Veranstaltungen

Let \(A\) be the group of positive diagonal \(d\times d\) matrices on \(SL_d(\mathbb{R})\) and \(U\cong \mathbb{R}^{d-1}\) be an abelian expanding horospherical group in \(SL_d(\mathbb{R})\), where \(d\ge 2\). Denote by \(A^{+}\) the expanding cone in \(A\) associated to \(U\). We say that \(x\in SL_d(\mathbb{R})/SL_d(\mathbb{Z})\) is \(A^{+}\)-divergent on average if for any compact set \(K\) the orbit \(A^{+}x\) escapes \(K\) on average. One may ask how large the set of points which are \(A^{+}\)-divergent on average is. In this talk, I will discuss upper and lower bounds for the Hausdorff dimension of the set of points which are \(A^{+}\)-divergent on average.

A major tool in the study of the Dehn twist along a Lagrangian sphere is Seidel'slong exact sequence. This sequence comes with a distinguished element A in the Floer homology group of the Dehn twist. In this talk we will discuss a property of A in case the Dehn twist is a monodromy in a real Lefschetz fibration. We will see that the real structure induces an automorphism on the Floer homology group of the Dehn twist and that A is a fixed point.

This talk is about algebraic K-theory groups (defined by Quillen in the early 1970s). We will review the definition and motivation behind those groups and explain some applications. Then we try to summarise what is known in terms of computations and explain some recent breakthroughs (based on so-called trace methods). One of the central tools used to achieve this progress is`higher categorical algebra' in the sense of Waldhausen, Lurie and others. As an sample application we cover the recent results on the K-theory of Z/p^n obtained in joint work with Anteau and Krause.

More information: https://math.ethz.ch/news-and-events/events/research-seminars/zurich-colloquium-in-mathematics.html?s=hs22call_made

One common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. In his study of reflection subgroups of Coxeter groups, Dyer introduces a family of groups, Dyer groups, which also have the same solution to the word problem as Coxeter groups. I will introduce this family of groups and give some of their properties. Then I explain how to construct actions of Dyer groups on CAT(0) spaces that extend those of Coxeter groups on Davis–Moussong complexes and those of right-angled Artin groups on Salvetti complexes.

Disordered quantum systems exhibit a variety of spectral phases, characterized by the extent of spatial localization of the eigenvectors. Through their adjacency matrices, random graphs provide a natural class of models for such systems, where the disorder arises from the random geometry of the graph. The simplest random graph is the Erdös-Rényi graph G(N,p), whose adjacency matrix is the archetypal sparse random matrix. The parameter d=pN represents the expected degree of a vertex. A dramatic change in behaviour is known to occur at the scale d \sim \log N, which is the threshold where the degrees of the vertices cease to concentrate. Below this scale the graph becomes inhomogeneous and develops structures such as hubs and leaves which accompany the appearance of a localized phase. I report on recent progress in establishing the phase diagram for G(N,p) at and below the critical scale d \sim \log N. We show that the spectrum splits into a fully delocalized region in the middle of the spectrum and a semilocalized phase near the spectral edges. The transition between the phases is sharp in the sense of a discontinuity in the localization exponent of eigenvectors. Furthermore, we show that the semilocalized phase consists of a fully localized region and in addition, for some values of d, a complementary region that we conjecture to be nonergodic delocalized. Joint work with Johannes Alt and Raphael Ducatez.

The standard mapping class groups are fundamental groups of moduli spaces/stacks of pointed Riemann surfaces: they thus encode much information about the topology of the deformations of such surfaces. Recently this story has been extended to wild Riemann surfaces, which generalise pointed Riemann surface by adding local moduli at each marked point -- the irregular classes. The new parameters control the polar parts of meromorphic connections with wild/irregular singularities, defined on principal bundles, and importantly provide an intrinsic viewpoint on the `times' of isomonodromic deformations. <br> In this talk we will explain how to compute the fundamental groups of (universal) spaces of deformations of irregular classes, related to cabled versions of braid groups, which thus play the role of `wild' mapping class groups. This is joint work with P. Boalch, J. Douçot and M. Tamiozzo. If time allows we will sketch a relation with bundles of irregular conformal blocks in the Wess-Zumino-Witten model, in joint work with G. Felder (past) and G. Baverez (in progress).

Abstract: We will capture the asymptotic behavior of the solutions to the Vlasov-Maxwell system arising from sufficiently small and regular data. Our analysis is based on vector fields methods, allowing us to exploit the null structure of the equations, and the Glassey-Strauss decomposition of the electromagnetic field. In particular, we will see that the electromagnetic field approaches, for large time, a solution to the vacuum Maxwell equations. Due to the long-range effects of the Lorentz force, the Vlasov field converges along logarithmic corrections of the linear characteristics.

Topological quantum field theories are a rich field of study, at the crossroads of many areas of mathematics and physics. Informally, a TQFT is a map that translates a geometric structure into an algebraic setting. More precisely, it is a symmetric monoidal functor from a category of cobordisms to a category of vector spaces. In this talk, we will introduce 1- and 2-dimensional TQFTs and see how they can be used to define some important invariants of knots, links and 3-manifolds.

Deep neural networks are usually trained by minimizing a non-convex loss functional via (stochastic) gradient descent methods. Unfortunately, the convergence properties are not very well-understood. Moreover, a puzzling empirical observation is that learning neural networks with a number of parameters exceeding the number of training examples often leads to zero loss, i.e., the network exactly interpolates the data. Nevertheless, it generalizes very well to unseen data, which is in stark contrast to intuition from classical statistics which would predict a scenario of overfitting. A current working hypothesis is that the chosen optimization algorithm has a significant influence on the selection of the learned network. In fact, in this overparameterized context there are many global minimizers so that the optimization method induces an implicit bias on the computed solution. It seems that gradient descent methods and their stochastic variants favor networks of low complexity (in a suitable sense to be understood), and, hence, appear to be very well suited for large classes of real data. Initial attempts in understanding the implicit bias phenomen considers the simplified setting of linear networks, i.e., (deep) factorizations of matrices. This has revealed a surprising relation to the field of low rank matrix recovery (a variant of compressive sensing) in the sense that gradient descent favors low rank matrices in certain situations. Moreover, restricting further to diagonal matrices, or equivalently factorizing the entries of a vector to be recovered, leads to connections to compressive sensing and l1-minimization. After giving a general introduction to these topics, the talk will concentrate on results by the speaker on the convergence of gradient flows and gradient descent for learning linear neural networks and on the implicit bias towards low rank and sparse solutions.

Renormalization is a key technique in dynamics, which provides the mathematical tools to explain many (approximate) self-​similarity phenomena. In this colloquium we will discuss two problems: the study of diffusion phenomena in the Ehrenfest model in mathematical physics and the description of symbolic sequences of very low complexity, which often appear also in the theoretical computer science literature. In both, we will explain some results which were proved using renormalization on geometric surfaces and offer an insight on how they are obtained.

More information: https://eth-its.ethz.ch/activities/its-science-colloquium.html#call_made

We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning problems. The goal is to challange the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The proposal is to use a joint embedding for all categorical features, instead of separate embeddings for each categorical features, and initalize the parameters of the neural network with parameters trained with a denoising autoencoder in a unsupervised learning problem, instead of random initialization. In other words, we propose a special initialization strategy for taining neural networks. We illustrate our ideas with experiments on a data set with insurance claim numbers. We demonstrate that we can achieve a predictive power on an independent test set higher than in the current approach. Moreover, we also show with experiments that our special initialization strategy performs better than other initialization strategies (with PCA, MCA, GLM) which can be alternatively proposed in our setting. Finally, we investigate the predictive power vs the bias of the predictions.