Events

In this series of talks, I will provide an overview of Viterbo's volume-capacity conjecture, a symplectic isoperimetric-type inequality concerning symplectic capacities of convex domains in the classical phase space. Specifically, in the first talk, we will explore Viterbo's conjecture from the perspective of asymptotic geometric analysis. The second talk will focus on Minkowski billiard dynamics, and the characteristic foliation on convex polytopes. Finally, in the third talk, we will present a counterexample to Viterbo's conjecture and discuss potential future research directions. These talks are based on joint work with S. Artstein-Avidan, P. Haim-Kislev, E. Gluskin, R. Karasev, and V. Milman. Zoom: https://ethz.zoom.us/j/64239710101

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In this series of talks, I will provide an overview of Viterbo's volume-capacity conjecture, a symplectic isoperimetric-type inequality concerning symplectic capacities of convex domains in the classical phase space. Specifically, in the first talk, we will explore Viterbo's conjecture from the perspective of asymptotic geometric analysis. The second talk will focus on Minkowski billiard dynamics, and the characteristic foliation on convex polytopes. Finally, in the third talk, we will present a counterexample to Viterbo's conjecture and discuss potential future research directions. These talks are based on joint work with S. Artstein-Avidan, P. Haim-Kislev, E. Gluskin, R. Karasev, and V. Milman. Zoom: https://ethz.zoom.us/j/65744003648

We introduce the concept of samplets, extending the Tausch-White multi-wavelet construction to scattered data. This results in a multiresolution analysis of discrete signed measures with vanishing moments, enabling efficient data compression, feature detection, and adaptivity. The cost for constructing the samplet basis and applying the fast samplet transform is linear in the number of data sites N. We apply samplets to compress kernel matrices for scattered data approximation, achieving sparse matrices with only O(N log N) non-zero entries in the case of quasi-uniform data. The approximation error is controlled by the number of vanishing moments. We demonstrate two applications: a multiscale interpolation scheme for improved conditioning of kernel matrices and a dictionary learning approach with sparsity constraints.

In this talk, I will present a geometric (to say that it also works in the case of ambient Riemannian manifolds) notion of codimension-two fractional mass that Gamma-converges to the (n-2)-dimensional Hausdorff measure. I will also discuss possible extensions to higher codimension and applications to the construction of minimal surfaces in codimension two. The talk is based on a joint work with Mattia Freguglia and Nicola Picenni.

In this series of talks, I will provide an overview of Viterbo's volume-capacity conjecture, a symplectic isoperimetric-type inequality concerning symplectic capacities of convex domains in the classical phase space. Specifically, in the first talk, we will explore Viterbo's conjecture from the perspective of asymptotic geometric analysis. The second talk will focus on Minkowski billiard dynamics, and the characteristic foliation on convex polytopes. Finally, in the third talk, we will present a counterexample to Viterbo's conjecture and discuss potential future research directions. These talks are based on joint work with S. Artstein-Avidan, P. Haim-Kislev, E. Gluskin, R. Karasev, and V. Milman. Zoom: https://ethz.zoom.us/j/68488661786

Introduction to log abelian varieties

One of the long-standing conjectures in quantum chaos is that the spectral statistics of quantum systems with chaotic classical limit are governed by random matrix theory. Despite convincing heuristics, there is currently not a single example where this phenomenon can be established rigorously. Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this lecture we will review Rudnick's approach and extend it to explain the emergence of the Gaussian Unitary Ensemble for twisted Laplacians (which break time-reversal symmetry) and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high genus random covers of a fixed compact surface. This lecture is based on joint work with Laura Monk (Bristol).

The Hilbert metric is a distance defined on properly convex subsets of real projective space. It generalizes the hyperbolic metric in Klein's model of hyperbolic space, where the convex set is the unit ball. The study of degenerations in Hilbert geometry leads to replacing the reals by a non-archimedean ordered valued field. The aim of this talk is to introduce convex sets over such fields, describe their Hilbert metrics through several examples and view how they arise as limits of classical Hilbert geometries. This is joint work with Anne Parreau.

In his celebrated theory of turbulence, created in 1941 and known as the K41 theory, A.N.Kolmogorov considered the velocity u(t,x) of a turbulent flow of fluid. Assuming that the Reynolds number of the flow is large, he studied the increments u(t,x+r)-u(t,x) of u and heuristically examined their statistical properties, summarised in a number of law of the K41 theory. I will talk about fictitious 1d fluid, described by the stochastic Burgers equation, consider increments of its velocity field and will rigorously derive for them analogies of the corresponding laws from Kolmogorov's theory, strikingly close to these laws.

For a probability-preserving ergodic dynamical system (X, T, u) and an integrable function f, the asymptotic almost sure behaviour of the ergodic sums S_N (f ) is described by the Birkhoff Ergodic Theorem. The situation is much more complicated if f is not integrable, a result by Aaronson forbids almost sure Limit Theorems. Instead, the notion of trimming is introduced, by excluding the largest observations from S_N(f) . Trimmed limit Theorems are well-studied for iids. In the dynamical setting, results are only known for systems exhibiting strong mixing behaviour. We study trimming for irrational rotations in and functions with polynomial singularities.

We give an introduction to Topological Data Analysis (TDA), focusing on Persistent Homology. This is a technique where one filters a topological space with respect to the sublevel sets of a given function and then considers the changes in homology of these subsets. We discuss connections to Morse theory, where the topology of a manifold is related to the critical points of a function on the manifold. We introduce Morse-Smale vector fields, which are a class of vector fields with good structural properties. Finally, we discuss some new approaches in TDA, like parametrized chain complexes, and show how one might apply them to the study of vector fields.

The exposure of an individual i often affects the outcome of another individual j. A prominent example occurs in infectious disease settings, where vaccinating one individual can reduce disease transmission and thereby affect the health outcomes of others. This spillover effect is a type of interference, which implies that individuals cannot plausibly be perceived as independent and identically distributed (iid). Extensive methodological research has recently been motivated by interference problems and the violation of conventional iid assumptions. However, despite the growing interest in the topic, there remains controversy over whether and when existing methods capture effects of practical interest. In this talk, I will present causal methodology—motivated by infectious disease settings—for addressing interference. The central idea is to consider estimands that are insensitive to the interference structure. I will argue that these estimands have a clear interpretation and can be used to guide decisions made by, for example, doctors and patients facing infectious diseases. The methodology will be illustrated with examples of the effects of vaccines against HIV and influenza.

In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. For mu,nu probability measures on R^d, increasing in convex order, stretched Brownian motion provides an analogue for the martingale version of this problem. We provide a characterization in terms of gradients of convex func- tions, similar to the characterization of optimizers in classical optimal transport. We use stretched Brownian motion to extend Kellerer’s theorem on the existence of Markov martingales for given marginal distributions to the d-dimensional case. While this celebrated theorem is known for 50 years in the one-dimensional case, this is the first version pertaining to the d-dimensional case. We also provide a gradient flow to find the optimizer of stretched Brownian motion in the irreducible case. Joint work with M. Beiglboeck, J. Backhoff, B. Robinson, and B. Tschiderer.

In the 17th century, Pierre de Fermat stated – and claimed to have proved – an elegant mathematical theorem, stating that a certain equation has no solutions in the whole numbers. Generations of mathematicians tried to find a proof of this theorem, but the problem resisted attack for more than 350 years, until it was solved in 1995 by Andrew Wiles and Richard Taylor. Professor David Loeffler will explain the problem, and some of the beautiful and intricate ideas that played a role in its solution; and he will explain some more recent mathematical developments arising from the same circle of ideas which are still the focus of intense research today.

More information: https://fernuni.ch/mathematik-und-informatik/event/lecon-inaugurale-david-loefflercall_made

A few months ago, Etienne fouvry, Emmanuel Kowalski and myself initiated the study of "mixed" moments of $L$-functions of Dirichlet characters of prime modulus $q$. The mixed second moments are of the shape $$\sum_{\chi\mod q}L(\chi^a,1/2)L(\chi^b,1/2)$$ with $a$ and $b$ non-zero integers not necessarily equal (hence the term "mixed"; we could establish an asymptotic formula for these moments when $q$ is large, with a power saving error term in $q$. In this talk, I will explain what we (EF, EK, myself together with Will Sawin) can currently prove in the much more complicated case of cubic mixed moments: moments of the shape $$\sum_{\chi\mod q}L(\chi^a,1/2)L(\chi^b,1/2)L(\chi^c,1/2).$$ Our methods belong to the area of "applied $\ell$-adic cohomology" and combine a variety of techniques from analytic number theory with the theory of algebraic exponential sums, in particular the properties of Kloosterman and hypergeometric sheaves studied by Katz.

PDEs are considered to be language of physics as they provide mathematical descriptions of a whole range of physical phenomena. The complexity and prohibitive computational cost of traditional physics-based numerical schemes necessitates the search for fast and efficient surrogates, based on machine learning. In this lecture, we survey recent developments in the field of learning solution operators for PDEs by focussing on structure preserving neural operators and on foundation models for sample efficient and generalizable multi-operator learning. We also briefly discuss graph neural network based learning of PDEs on arbitrary domain geometries and conditional Diffusion models for learning multi-scale physical systems such as Turbulent Fluid Flows.