Stéphane Gaubert (INRIA and CMAP, Ecole Polytechnique)
The competitive spectral radius of families of nonexpansive mappings abstract
Abstract:
We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of invertible matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Mañe’s lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the author, concerning the asymptotic behavior of nonexpansive mappings (generalizations of the Wolff-Denjoy theorem). We also show that the value of the game admits a dual characterization when the nonexpansive maps are quasi-isometries, or when they are order preserving and positively homogeneous self-maps of a cone equipped with Funk’s and Thompson’s metrics.This is based on joint work with Marianne Akian and Loic Marchesini,arXiv:2410.21097
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Antoine Usseglio Carleve (University of Avignon)
Accurate inference about extreme expectiles and Expected Shortfall abstract
Abstract:
The expectile and Expected Shortfall are prime candidates for being standard risk measures in actuarial and financial contexts, for their ability to recover information about probabilities and typical behavior of extreme values, as well as their axiomatic properties. A series of recent papers has focused on their estimation at extreme levels and has obtained the asymptotic normality of the proposed estimators. The obtention of accurate confidence intervals for extreme expectiles and Expected Shortfall is paramount in any decision process in which they are involved, but actual inference on these tail risk measures is still a difficult question due to their sensitivity to tail heaviness. This talk focuses on asymptotic Gaussian inference about tail expectiles in the challenging context of heavy-tailed observations, and tail Expected Shortfall in the general max-domain of attraction. We use an in-depth analysis of the proofs of asymptotic normality results to derive bias-reduced and variance-corrected Gaussian confidence intervals. We illustrate the usefulness of our construction on several sets of financial and insurance claims data.
14:00 • EPF Lausanne, Extranef 125
Andrey Pilipenko (UNIGE)
Diffusions in a media with semipermeable membranes abstract
Abstract:
Diffusions in a medium with semipermeable membranes naturally arise in models of material science, chemistry, biology, telecommunications, etc. We will discuss them from the following points of view: (a) macroscopic, (b) stochastic, (c) PDE.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Matthias Christandl (Copenhagen)
Asymptotic tensor rank is characterized by polynomials abstract
Abstract:
Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the 2×2 matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other hand, Strassen\'s asymptotic rank conjecture makes the bold claim that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable.We prove that asymptotic tensor rank is "computable from above", that is, for any real number r there is an (efficient) algorithm that determines, given a tensor T, if the asymptotic tensor rank of T is at most r. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank.As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any upper bound on the matrix multiplication exponent that is close enough, will "snap" to it. Previously such discreteness results were only known for finite field or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers.
15:00 • Université de Genève, Conseil Général 7-9, Room 1-07
Marco Baracchini (Università di Genova)
BGG decomposition of some de Rham complexes over flag varieties abstract
Abstract:
The aim of this talk is to describe how BGG theory can be used to compute the de Rham cohomology over flag varieties. As motivational example we will see the GL_2-case computing the finite slope part of the de Rham cohomology of the sheaf \\mathbb{W}_k built in Andreatta-Iovita, interpolating the symmetric powers of the de Rham cohomology of the universal elliptic curve. Let G be a reductive group with a parabolic subgroup P, let g:=Lie(G) and V a finite-dimensional, irreducible g-module. Using BGG theory, we obtain a decomposition of the Koszul complex associated with V. Moreover we can construct a de Rham complex linked to the dual of V and the induced g-action. Faltings and Faltings-Chai discovered a relation between the Koszul and de Rham complexes, investigating how the BGG decomposition can be translated into a decomposition of the de Rham complex, ultimately enabling the computation of de Rham cohomology.
Dr. Mitchell Taylor (ETH Zürich, Switzerland)
Stable phase retrieval in function spaces abstract
Abstract:
A subspace $E \\subset L2(\\mu)$ is said to do stable phase retrieval (SPR) if there exists a constant $C \\geq 1$ such that for any $f,g \\in E$ we have$\\inf_{|| \\lambda || =1} || f- \\lambda g || \\leq C || |f| - |g| ||$ In this case, if |f| is known, then f is uniquely determined up to an unavoidable global phase factor$\\lambda$ moreover, the phase recovery map is C-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this talk, I will present some elementary examples of subspaces of L2(µ) which do stable phase retrieval and discuss the structure of this class of subspaces. The material in this talk is based on joint work with M. Christ and B. Pineau as well as with D. Freeman, T. Oikhberg and B. Pineau.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Susanna Heikkilä (University of Jyväskylä)
De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean abstract
Abstract:
In this talk, we discuss a result stating that, if a closed manifold admits a non-constant quasiregular map from a Euclidean space, then the de Rham cohomology of the manifold embeds into the Euclidean exterior algebra as a subalgebra. This embedding of algebras yields a topological classification of closed simply connected orientable 4-manifolds admitting a non-constant quasiregular map from the four-dimensional Euclidean space. The talk is based on joint work with Pekka Pankka (University of Helsinki).
15:15 • Université de Fribourg, PER23 room 0.05
Thomas Coolican Fraser (Copenhagen)
Eigenvalues of tetrahedral sums of Hermitian matrices abstract
Abstract:
Horn\'s problem aims to determine set of possible eigenvalues (a,b,c) for a triple of n x n Hermitian matrices (A,B,C) satisfying A+B=C. This talk is based on joint work with A. Alekseev and M. Christandl concerning a generalization of Horn\'s problem called the tetrahedral Horn problem. The tetrahedral Horn problem aims to determine the set of possible eigenvalues (a,b,c,d,e,f) for a sextuplet of n x n Hermitian matrices (A,B,C,D,E,F) satisfying A+B=C, B+D=F, C+D=E, and A+F=E. We will derive the complete solution for the tetrahedral Horn problem for n=2 and see how, unlike Horn\'s original problem, the solution space is neither convex nor a polytope.For general n, we describe various inequality conditions and symmetries of the tetrahedral Horn problem, as well as prove how the set of tetrahedral eigenvalues is characterized by the asymptotics of the Wigner 6j symbol for the unitary group U(n), generalizing the well-known correspondence between the Littlewood-Richardson coefficients and Horn\'s original problem.
16:10 • Université de Genève, Conseil Général 7-9, Room 1-07
Stefano Galanda (Dipartimento di matematica - DIMA, Universita di Genova)
What is... Algebraic Quantum Field Theory? abstract
Abstract:
The Algebraic approach to quantum theories was developed in the end of last century in order to provide a solid mathematical description to the theory of the standard model of particle physics and especially its generalization on curved spacetime backgrounds. Despite its original purpose, the formalism also emerged as a suitable framework for formulating the thermodynamic limit of models in statistical mechanics. The goal of this talk is to provide a gentle introduction to the algebraic approach by first presenting its main motivations and then outlining some of its mathematical foundations. In particular, we will present this approach as an abstract generalization of the mathematical framework of Quantum Mechanics.
16:30 • UZH Zentrum, Building KO2, Room F 150