is to discuss issues related with the Cahn-Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials. In particular, we discuss the well-posedness and the asymptotic behavior of the system. Furthermore, we consider both the usual Neumann boundary conditions and the newly proposed dynamic boundary conditions.

La buona posizione di certe equazioni alle derivate parziali, in particolar modo le equazioni di Navier-Stokes in dimensione 3,
è uno dei problemi aperti più studiati. Per equazioni differenziali ordinarie, è ben noto che la aggiunta di un white noise può rendere ben posta una equazione che altrimenti non lo è (in questo senso parliamo di "regolarizzazione" da parte del noise). Si deve allora capire se questo fenomeno si estende alle PDE.
Il seminario riassumerà alcuni fatti di base sulla regolarizzazione dovuta al moto browniano in dimensione finita, concludendo con alcuni
recenti progressi per certe PDE; pur restando aperto il caso delle equazioni di Navier-Stokes.

Lo scopo del seminario è quello di presentare alcuni risultati riguardanti le proprietà geometriche e di simmetria delle interfacce che compaiono nello studio di alcuni modelli variazionali introdotti per descrivere le transizioni di fase. Facendo uso di tecniche di teoria delle equazioni alle derivate parziali, di calcolo delle variazioni, e di un pò di geometria,
dimostreremo la simmetria delle interfacce, in accordo con una famosa congettura di E. De Giorgi.

A collection of arguments will be presented in order to show that $L^1$ is the natural functional framework for convex gradient flows. In particular, Lipschitz continuous data dependence is not available in general except possibly for the $L^1$ norm.
Examples from solid mechanics and phase transitions illustrate how this information can be exploited for existence and uniqueness of solutions to coupled problems.

I will explain how one could construct a birational invariant from the derived category of an algebraic variety. An application to characterization of rational cubic fourfolds will be discussed.

Hypercomplex analysis is a generic name for those generalizations of one-dimensional complex analysis which involve hypercomplex numbers. Quaternionic analysis is the oldest and the most known version of it, so that it will be discussed, first of all, in which sense it is a "proper" or a "closest" version in low dimensions which includes, as particular cases, such classic theories as vector analysis and holomorphic mappings in two complex variables, as well as some systems of partial differential equations. This allows to one, by developing quaternionic analysis, to obtain new results for the above classic theories and to refine known ones; some applications of this approach will be presented. Some comments on Clifford analysis and its applications will be also made.

Moduli spaces of spin curves parametrize roots of canonical bundles of curves (theta-characteristics), and are highly interesting
covers of the moduli space of curves. Their geometry is of interest both from an algebraic-geometric, as well as from a string-theoretic point of view. Recently, in joint work with A. Verra we have achieved a complete classification of these spaces from the point of view of birational geometry. For instance, the even spin moduli space is a unirational variety for g8, and a very mysterious variety of Calabi-Yau type for g=8.

A control system is a dynamical system on which one can act by using ``controls'. A control system is controllable if, for every pair of states, one can steer the control system from the first state to the second one. We survey in this talk some methods to check if a control system is controllable, specially if the control is modelled by partial differential equations and in the case where the nonlinearity plays a crucial role. Applications are presented to control systems in fluid dynamics.

La teoria classica di Calderon-Zygmund, nei suoi aspetti classici, risale agli anni 50 e permette di determinare le proprietà di integrabilità ottimali delle soluzioni di equazioni ellittiche e paraboliche in termini dell integrabilità del dato, nel caso in cui i problemi considerati siano lineari. Questa parte riposa su rappresentazioni esplicite via operatori di convoluzione integrali, e tecniche di Analisi Armonica. La teoria di Calderon-Zygmund nel caso
di equazioni non-lineari è cosa più recente. Tenterò di delineare alcuni sviluppi recenti che permettono, da un lato, di evitare completamente l uso dell Analisi Armonica, e dall altro, di stabilire paralleli più precisi tra gli aspetti lineari e quelli non lineari.

The problem of front propagation in a stirred medium is addressed
in the case of laminar and turbulent flows in three different regimes:
slow reaction, fast reaction and geometrical optics limit. It is well
known that a consequence of stirring is the enhancement of front speed
with respect to the non-stirred case. By means of numerical
simulations and theoretical arguments we describe the behavior of
front speed as a function of the stirring intensity. The large scale
of the velocity field mainly rules the front speed behavior even in
the presence of smaller scales. In the unsteady (time-periodic) case,
the front speed displays a phase-locking on the flow frequency and,
albeit the Lagrangian dynamics is chaotic, chaos in front dynamics
only survives for a transient. Asymptotically the front evolves
periodically and chaos manifests only in the spatially wrinkled
structure of the front.

This talk aims to introduce some general ideas relating properties from algebraic
geometry to concepts from metric geometry, in particular that of Gromov-Hausdorff limits
of metric spaces.
Let $X$ be a Calabi-Yau manifold of dimension $n$, that is a complex projective manifold
which admits a nowhere vanishing holomorphic $n$-form, and no holomorphic $i$-forms for
$0 < i < n$. By a famous theorem of Yau, for each K"ahler class in the real second cohomology,
there exists a unique Ricci
at K"ahler metric on X with K"ahler form in the given
class, the Calabi Yau metric; hence there is a well-defined metric space structure on $X$.
A natural question then arises: if we degenerate either the complex or Kähler structures
on X in the sense of algebraic geometry, obtaining a singular projective variety, what
can be said about the metric limits (in the sense of Gromov-Hausdorff) of the corresponding Ricci
at K"ahler manifolds? We will suggest some answers to this question and explain
their relevance for the geometry of Calabi-Yau manifolds.

In mechanical engineering distortion means undesired alterations of size and/or shape of a workpiece. A typical cause for distortion are solid-solid phase transitions as they occur during the heat treatment of steel.
In my talk I will present a thermomechanical model of phase transitions in steel, show some applications in the heat treatment of steel and discuss possible optimal control approaches to compensate distortion.

In a non-cooperative differential game, two or more players can influence the time evolution of a system. Each player chooses a feedback control $u=u(x)$ in order to minimize his own cost functional.
If a Nash equilibrium solution exists, under suitable regularity conditions the value functions satisfy a (highly nonlinear) system of Hamilton-Jacobi equations.
After a general overview, the talk will present a
new analytical approach to the study of these PDEs, based on a homotopy method.
Examples show that this approach is also naturally motivated by some economic models.

Un piano proiettivo finto è una varietà complessa, compatta, liscia, e senza frontiera, di dimensione complessa ~2, la quale ha gli stessi numeri di Betti del piano proiettivo complesso.
Il primo piano finto è stato costruito da [Mumford, 1979]. Cartwright e Steger sono riusciti a portare a termine il lavoro
cominciato in [Prasad, Yeung, 2007], e hanno elencato tutti i piani finti. A meno di equivalenza olomorfa e antiolomorfa, ce ne sono ~50. Il lavoro dipende fondamentalmente da diversi calcoli numerici e simbolici a proposito di certi sottogruppi di matrici di $U(2,1)$, calcoli eseguiti con l aiuto degli elaboratori elettronici.

By revisiting Gauss work on magnetism we provide [1] a plausible reconstructionof what might have been Gauss own derivation of the linking number
concept and formula. Information on linking number and crossing number are fundamental
in a topological approach to field theory, topology providing a lower bounds on the
energy of the physical system [2]. In the case of knots, for instance, an exact analytical
expression for the minimum energy is found and the energy spectrum determined [3].
These results are fundamental in the classification of knots and links by energy methods and in the search for ground-state energy information in topological field theory [4].
[1] Ricca, R.L., Nipoti, B. On the original derivation and modern interpretations of
Gauss' linking number. To be submitted to J. Knot Theory and Its Ram.
[2] Ricca (2008) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293-300.
[3] Maggioni, F. and Ricca, R.L. (2009) On the groundstate energy of tight knots. Proc. R. Soc. A 465, 2761-2783.
[4] Ricca, R.L. (Ed.) (2009) Lectures on Topological Fluid Mechanics. Springer-CIME
Lecture Notes in Mathematics 1973. Springer-Verlag. Heidelberg, Germany.

We consider the elliptic system
$$ \Delta u = W_u(u), \quad
x\in\mathbb{R}^n, $$
for a class of potentials $W : \mathbb{R}^n \to\mathbb{R}$ that possess
several global minima and are
invariant under a finite or discrete reflection group $G$ acting on
$\mathbb{R}^n$. We establish existence of nontrivial $G$−equivariant
entire solutions $u : \mathbb{R}^n \to\mathbb{R}^n$ connecting the
global minima of $W$. If $G$ is a discrete (infinite) group the solution
$u$ has a kind of crystalline periodic structure and existence is ensured
provided the elementary cell contains a ball of radius $R^*$
with $R^*$ a constant that depends only on $W$.

Given a bounded open set $\Omega$
in $\mathbb{R}^2$ or in a Riemannain manifold
and a partition of $\Omega$
by $k$ open sets $\omega_j$ , we can consider the quantity
$\max_j \lambda(\omega_j)$ where $\lambda(\omega_j)$ is the ground state energy of
the Dirichlet realization of the Laplacian in $\omega_j$. If we denote by
$\mathcal{L}_k(\Omega
)$ the infimum over all the $k$-partitions of
$\max_j \lambda(\omega_j)$, a minimal $k$-partition is then a partition which
realizes the infimum.
Although the analysis is rather standard when $k=2$ (we find the nodal domains of a second
eigenfunction), the analysis of higher $k$ becomes non trivial and quite interesting.
In this talk, we would like to discuss the properties of minimal spectral partitions,
illustrate the difficulties by considering simple cases like the disc, the rectangle
or the sphere ($k = 3$) and also exhibit the possible role of the hexagone in the
asymptotic behavior as $k \to \infty$ of
$\mathcal{L}_k(\Omega
)$. We will also explain the link of these questions with spectral
properties of some Aharonov-Bohm hamiltonians.
We will finally discuss other definitions of minimal partitions.
This work has started in collaboration with T. Hoffmann-Ostenhof and has been continued
(published or to appear) with the coauthors V. Bonnaillie-Noel, T. Hoffmann-Ostenhof, S. Terracini
and G. Vial.

We discuss some recent development in Hamiltonian PDE. The Hamiltonians here are typically non convex and moreover in infinite dimensions. This new approach is local. It eliminates the convexity condition and is natural for PDE.

Thirty years old, the famous Italian mathematician De Giorgi asked if entire solutions to certain nonlinear PDEs in the whole space are indeed ODE solutions, provided certain monotonicity assumption is assumed.
This conjecture has attracted lots of studies. In this talk, I will discuss the status of this conjecture, its extensions, and moreover the intricate connections between differential geometry and nonlinear PDE.

We discuss some recent results in the mathematical theory of complex fluid systems, in particular, the concept of weak solution and its relevance to a proper formulation of balance laws in fluid mechanics.
A rigorous asymptotic analysis of these systems is developed and several model problems discussed.
In particular, the following topics will be addressed in detail:
functional analytic framework and the basic ideas of the mathematical theory of continuum fluid mechanics;
thermodynamic stability and equilibrium states, behavior of energetically insulated systems for large time;
low Mach number limits and propagation of acoustic waves in thermally conducting viscous fluids.

The lecture introduces a new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers. It is based on the principle The part is less than the whole introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework different from those of the non-standard analysis and surreal numbers. The new approach gives possibility to introduce a new type of a computer - the Infinity Computer - able to operate numerically not only with finite numbers but also with infinite and infinitesimal ones (European Patent has been granted recently). The Infinity Calculator using the Infinity Computer technology is presented during the talk. An additional information can be downloaded from the page http://www.theinfinitycomputer.com

A flow is a pair $(S,X)$, where $X$ is a compact Hausdorff space and $S$ is a semigroup of continuous
maps of $X$ into itself. In this joint work with Ali Jabbari, we analyze the family of special flows.
Let ${\mathbb{T}}=\{z\in{\mathbb{C}}:|z|=1\}, k\in{\mathbb{N}}$ and $\lambda=e^{2\pi it}$ with $t$ a
fixed irrational real number and let $f\in{{\mathbb{T}}^{\mathbb Z}}$ be defined by $f(n)=\lambda^{n^k}$.
As usual we define the shift operator $U:{\mathbb T}^{\mathbb Z}\rightarrow {\mathbb T}^{\mathbb Z}$
by $Ug(n)=g(n+1)$ for each $g\in{\mathbb T}^{\mathbb Z}$.
Let $X_f$ be the closure of the orbit $\{U^nf:n\in\mathbb N\}$.
Then the flow we are interested is of the form $(\{U^n:n\in\mathbb N\},X_f)$.
Previously the case $k=2$ was treated by us in 1984, and the case $k=4$ was considered by P.Milnes.

"Cluster categories were introduced in the paper ""Tilting theory and cluster combinatorics"" in order to better understand the combinatorics of cluster algebras, by giving new, module theoretic and categorical meanings to the combinatorics of the well known Cluster algebras of Fomin and Zelevinsky. Subsequently, we gave a very precise correspondence between the notions in these two areas. This proved to be quite useful and productive approach with even further connections to semi-invariants of quivers. However, in order to get this connection, we define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem (determinantal), the Saturation theorem and the Canonical Decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between the supports of the semi-invariants and the Tilting triangulation of the (n-1) sphere."

This talk will be centered around the three related partial orderings on the space of d-dimensional representations of a finitely generated associative algebra over an algebraically closed field given by degenerations, what is called virtual degenerations and an order induced by the dimensions of the spaces of homomorphisms. These partial orders are on the outset of a geometrical nature. However, it is usually more convenient to express them in pure algebraic terms using homological properties. Therefore these notions can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of the field altogether. The notions will be introduced by using familiar examples from linear algebra and matrices, and some basic algebra.

Vito Volterra's mathematical work stretches over the period 1881 to 1940. His first paper was written while still an undergraduate student and not yet twenty years of age, while his last was published only months before his death in 1940 at the age of eighty. His work stretches over a long period, during which great mathematical strides were made, and mathematical styles changed considerably. Likewise, he danced from one subject to another, covering wide swathes of continuum mechanics, astronomy, differential equations and mathematical biology. Unlike many mathematical colleagues, he did not settle into a happy but intellectually unproductive old age. He wrote several masterly textbooks, some parts of which give summaries of the field which cannot be bettered even today. Finally, he was no mean historian of science, as can be seen from some of the semi-popular essays he wrote. Meanwhile he had an active life as a politician and public figure. In this talk I shall give a brief outline of Volterra's life, career and mathematics, concentrating particularly on areas in which his work, sometimes accidentally, had lasting influence.

In this seminar we consider the problem of constructing confidence regions for the model parameters of dynamical systems from observed data. Taking a major departure from previous methods, we introduce a new approach called Leave-out Sign-dominant Correlation Regions (LSCR) which delivers confidence regions with guaranteed probability. Based on subsampling techniques, we derive the exact probability that the true parameters belong to certain regions in the parameter space. By intersecting these regions, a confidence set containing the true parameters with guaranteed probability is obtained. All results hold true for any finite number of data point. Moreover, prior knowledge on the noise affecting the data is reduced to a minimum. The approach will be illustrated on simulation examples, showing that it delivers practically useful confidence sets with guaranteed probabilities.

"How many different objects can be obtained by gluing together in pairs the faces of an octahedron? After deciding what ""object"" and ""different"" mean, this is an apparently very elementary question, but the answer is not quite so. Before facing it I will go one dimension down and consider the gluings of the edges of a polygon, discussing surface topology and showing that there are extremely few surfaces one can get from a given polygon compared to the number of inequivalent gluing patterns. Then I will introduce the notions of curvature and hyperbolic geometry in two and three dimensions, I will discuss rigidity and I will sketch how this applies to the original question, yielding the fact that the number of different results is indeed quite big."

We point out the strict connection between the convexity property, representation formula of solutions of quasilinear elliptic problems and related Liouville theorems.

A general formula is established for translation speed of an axisymmetric vortex ring whose core is not necessarily thin. We rely on Lamb-Saffman-Rott-Cantwell's method of calculating the total kinetic energy of fluid in two ways. Combined with the Navier-Stokes equations, we can skip the detailed solution for the flow field to extend Saffman's velocity formula of a viscous vortex ring to third order in the ratio of the core radius to the ring radius, a small parameter, for the entire range of the Reynolds number. At small Reynolds numbers, a solution that describes the whole life of a vortex ring is available. For inviscid motion, a further simplification is achieved by resorting to the variation, under the topological constraints, of the kinetic energy with respect to the hydrodynamic impulse. This principle bears similarity with the variational principle for a vortex ring governed by the Gross-Pitaevskii equation. Similarity is also found with Rasetti-Regge's theory for the three-dimensional motion of a vortex filament.

Nel seminario si presenta una nuova teoria delle funzioni regolari sui quaternioni (e sugli ottetti), nata da una recente interpretazione geometrica di una definizione dovuta a Cullen. La nuova teoria è interessante perché accoglie tra le funzioni regolari i naturali polinomi e le serie di potenze, escluse dalle principali teorie esistenti, compresa quella dovuta a Fueter. Gli analoghi dei risultati basilari della teoria delle funzioni olomorfe vengono dimostrati in questo ambito, e si trova una nuova ed interessante proprietà geometrica caratterizzante per il luogo di zeri (e di punti singolari) di una funzione regolare.

Nonsmooth analysis is the subject that develops differential calculus for functions that don't have derivatives in the usual sense. Its applications are wide-ranging, from basic analysis and optimization to mechanics and control theory. We give a non-technical account of its principal ideas, together with some of its applications. We also reveal that nonsmooth analysis is an Italian invention.

An informal review is presented of the problem of viscous flow in a corner between two intersecting plane rigid boundaries, with particular attention to the behaviour (i) when the stirring that generates the flow is remote and sinusoidal, and (ii) when the stirring is provoked by torsional oscillation of the fluid domain about the line of intersection of the two boundaries. In each case, weak inertial effects lead to a lag of the response of the fluid behind the forcing mechanism. Conditions determining the existence and evolution of a geometric sequence of eddies in the corner are determined, and the manner in which the associated streamline pattern reverses during each half-period of the flow is described.

The study of shape optimization problems is a very wide field, both classical, as the isoperimetric problem and the Newton problem of the best aerodynamical shape show, and modern, for all the recent results obtained in the last two, three decades. The interesting feature is that the competing objects are shapes, i.e. domains of RN, instead of functions, as it usually occurs in problems of the calculus of variations. This constraint often produces additional difficulties that lead to a lack of existence of a solution and to the introduction of suitable relaxed formulations of the problem. However, in some few cases an optimal solution exists, due to the special form of the cost functional and to the geometrical restrictions on the class of competing domains. The purpose of this talk is to make a short survey of this question.

Recent applications such as control over networks, system state estimation using networks of sensors and real-time embedded control systems have blurred the boundaries of the disciplines of communications, control and computation. Increasingly, communication, control and computation take place through interconnection of systems leading to desirable interactions. From a methodological point of view, simple models where the nature of these desirable interactions can be studied in some depth are needed. We examine the structure of interaction between sensing, communicating and controlling in the context of statistical filtering for signals described as Hidden Markov processes and in the context of stabilization of an unstable control system where the sensor and controller are linked via a noisy communication channel. We argue amongst other things, that a fundamental reexamination of information theory is needed to address these questions. These ideas appear to have nontrivial connections to recent work in nonequilibrium statistical mechanics.

In my talk I give a survey of the existence results for the Neumann problem involving Sobolev and Sobolev - Hardy exponents. This also includes problems with the critical exponent on the boundary. A joint effect of the shape of the indefinite weight and the mean curvature on the existence will be discussed.

"Le equazioni a derivate parziali stocastiche permettono di introdurre elementi di incertezza nella modellizzazione matematica: ciò può essere realizzato considerando come variabili aleatorie i coefficienti dell'equazione, i termini noti e lo stesso dominio in cui l'equazione è posta. Lo studio e la discretizzazione numerica di tali equazioni pone problemi interessanti e non banali. Focalizziamo l'attenzione sul trattamento dell'aleatorietà del dominio, che nasce ad esempio in certi processi tecnologici avanzati. Consideriamo due diversi approcci (""mapping"" e ""fictitious domain""): in entrambi i casi, il problema aleatorio viene trasformato in un problema deterministico attraverso l'uso di espansioni di Wiener (generalizzate), dette anche ""caos polinomiale"". La discretizzazione è effettuata attraverso metodi di proiezione di tipo Galerkin, eventualmente accompagnati da integrazione numerica (collocazione). Il confronto teorico e sperimentale con tecniche Monte Carlo mette in luce l'efficienza dei metodi proposti."

The approach based on monotonicity and non-degeneracy to study the regularity of free boundaries will be discussed. In particular recent results obtained jointly with H. Shahgholian and G. Weiss will be presented. For two-phase obstacle problem it is proved that if two phases are close enough to each other at the origin then in some ball Br(0) their boundaries are C1 graphs.

Long, filamentary structures which are disordered, knotted or tangled are ubiquitous in Nature. Examples range from long molecules and DNA, to magnetic field lines in astrophysical plasmas, to vortex structures or patticle paths in turbulent flows. The aim of this talk is to show that simple ideas from knot theory can be used to characterize the complexity of these structures. The two examples which I shall discuss are turbulence in superfluid helium and classical turbulence in an ordinary viscous fluid. The former is a particularly nice benchmark to apply these ideas, because it consists of a tangle of discrete vortex lines.

Il Professor Malliavin è noto nell’ambito del calcolo stocastico per avere sviluppato un calcolo differenziale stocastico infinito dimensionale nel contesto dei processi stocastici a traiettorie continue (calcolo di Malliavin). Recentemente questo strumento è entrato a fare parte del bagaglio di conoscenze di coloro che si occupano di finanza matematica in quanto è uno strumento flessibile e potente che permette di ricavare soluzioni in forma chiusa di strategie di copertura e della densità di prezzo in contesti non classici (non markoviani). Lo strumento si è rivelato utile anche nelle simulazioni Monte-Carlo per il pricing di titoli derivati. Più recentemente il prof. Malliavin ha conseguito interessanti risultati in finanza matematica riguardo alla stima della volatilità dei mercati finanziari e allo studio delle proprietà geometriche della curva dei tassi di interesse in modelli HJM. Questo argomento sarà l’oggetto del suo seminario.

Sia X una superficie fibrata su una curva B, e sia g il genere della fibra. Se la fibrazione è relativamente minimale la classica disuguaglianza di slope dice che il rapporto tra l'autointersezione del fascio canonico relativo e il grado dell'immagine diretta di questo stesso fascio è limitata inferiormente da 4(g-1)/g. Questa disuguaglianza è ottimale, ma è naturale ritenere che ponendo limitazioni sul tipo di fibrazione che si considera si possano ottenere disuguaglianze migliori. Dopo aver accennato una breve dimostrazione della disuguaglianza di slope discuterò il problema in alcuni casi specifici, in particolare in quello delle fibrazioni non albanese, in cui ci sono stati progressi recenti.

The first part of this lecture begins with a brief description of the nonlinear equations of continuum physics emphasizing the fundamental role of constitutive equations and the difficulties in determining them. Here the connection between experimentation, inverse methods, and qualitative methods is discussed. Some pressing fundamental open problems for the differential equations are presented. The second and major part of the lecture treats the solutions of a collection of conceptually simple specific problems illustrating the general themes raised in the first part: studies of of thresholds in material response, inverse problems for constitutive behavior, design and control problems for smart materials, asymptotic justification of quasistaticity, admissibility conditions for shock and other discontinuities, etc.

Dirac operators on spin manifolds have been studied extensively. Further Dirac operators in euclidean space have also been studied extensively. In this talk some attempt will be made to bring the two perspectived together. The context is that of conformally flat manifolds. Automorphic forms in n real variables will be used to develop aspects of function theory and classical harmonic analysis in this context.

I shall discuss some results, old and new, about solvability of elliptic boundary value problems with a power non-linearity. Of particular interest is the case when the power lies above Sobolev’s critical exponent, where variational methods do not apply, and strong obstructions for solvability are present. I will concentrate on a new existence result for the classical Lane-Emden-Fowler equation in a exterior domain.

In this lecture, I shall give a brief review of some developments in KAM theory and the problem of Arnold diffusion, such as the lower dimensional tori in resonant regions, natural parametrization and Russmann's condition and variational construction of diffusion orbits.

Anche senza scomodare il Teorema di De Giorgi, ´e semplice provare l’esistenza del minimo per funzionali integrali del Calcolo delle Variazioni della semplice forma (0.1) J(v) = Z |rv|2 −: 2 Z fv ( aperto limitato di IRN). Essendo l’ambiente adeguato W1,2 0 ( ) e appartenendo v a L 2N N−:2 ( ), serve che f stia in L 2N N+2 ( ). Il minimo u esiste, ´e unico e verica la forma debole dell’equazione di Eulero- Lagrange (0.2) ( −:
u = f , u = 0 @ Poich´e il precedente problema al contorno ammette soluzioni distribuzionali anche se il dato f appartiene anche solo L1( ): (0.3) u 2 W1,q 0 ( ), q < N N −: 1 : Z rur
= Z f
, 8
2 D( ) potrebbe sorgere la domanda (a me ´e sorta!): il problema (0.3) ´e legato a un qualche problema di minimo, anche se f 2 L1( ) comporta che l’estremo inferiore di J in (0.1) ´e meno infinito? Funzionali convessi pi´u generali del tipo C(v) = Z j(x,rv) −: Z fv Funzionali non convessi del tipo I(v) = Z j(x, |v|,rv) −. Z fv 2 LUCIO BOCCARDO Funzionali vettoriali (Sistemi): lavori in corso. Altri problemi. Dipartimento di Matematica, Universit`a di Roma 1, Piazza A. Moro 2, 00185 Roma tel: 00 39 06 49913202 E-mail address: boccardo@mat.uniroma1.it

This talk will give elementary proofs (due to Jay Goldman, Sofia Lambropoulou and LK) of the Conway classification of rational tangles, and we will sketch the classification of rational knots and links by using methods developed with Sofia Lambropoulou. This is very beautiful elementary topoloogy that applies to the study of DNA recombination. The talk will discuss how this knot theory is used to study DNA.

In the talk I'll give a definition of the the Fourier transform on the space S(V) of locally constant functions on with compact support on a finite dimensional vector space V over the field of p-adic numbers and discuss the property of the induced Fourier transform on the space of distributions.

Modules (representations) play a key role in the structure theory of associative rings (algebras). Starting from Morita theorem characterizing equivalence of full module categories, we will proceed to Brenner-Butler theorem (1-dimensional case), and the Miyashita theorem (n-dimensional case). In all these theorems, the representing tilting modules are finitely generated. Then we will introduce the recent notion of an infinitely generated tilting module and show its relation to module approximations. Though infinitely generated, the tilting modules are of `finite type', and this fact makes it possible to classify them over particular rings. We will finish by presenting two recent applications of infinitely generated n-tilting modules: (i) for n=1, to describing the structure of Matlis localizations of commutative rings, and (ii) for an arbitrary n, to proving finitistic dimension conjectures for (non-commutative) Iwanaga-Gorenstein rings.

The macroscopic plasticity of ductile crystals is the net result of the collective motion of large numbers of crystal lattice defects, most notably, glissile dislocations, with structures forming at multiple length scales. These include the scale of the lattice, where dislocations may be regarded as discrete topological defects: the scale of the mean distance between dislocations, where the dynamics of the dislocation ensemble is of primary interest: and the sub-grain scale, where dislocations form characteristic patterns. The development of mathematical links between the behaviors at all scales, and the characterization of the effective macroscopic behavior of ductile crystals, remains a central and long-standing problem in physical metallurgy. Tools of the calculus of variations such as relaxation and Gamma convergence prove powerful and convenient in forging those links. I plan to review a number of mathematical problems that arise in that context, and some recent results pertaining to those problems, including: the formulation of a geometrical mechanics of discrete dislocations and the passage to the continuum: the effective energetics and dynamics of dislocation ensembles: and the relaxation and optimal scaling properties of single-crystals.

"n questo seminario verranno esposti alcuni risultati (ottenuti in collaborazione con Alberto Besana (Dipartimento di Matematica ""F.Enriques"", Università di Milano)) riguardanti un'interpretazione meccanica del writhing number (o writhe, o numero di avvitamento, o di Tait) di un nodo, rispetto ad una sua proiezione regolare piana, ricorrendo ad un analogo infinito-dimensionale della teoria di Maslov- Hörmander. Il cambiamento di writhe corrisponde al passaggio attraverso il ciclo di Maslov (caustica), rappresentato da nodi con un solo punto doppio in un'opportuna sottovarietà lagrangiana della varietà simplettica dei nodi (debolmente singolari) introdotta da J.L. Brylinski, ottenuta tramite una semplice costruzione geometrica (nodi che si avvitano su un cono avente come direttrice una prefissata proiezione piana). La funzione generatrice è rappresentata da un'azione di Chern-Simons abeliana (elicità, in meccanica dei fluidi) con l'aggiunta di un termine di sorgente, dipendente dal nodo. Le connessioni (abeliane) forniscono le variabili ausiliarie. L'elicità gioca il ruolo della segnatura dell'operatore Hessiano rispetto alle suddette variabili. Verranno inoltre discusse, tempo permettendo, questioni di quantizzazione geometrica e si mostrerà come la condizione di quantizzazione di Feynman-Onsager della teoria quantistica dei vortici emerga quale condizione di Bohr-Sommerfeld imposta ad una diversa sottovarietà lagrangiana (nodi sulla sfera unitaria)."

We present a new approach for proving of existence and finite-dimensionality of global attractors for infinite-dimensional dissipative systems generated by abstract nonlinear second order in time evolution equations. This approach is based on far reaching generalizations of the Ceron-Lopes theorem on asymptotic compactness and Ladyzhenskaya's theorem on the dimension of invariant sets. An application of our results to nonlinear damped wave equations allow us to obtain new results pertaining to structure and properties of global attractors for nonlinear waves.

The point of the talk is that of providing purely algebraic definitions and proofs of otherwise known results on integrable systems of linear di erential equations with only a finite dimensional space of local solutions over a smooth algebraic manifold. This talk represents work in collaboration with Y. Andr´e and it is based on our joint book: “De Rham Cohomology of Di erential Modules on Algebraic Varieties”, Progress in Mathematics Vol. 189, Birkhaueser (2001). We correct some statements in that book. The result is still incomplete. Summary: 1. Review of the notion of regular singular point for a linear ordinary di erential equation with meromorphic coecients. Global counterpart “Fuchsian equations”) on a Riemann surface. 2. Examples: Hypergeometric equations on the projective complex line, with 3 singular points. 3. Generalisation of the notion of regular singularity along a divisor to integrable overdetermined systems of linear PDE’s over a complex algebraic manifold, and of the notion of fuchsian connection on an algebraic vector bundle. 4. Deligne’s canonical extension of a Fuchsian connection on a smooth complex algebraic variety, as a logarithmic connection with singularities along a divisor with normal crossings on a Hironaka compactification. 5. Deligne’s criterion of regularity on a normal (not necessarily smooth!) compactification: one only needs to consider the behaviour along the divisors at infinity. 6. (Counter)examples of J. Bernstein to some statements in loc. cit.. 6. Reduction process in a purely algebraic proof of 5. 1

"1-Algebraic 2-In terms of the assymptotic properties of the invariant difusion 3-In terms of the geometry ""at infinity"""

Starting with the classical result of Ambrosetti and Prodi, I will review the history of an area that can be summarized as: asymmetric nonlinearities with positive loads have many solutions... the more asymmetric the nonlinearity, the more solutions. We shall look at the history for semilinear elliptic equations, and also some applications to simple mechanical systems such as cable-suspended masses.

Nel seminario verrà illustrato il legame fra il problema dell'unicità della soluzione positiva di un problema ellittico semilineare con il termine non lineare di tipo potenza e quello dello studio delle proprietà geometriche della seconda autofunzione del problema linearizzato. Da questa questione ha origine lo studio del comportamento asintotico di tali autofunzioni e dei relativi autovalori quando l'esponente non lineare tende all'esponente critico, in dimensione N >= 3.

In questo seminario si discute la stretta correlazione che intercorre tra la dinamica caotica e la teoria dell'informazione. Si mostra che la quantità media di informazione necessaria per descrivere un orbita caotica è uguale all'entropia di Kolmogorov-Sinai del sistema dinamico. Questa analisi porta a definire nuovi indicatori di caoticità per i sistemi dinamici di entropia nulla. Se il tempo lo permetterà, discuteremo brevemente alcune applicazioni di questa teoria allo studio di serie temporali di natura biomedica.

"Foams present a fascinating variety of interdisciplinary challenges to our understanding, from the mathematics of minimal surfaces to the behaviour of the head on a pint of beer. We will discuss recent research, which is moving from questions of static structure to more dynamical effects, in essence ""the fluid dynamics of foams"". We will go back in history to Kelvin and Plateau (and beyond), and also forward to 2008 when the Beijing Olympics will feature an astonishing manifestation of the theory of foams..."

In this seminar, a new design procedure for H2 optimal robust filtering is presented. The robust filter is determined from the equilibrium solution of a minimax programming problem where the H2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all full order, linear, rational and causal filters. It is shown that for the class of parameter uncertainty considered, the equilibrium solution of the aforementioned minimax problem can be exactly determined. In contrast to the design methods available in the literature to date, the proposed one does not include any degree of conservatism. The classical static linear approximation problem as well as the filter design problem corresponding to continuous and discrete time linear systems are considered. An illustrative example is presented. The seminar ends with a discussion on a problem still unsolved.

In this talk, I shall describe the geometric approach to solve the Schrödinger equation for various physically meaningful three body systems such as He, H2+, H-, three bosons in R2 with d-function potential etc. The configuration space of the three body system in R3(resp. R2) (with center of gravity fixed at the origin) is an R6 (resp. R4) equipped with an SO(3) (resp. SO(2)) symmetric kinematic metric, while the potential function U is also SO(3) (resp. SO(2)) invariant. The first step is to fully utilize the SO(3) (resp. SO(2)) symmetry to reduce the Schrödinger equation to an equation solely defined at the level of the orbit space (i.e. R6/SO(3) (resp. R4/SO(2))) equipped with the orbital distance metric. One needs to make effective use of both group representation theory and equivariant differential geometry to achieve such a reduction. The orbit space of a three body system in R3 (resp. R2) equipped with the orbital distance metric is always isometric to the Riemannian cone over S2+ (1/2) (resp. S2(1/2))), namely the Euclidean hemisphere (resp. sphere) of radius 1/2. This remarkable fact (i.e. sphericality) enables us to bring in the spherical harmonics and their generalizations (namely, Jacobi polynomials and monopole harmonics) to greatly simplify the analysis of the angular part of the reduced equation. I will use the simpler case of the boson system to illustrate this step which enables us to further reduce the Schrödinger equation to an ODE solely in the radial direction. Such an ODE can be thoroughly analyzed and I will discuss the physical significance of these solutions so obtained for the three boson system. Bibliography Wu-Yi Hsiang. Kinematic geometry of mass-triangles and reduction of Schr¨odinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters. Proc. Nat. Acad. Sci. U.S.A., 94(17):8936–8938, 1997. Wu-Yi Hsiang. On the kinematic geometry of many body systems. Chinese Ann. Math. Ser. B, 20(1):11–28, 1999. A Chinese summary appears in Chinese Ann. Math. Ser. A 20 (1999), no. 1, 141.

"Nonlinear dynamical systems that satisfy hypothesis such as ergodicity and exponentially quick mixing are well known to be adequately studied in terms of the Boltzmann-Gibbs entropy and its corresponding statistical mechanics. These simplifying hypothesis are however NOT satisfied in vast classes of systems such as the so called ""complex systems"", ubiquitously emerging in physics, mathematics, economics, linguistics, chemistry, astrophysics, geophysics, biology, computer networks, engineering and elsewhere. A nonextensive entropy (characterized by an entropic index q, which reproduces the Boltzmann-Gibbs expression for q = 1) and its corresponding statistical mechanics provide an answer for at least part of such anomalous systems. A brief introduction will be given to the subject, followed by a survey on its dynamical foundations, which enable in particular the calculation, from first principles, of the index q associated with specific systems. Recent bibliography: ""Nonextensive Entropy - Interdisciplinary Applications"", M. Gell-Mann and C. Tsallis, eds. (Oxford University Press, New York, 2004) Full bibliography"

Let G be any compact Lie group. The aim of this lecture is to present the degree theory for G-equivariant gradient maps and to point out some applications of this degree to Hamiltonian systems. We will start with some remarks concerning the Brouwer degree. Moreover, using the Brouwer degree, we will classify homotopy classes of continuous gradient maps. Next, we will present properties of the degree for G-equivariant gradient maps and, using this degree, classify homotopy classes of continuous G-equivariant gradient maps. Additionally, we will show how to compute this degree. Finally, we will study the existence of periodic solutions of Hénon-Heiles nad Yang-Mills Hamiltonian systems in a neighborhood of an isolated, degenerate stationary solution. Moreover, we are going to study continuation of nonstationary periodic solutions of autonomous Newtonian systems. We will finish this lecture with some remarks and open questions concerning the degree for G-equivariant gradient maps.

Presentiamo alcuni risultati dati in [PGMS], come proprietà qualitative e teoremi di unicità per stati fondamentali a simmetria radiale, positivi o a supporto compatto in Rn, che sono soluzioni di equazioni ellittiche quasilineari, possibilmente singolari o degeneri, con pesi. Le ipotesi sui pesi consentono di includere interessanti ed attuali modelli ben noti e ampiamente studiati in letteratura, quali quelli soggetti a equazioni ellittiche variazionali del tipo di Matukuma e di Batt–Faltenbacher–Horst nella dinamica stellare. È notevole che anche se per l’equazione Du + f(u) = 0, e per varie equazioni ad essa collegate, la questione dell’unicità è stata ampiamente studiata, tranne che in pochissimi casi (cf. [PS]), ben poco è noto nel caso di equazioni dipendenti dalla variabile spaziale. In questo studio affrontiamo per la prima volta la questione dell’unicità di soluzioni per tali equazioni. Poiché il problema è già abbastanza difficile, ci è sembrato ragionevole considerare, in questo primo lavoro, solo soluzioni radiali ed equazioni per il p–Laplaciano, p > 1. Un’altra questione delicata affrontata in [PGMS] riguarda il fatto che la funzione f può essere indefinita in u = 0. Questo caso era stato precedentemente studiato solo in [PS], ma senza prestare troppa attenzione alle difficoltà insorgenti anche dalle possibili singolarità di f in u = 0. In [PGMS] formuliamo inoltre una precisa definizione di soluzione per tali problemi, che consente di trattare in modo formalmente elegante l’intera teoria. Verranno presentati anche teoremi di esistenza dati recentemente in [CFP], utilizzando condizioni di sottocriticità per f introdotte in [AP] e [MP] per equazioni ellittiche quasilineari senza pesi. Bibliografia [AP] B. Acciaio e P. Pucci, Existence of radial solutions for quasilinear elliptic equations with singular nonlinearities, Adv. Nonlinear Stud., 3 (2003), 511-539. [CFP] E. Calzolari, R. Filippucci e P. Pucci, Existence of radial ground states for p–Laplacian elliptic equations with weights, in preparazione. [MP] E. Montefusco e P. Pucci, Existence of radial ground states for quasilinear elliptic equations, Adv. Differential Equations 6 (2001), 959–986. [PGMS] P. Pucci, M. García-Huidobro, R. Manásevich e J. Serrin, Uniqueness and other properties of radial ground states of singular quasilinear elliptic equations with weights, in corso di stampa in Ann. Mat. Pura Appl., pagine 34. [PS] P. Pucci e J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Math. J. 47 (1998), 501–528.

Molti Autori hanno avuto occasione di osservare che, nella teoria cinetica dei gas, le velocità peculiari sono definite in modo non del tutto appropriato. Come conseguenza, teorie di secondo ordine (ottenute attraverso l'iterazione di Chapman-Enskog, ad esempio) contengono termini dipendenti dall'osservatore. Gli effetti sgraditi possono diventare meno riposti nello studio di gas granulari, corpi nei quali le velocità peculiari hanno valori molto minori e quindi una eventuale rotazione dell'osservatore ha conseguenze più vistose. La correzione più radicale consiste nel definire un riferimento locale medio, piuttosto che soltanto una velocità locale media, e nel trarne le conseguenze, sia cinetiche che dinamiche (vedi lavoro con George Mullenger in corso di stampa, ma già disponibile sul sito dei Rend. Sem. Mat. Univ. Padova). Si aprono prospettive molto ampie, alcune di esse controverse. Nel seminario, comunque, esse vengono presentate nel contesto per altri versi più semplice, quello conservativo.

Partial differential equations with highly oscillatory solutions occur in many areas of science like quantum mechanics and acoustics, with important spin-offs to semiconductors, nanotechnology and low-temperature physics. These equations pose a great challenge to mathematical and numerical analysis. Recently a new mathematical technique has been developed to treat highly oscillatory PDEs, namely Wigner transforms. They allow deep new insights into high-frequency asymptotics. The state-of-the-art is reported in the lecture, with emphasis on applications in Bose­Einstein condensation.

We consider Carleman type estimates for general second order partial differential operators with real-valued coefficients. We include boundary terms. We give a derivation of these estimates, examples for elliptic and hyperbolic equations. As applications we consider uniqueness and stability in the Cauchy problem (in particular increased stability for the Helmholtz equation) and identification of coefficients and source terms of hyperbolic equations from the lateral Cauchy data.

Nella teoria delle C*-algebre i limite diretti di algebre semi-semplici di dimensione finita hanno un ruolo particolare perché ogni tale algebra è determinata dal suo gruppo di Grothendieck. Esempi non-banali sono le algebre gruppali di gruppi localmente finiti. Certe misure di probabilità sul grafo di ramificazione di queste algebre corrispondono alle tracce dell'algebra. Usando la teoria dei caratteri dei gruppi finiti da un punto di vista asintotico si descrivono le tracce delle algebre gruppali di gruppi semplici che sono limiti diretti di gruppi alterni finiti.

Sub-Gaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. This is a joint work with Martin Barlow and Takashi Kumagai.

Nell'ambito dell'analisi funzionale, la teoria dei semigruppi di operatori fornisce uno strumento molto potente per lo studio delle equazioni di evoluzione. La prima parte del seminario dà una breve introduzione di questa teoria. In particolare vengono richiamati risultati sulla generazione e sul comportamento asintotico di semigruppi, partendo da alcuni spunti storici. Nella seconda parte viene dimostrato come questi risultati astratti possono essere applicati a problemi concreti come equazioni integro-differenziali, equazioni differenziali con ritardo oppure sistemi con condizioni dinamiche al bordo. In particolare vengono presentati risultati recenti sulla buona positura ed il comportamento asintotico delle soluzioni dei sistemi sopra citati.

This seminar deals with fluxes of quantities across parts of the boundary of a body R with fractal boundary. A broad interpretation of the notion of a fractal boundary is employed here to mean any bounded subset R of Rn whose boundary dR is so complicated that the outer normal n to R is not defined for a.e. point of dR. Here a.e. means almost everywhere with respect to the n-1 dimensional Hausdorff measure (area) Hn-1. For a fractal body, Hn-1(dR) is infinite, since otherwise R is a set of finite perimeter with an Hn-1 a.e. defined normal. The paper is concerned with determining the net flux F(q,T) of a scalar quantity (such as the heat flux) across a subset T of dR, where the quantity is represented by a continuous field of the flux vector q on Rn with integrable distributional divergence. The paper examines basic properties of the functional F: (1) On the negative side, it is shown that if Hn-1(dR) is infinite, then F(q, . ) does not extend to a measure unless q is in some sense trivial. (2) On the positive side, it is proved that each R can be approximated by a sequence Rj of sets of finite perimeter such that the classical Cauchy formula holds in some limiting sense. (3) Consequences are derived of the situation when a given T insulates under q in the sense that the flux through each trace S of T vanishes. (4) Conditions are given on dR for the locality of F (so that the value F(q,T) depends on the values of q on T). (5) Classes of flux vector fields are described to which F(q,T) can be extended.

Il gruppo di Heisenberg costituisce l'esempio più semplice di varietà dotata di metrica sub-riemanniana. Tale struttura è stata ampiamente studiata per il suo interesse in analisi armonica, analisi complessa, geometria conforme, teoria geometrica della misura. Per altri aspetti, diversi lavori di geometria differenziale trattano un'altra metrica sul gruppo di Heisenberg, questa volta riemanniana, che è strettamente collegata alla precedente. In un certo senso, le due metriche hanno lo stesso andamento asintotico. Questa affinità viene riscontrata anche nell'analisi degli operatori differenziali più significativi in ciascuna struttura: il sub-laplaciano nel primo caso, e l'operatore di Laplace-Beltrami nel secondo. Più complessa è la relazione tra le due strutture quando si passa ad analizzare operatori che agiscono su forme differenziali. Verranno indicati alcuni risultati recenti e problemi aperti di teoria spettrale in questo contesto.

"Viene illustrata la costruzione unificata di somme statisiche invarianti per 3-varietà munite di PSL(2,C)-fibrati principali piatti, basate sul diligaritmo classico di Rogers, o sui dilogaritmi matriciali di Faddeev-Kashaev. Si discutono tali somme statistiche come parte di una teoria di campo ""esatta"" per la gravità 3-dimensionale. [Per maggiori dettagli, si rimanda agli articoli in collaborazione con S. Baseilhac, math.GT/0306283, math.GT/0306280, math.GT/0211053]."

The well-known notion of the dual quadratic form can be extended to homogeneous polynomials of arbitrary even degree. Following the work of S. Mukai, I will discuss how this is used for an explicit description of the varieties parametrizing representations of a homogeneous polynomial as a sum of powers of linear forms.

Dispersive geometric evolution equations have received much attention in the past few decades. Some of these equations describe the dynamics of micromagnetics, vortex motion in liquid crystals, or vibrations of certain crystalline lattices. In this talk we will present some recent results on the equations describing ferromagnetic and anti-ferromagnetic materials concerning existence of solutions and the relation between anti-ferromagnetic equations (Schrodinger maps) and wave maps.

Control theory makes significant progress, since nonlinear systems and differential geometry have been brought together about 30 years ago. The main advantage is that a description of the control system is avalaible, which does not require the choice of special coordinates. Now, many important properties of dynamic systems can be characterized in a geometric way. Within this framework nonlinear systems are considered as geometric objects, defined on smooth manifolds. E.g., a nonlinear control system can be identified with a submanifold of a certain geometric structure. Non observable or non accesible systems generate a foliation, and so on. This talk gives an introduction to the geometric description of nonlinear control systems, described by a set of ordinary differential equations. It starts with a short overview, where time invariant and time variant systems with or without control input are characterized by its geometric properties. At the same time a short introduction to the required mathematical tools will be presented. Having this preliminaries at our disposal we develop the concept of accessibility and observability by geometric ideas. It will turn our that this approach is not confined to explicit systems, on the contrary it can be generalized to more complex ones. Fortunately these methods are not only useful for the analysis of systems, where equivalence means that solutions of one system can be transformed to solutions of the other and vice versa. Using the idea that a dynamic system is also a certain sybmanifold, we show that equivalent systems describe the same submanifold in different coordinates only. From a systems designer's point of view, it is important to construct equivalent systems such that the control loop design can be reduced to an already solved problem Finally, this talk finishes with an industrial application of the present methods. The hydraulic gap control of stands in steel rolling mills is a challenging task because of the intrinsic nonlinear behaviour of these systems. We present new design ideas and controllers and prove the performance by simulations and measurements taken in mills located in the US and Europe.

"By using the concept of nonlinear capacity we suggest the general approach to the blow-up phenomena in nonlinear problems. This approach does not use the maximum principle arguments and properties of the fundamental solutions of related differential operators. That fact gives the possibility to consider a wide classes of nonlinear problems and obtain the sharp results for blow-up problems including the estimates for the ""blow-up time"". Finally we construct the ""Mendeleev table"" for nolinear problems. I am going to demonstrate this approach for nonlinear (multidimensional) hyperbolic problems. The results are obtained jointly with E.Mitidieri, L.Veron and A.Tesei."

A priori bounds for positive solutions of a class of nonvariational elliptic systems is obtained by the blow-up method. Some Liouville type results are discussed.

I will consider the FitzHugh-Nagumo PDE. It is well-known in neuroscience and is used to describe the propagation of voltage impulse through a nurve axion. Its discrete version provides a competing model that I discuss in the talk. I present some results on the dynamics of the evolution operator on the space of traveling wave solutions and in particular, show that this dynamics changes from Morse-Smale type to a chaotic attractor to a horseshoe as a leading parameter (corresponding to the Reynolds number) of the system varies.

Guided waves in nonlinear optical fibres are modelled as special solutions of Maxwell's equations in an axi-symmetric dielectric medium whose refractive index depends on the intensity of the electric field. I shall present some situations in which the problem can be reduced to a nonlinear eigenvalue problem and then I shall summarise some of the rigorous results that have been obtained and how they relate to the underlying physical problem.

Suppose that an experiment with an unknown, possibly infinite, number of outcomes is performed and that these outcomes occur according to some random mechanism. Suppose that n independent trials are carried out and that N distinct outcomes have been observed. We attempt to answer the following questions: What is the probability that, on the next trial, an outcome not observed before occurs? (This is called the problem of unobserved probability.) What is the total number of outcomes not observed? This second problem has a long history going back to Turing and is, apart from its mathematical interest, important in many areas such as biology (species sampling), numismatics, and literary scholarship. We'll give a brief survey of past work and also discuss recent joint work with Alberto Gandolfi in which a Bayes-like estimator for the number of unobserved outcomes is derived. This has the advantage over the existing estimators -- due to Chao and Lee and others -- in that, modulo the fact that Turing's ansatz is used (it is used by everyone else as well), it is derived from first principles, without any ad hoc assumptions, and includes previous estimators as special cases. We'll also briefly discuss an almost complete classification of infinite discrete probability measures, which emerges as a by-product of a solution we have obtained for the problem of unobserved probability.

Direct detection of gravitational waves from cosmic sources is one of the great challenges of contemporary experimental physics. The principles of operation and the status of the present detectors and of the main future projects will be reported. The recent results of a search for gravitational wave bursts, using the data collected by the ROG Collaboration cryogenic bar detectors EXPLORER (at CERN) and NAUTILUS (at LNF), will be discussed.

Saranno discussi vari aspetti della teoria dei vortici di Chern-Simons relativamente a diversi modelli con struttura autoduale. A tale scopo si vedrà come sia possibile formulare le equazioni autoduali corrispondenti in termini di problemi ellittici nonlineari di tipo Liouville, Toda etc. Inoltre saranno evidenziate interessanti questioni analitiche relative allo studio dei suddetti problemi ellittici.

The lecture will attempt to present both the historical roots, as well as some of the principal developments and applications to other fields of Mathematics of this subject within the past three decades, underlying the role of Linear Algebra. It will concentrate on explicit presentations of representations of a given algebra, employing some related concepts such as classification problems (Brauer-Thrall conjectures, finite, tame and wild types), hereditary algebras and graphs, Coxeter functors and their linear transformations, as well as more recent results on quasi-hereditary algebras in relation to semi-simple complex Lie algebras. These concepts will be illustrated on solving some classification problems of geometry (von Staud's pairs and Kronecker's pairs of matrices), modular representations of groups (A4 over GF2) and C*-algebras (Jones index)..

Si illustrerà il problema di caratterizzare i limiti deboli di successioni di applicazioni a valori sul cerchio con energie equilimitate. In particolare, si discuteranno i casi dell'integrale di Dirichlet, del funzionale area e dell'energia cinetica relativistica.