L6

Training deep learning models involves searching for a good model over the space of possible architectures and their parameters. Discovering models that exhibit robust generalization to unseen data and tasks is of paramount for accurate and reliable machine learning. Generalization, a hallmark of model efficacy, is conventionally gauged by a model's performance on data beyond its training set. Yet, the reliance on vast training datasets raises a pivotal question: how can deep learning models transcend the notorious hurdle of 'memorization' to generalize effectively? Is it feasible to assess and guarantee the generalization prowess of deep neural networks in advance of empirical testing, and notably, without any recourse to test data? This inquiry is not merely theoretical; it underpins the practical utility of deep learning across myriad applications. In this talk, I will show that scrutinizing the training dynamics of neural networks through the lens of topology, specifically using 'persistent-homology dimension', leads to novel bounds on the generalization gap and can help demystifying the inner workings of neural networks. Our work bridges deep learning with the abstract realms of topology and learning theory, while relating to information theory through compression.

Aggregation-diffusion equations and systems have garnered much attention in the last few decades. More recently, models featuring nonlocal interactions through spatial convolution have been applied to several areas, including the physical, chemical, and biological sciences. Typically, one can establish the well-posedness of such models via regularity assumptions on the kernels themselves; however, more effort is required for many scenarios of interest as the nonlocal kernel is often discontinuous.

In this talk, I will present recent progress in establishing a robust well-posedness theory for a class of nonlocal aggregation-diffusion models with minimal regularity requirements on the interaction kernel in any spatial dimension on either the whole space or the torus. Starting with the scalar equation, we first establish the existence of a global weak solution in a small mass regime for merely bounded kernels. Under some additional hypotheses, we show the existence of a global weak solution for any initial mass. In typical cases of interest, these solutions are unique and classical. I will then discuss the generalisation to the $n$-species system for the regimes of small mass and arbitrary mass. We will conclude with some consequences of these theorems for several models typically found in ecological applications.

This is joint work with Dr. Jakub Skrzeczkowski and Prof. Jose Carrillo.

We model a tumor as an incompressible flow considering two antagonistic effects: repulsion of cells when the tumor grows (they push each other when they divide) and cell-cell adhesion which creates surface tension. To take into account these two effects, we use a 4th-order parabolic equation: the Cahn-Hilliard equation. The combination of these two effects creates a discontinuity at the boundary of the tumor that we call the pressure jump.  To compute this pressure jump, we include an external force and consider stationary radial solutions of the Cahn-Hilliard equation. We also characterize completely the stationary solutions in the incompressible case, prove the incompressible limit and prove convergence of the parabolic problems to stationary states.

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this talk, we will present a new approach by applying a natural scaling on the Boltzmann equation. The proof is based on an improved well-posedness theory for the Boltzmann equation without angular cutoff in the regime with an optimal range of parameters so that the grazing limit can be justified directly that includes the Coulomb potential. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two parts. The first one converges to the Landau operator when the parameter of deviation angle tends to its singular value and the second one vanishes in the limit. Hence, the scaling and limiting process exactly capture the grazing collisions. The talk is based on a recent joint work with Yu-Long Zhou.

I present an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor
on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study
curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective
proved extremely fruitful in the Riemannian case (Alexandrov- and CAT(k)-spaces). After introducing the basics of our
approach, we report on recent progress in developing a Sobolev calculus for time functions on such non-smooth
Lorentzian spaces. This seminar talk can also be viewed as a primer and advertisement for my mini course in
May: Current topics in Lorentzian geometric analysis: Non-regular spacetimes

In the 1960s Shanks conjectured that the  ζ'(ρ), where ρ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.

We will present a matching upper and lower bound for the right tail probability of the maximum of a random model of the Riemann zeta function over short intervals.  In particular, we show that the right tail interpolates between that of log-correlated and IID random variables as the interval varies in length. We will also discuss a new normalization for the moments over short intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over short intervals.

I will discuss certain dynamics of interacting particles in interlacing arrays with inhomogeneous, in space and time, jump probabilities and their relations to conditioned random walks and random tilings of the Aztec diamond.

email Hannah Hughes for further information.