The liquid drop model

Rupert Frank

LMU Munich, Mathematics Institute

The liquid drop model was originally introduced in the nuclear physics literature in 1930 and has recently been studied extensively using techniques from the calculus of variations, geometric analysis, PDE and mathematical physics. The talk will discuss some new results and open problems concerning a certain isoperimetric-type question in this model. In addition, we present a first rigorous analysis of a nuclear pasta phase encountered in astrophysics.

Understanding Machine Learning through Mathematics: A Variational Approach

Charalambos Makridakis

IACM and University of Sussex

In recent years, substantial progress has been made in understanding the approximation properties of neural networks, with foundational results showing that a range of architectures can efficiently approximate both smooth and singular functions. However, these advances do not by themselves explain the behaviour of machine learning algorithms, especially in applications such as PDE solvers, uncertainty quantification, and generative models.

This talk presents a mathematical framework for analysing ML algorithms through their underlying variational principles and loss functionals. By studying function learning, PDE solvers, and probability-measure approximation from this perspective, we relate classical notions of numerical stability to the properties of the associated variational problems. This approach yields new insights into stability and convergence and motivates the development of new algorithms.

Unstable motions in Celestial Mechanics

Marcel Guardia

Universitat de Barcelona

One of the oldest problems in dynamical systems is the stability of the Solar System. That is, consider N bodies moving following Newton's law of gravitation, one of them with large mass (the Sun) and the others with small masses (the planets). If one neglects, the gravitational interaction between planets, the classical Kepler's laws assert that the planets move on ellipses. Then, one wants to understand whether the effect of the planet's mutual attraction causes long term changes on the shape and relative position of the Keplerian ellipses. Nowadays, it is known that the answer to the stability of the Solar system is rather nuanced and that stability and instability coexist for nearby initial conditions. In this talk I will explain how to construct unstable motions in this model, which lead to drastic changes in the semimajor axes, eccentricity and inclination of these ellipses.

Well posedness of ODE's for nonsmooth velocities and in non Euclidean ambient spaces: a survey

Luigi Ambrosio

Scuola Normale Superiore, Pisa

In this lecture I will make a survey of the theory of regular lagrangian flows, that allows to prove existence, uniqueness and stability for ODE's associated to vector fields with little regularity, beyond the classical Cauchy-Lipschitz theory. Initiated in a work by Di Perna and Lions who dealt with Sobolev vector fields, the RLF axiomatization provides natural links with Probability. More recent developments, including the regularity of the flow map, counterexamples and the case when even the ambient space is not smooth, will be discussed.