Bruno Duchesne (Université Paris-Saclay)
CANCELLED: Maximal representations and hermitian symmetric spaces of infinite dimension abstract
Abstract:
In this talk, a family of infinite dimensional Hermitian symmetric spaces will be introduced and a rigidity theorem for complex hyperbolic lattices acting by isometries on such symmetric spaces will be presented. The context of this result will be given in order to explain why one can be interested in such result.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Christian Urech (University of Basel)
Finitely generated subgroups of algebraic elements of plane Cremona groups abstract
Abstract:
Let S be a variety. A birational transformation of S is called algebraic if it is contained in an algebraic subgroup of the group of birational transformations Bir(S) of S. In this talk, I will explain, why a finitely generated subgroup of Bir(S) is itself contained in an algebraic subgroup of Bir(S). This answers a question of Charles Favre. We will apply this result to describe the degree growth of finitely generated subgroups of plane Cremona groups. This is joint work with Anne Lonjou and Piotr Przytycki.
10:30 • Universität Basel, Spiegelgasse 5, Seminarraum 05.001
Dr. Swann Tubach (ENS Lyon)
Higher enhancements of Nori motives and realisation functors abstract
Abstract:
Let k be a field of characteristic zero and X a quasi-projective k-variety. The category of perverse Nori motives over X is an abelian category modelled on perverse sheaves but instead of having coefficients in \\Q-vector spaces, they have stalks in the Tannakian category of motives constructed by Nori. They were constructed by Ivorra and S. Morel. Their work, together with the work of Terenzi provides the 6 operations for the derived category of perverse Nori motives. By adapting an argument due to Nori in the setting of constructible sheaves on the complex points of X, we show that the derived category of perverse Nori motives is the derived category of its constructible heart. This enables us to see each of the 6 operations as a right derived functor, which have natural higher categorical lifts. Thanks to the work of Drew, Gallauer and Robalo, the existence of those lifts gives us a comparison functor from the category of Voevodsky motives over X, compatible with the operations. Our arguments also work for mixed Hodge modules, providing a Hodge realisation of étale Voevodsky motives. If times permits, we will explain how to use higher categorical tools to expand perverse Nori motives, together with the 6 operations, from quasi-projective varieties to all finite type k-schemes, and even to all qcqs schemes of characteristic zero (for those, we do not extend all the operations).
13:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
Léopold Trémant (Université de Strasbourg)
High-order averaging and machine learning abstract
Abstract:
Highly-oscillatory phenomena combine the well-known numerical challenges of stiff equations and geometric problems, such as order reduction and energy preservation. These problems involve both fast oscillatory dynamics and slow drift dynamics, which interact to create complex dynamics. High-order averaging allows to asymptotically decouple these dynamics using formal calculations, which generates modified problems that can be solved with better numerical accuracy.In this talk, I will introduce high-order averaging using a (somewhat recent) closed form approach, and exhibit the improvement in numerical accuracy it allows. I will then present the geometric properties of the method, notably regarding the preservation of a Hamiltonian structure. The talk will end with some preliminary results regarding geometric neural networks, which can replace the offline symbolic computations of averaging with an offline training procedure.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Prof. Dr. Ivan Dokmanić (University of Basel)
Statistical Mechanics of Graph Convolution Networks abstract
Abstract:
Graph neural networks (GNNs) excel in modeling relational data such as biological, social, and transportation networks, but the underpinnings of their success are elusive. Traditional complexity measures from statistical learning theory fail to account for observed phenomena like the double descent or the impact of relational semantics on generalization error. Motivated by experimental observations of ``transductive\'\' double descent in key networks and datasets, we use analytical tools from statistical physics and random matrix theory to precisely characterize generalization in simple graph convolution networks on the contextual stochastic block model. Our results illuminate the nuances of learning on homophilic versus heterophilic data and predict double descent whose existence in GNNs has been questioned by recent work. We show how risk is shaped by the interplay between the graph noise, feature noise, and the number of training labels. Our findings apply beyond stylized models, capturing qualitative trends in real-world GNNs and datasets. As a case in point, we use our analytic insights to improve performance of state-of-the-art graph convolution networks on heterophilic datasets.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Maxime Fairon (Orsay)
Compatible Poisson structures on multiplicative quiver varieties abstract
Abstract:
Any multiplicative quiver variety is endowed with a Poisson structure constructed by M. Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus of this variety carries a corresponding symplectic form defined by D. Yamakama through quasi-Hamiltonian reduction. In this talk, I want to explain how to include this Poisson structure as part of a larger pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of (non-degenerate) Hamiltonian quasi-Poisson structures, whose construction can be adapted to various frameworks, e.g. in relation to character varieties. I will start by explaining the simpler analogous situation that leads to a pencil of Poisson structures on (additive) quiver varieties. Time allowing, I will comment on how this result can be applied to the spin Ruijsenaars-Schneider phase space; this shows the compatibility of two Poisson structures that appeared in independent works of Arutyunov-Olivucci (arXiv:1906.02619) and of Chalykh and myself (arXiv:1811.08727).
15:15 • Université de Genève, Section de mathématiques, 7-9 rue du Conseil-Général, Room 1-07
Chi Cheuks Tsang (CRM Montreal)
Pseudo-Anosov maps, dilatations, and veering trian- gulations abstract
Abstract:
A pseudo-Anosov map is a surface homeomorphism that acts with similardynamics as a hyperbolic element of SL2R on R2. A classical result ofNielsen and Thurston shows that these are surprisingly prevalent amongmapping classes of surfaces. The dilatation of a pseudo-Anosov mapis a measure of the complexity of its dynamics. It is another classicalresult that the set of dilatations among all pseudo-Anosov maps definedon a fixed surface has a minimum element. This minimum dilatationcan be thought of as the smallest amount of mixing one can performon the surface while still doing something topologically interesting. Theminimum dilatation problem asks for this minimum value.In this talk, we will present some recent progress on a version of theminimum dilatation problem concerning fully-punctured pseudo-Anosovmaps. In the first part of the talk, we will provide some backgroundof pseudo-Anosov maps and the minimum dilatation problem. In thesecond part of the talk, we will go into some technical details about oneingredient of the proof, namely, how one can obtain a bound on thenumber of tetrahedra in the layered veering triangulation associated toa fully-punctured pseudo-Anosov map with small normalized dilatation.
15:15 • Université de Fribourg, room Phys 2.52
Hjalti Isleifsson
What is... higher rank hyperbolicity? abstract
Abstract:
We will recall the now classical notion of Gromov hyperbolicity and then discuss results due to Wenger and Kleiner-Lang on how hyperbolic phenomena arise in dimensions greater than or equal to the rank of spaces which satisfy non-positive curvature conditions.
16:15 • UZH Zentrum, Building KO2, Room F 150
Erwan Lanneau (Institut Fourier, Université Grenoble Alpes)
Dynamics of billiard flows in polygons abstract
Abstract:
In this survey talk, I will give an overview of the study of rational polygonal billiards and some of the highlights of Teichmüller dynamics. This will be the occasion to talk on infinite billiards such as the Ehrenfest model, also known as the wind-tree model.
17:15 • Université de Fribourg, room Phys 2.52