Karoly Böröczky (Alfréd Rényi Institute of Mathematics)
The Isoperimetric inequality, the Brunn-Minkowski theory, and the Lp Minkowski problem
10:15 • Zoom
Marialaura Noce (Università Salento)
Lower central series of some groups acting on rooted trees abstract
Abstract:
A rapidly developing area of group theory studies the properties of branch groups, a special kind of groups acting on rooted trees. An important motivation for studying branch groups comes from the remarkable properties that some of these groups can possess, like intermediate growth, amenability, the congruence subgroup property, or providing a negative answer to the General Burnside Problem. Among other properties, some research in the last decade has been done regarding the lower central series of subgroups of the group of automorphisms acting on rooted trees.
In this talk, we first survey some known results about the lower central series in first Grigorchuk group, the Gupta-Sidki 3-group, and the Fabrykowski-Gupta group, and then we present new results and some open problems.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Mogens Steffensen (University of Copenhagen)
Optimal consumption, investment, and insurance under state-dependent risk aversion abstract
Abstract:
We formalize a consumption-investment-insurance problem with the distinction of a state-dependent relative risk aversion. The state-dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability. We also discuss whether the approach is appropriate to deal with uncertainty in relative risk aversion and consider some alternative ideas.
11:00 • EPF Lausanne, UniL campus, Extranef 125
Boaz Nadler (The Weizmann Institute of Science, Israel)
The Trimmed Lasso: Sparse Recovery Guarantees And Practical Optimization abstract
Abstract:
Consider the sparse approximation or best subset selection problem: Given a vector y and a matrix A, find a k-sparse vector x that minimizes the residual ||Ax-y||. This sparse linear regression problem, and related variants, plays a key role in high dimensional statistics, compressed sensing, and more. In this talk we focus on the trimmed lasso penalty, defined as the L_1 norm of x minus the L_1 norm of its top k entries in absolute value. We advocate using this penalty by deriving sparse recovery guarantees for it, and by presenting a practical approach to optimize it. Our computational approach is based on the generalized soft-min penalty, a smooth surrogate that takes into account all possible k-sparse patterns. We derive a polynomial time algorithm to compute it, which in turn yields a novel method for the best subset selection problem. Numerical simulations illustrate its competitive performance compared to current state of the art.
13:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Adrien Laurent (University of Bergen)
The aromatic bicomplex for the study of volume-preserving integrators abstract
Abstract:
While B-series are used to represent the Taylor expansion of the solution of an ODE, aromatic B-series appear when computing the divergence of such a Taylor expansion. In this talk, we define aromatic forms (in the spirit of differential forms), we extend the divergence operator on aromatic forms, and we introduce the aromatic bicomplex, in the spirit of the variational bicomplex in differential geometry. We prove the exactness of this bicomplex with the help of the Euler and the homotopy operators. We will describe the far-reaching implications of the exactness of the aromatic bicomplex in the study of volume preservation.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Tatiana Brailovskaya (Princeton University, Princeton, USA)
Universality and matrix concentration inequalities abstract
Abstract:
Random matrices frequently appear in many different fields — physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure — entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). However, current understanding of such complex random matrices remains lacking. In this talk, I will discuss recent results concerning the spectrum of sums of independent random matrices with a.s. bounded operator norms. In particular, I will demonstrate that under some fairly general conditions, such sums will exhibit the following universality phenomenon — their spectrum will lie close to that of a Gaussian random matrix with the same mean and covariance. No prior background in random matrix theory is required — basic knowledge of probability and linear algebra are sufficient. (joint with Ramon van Handel)Pre-print link:https://web.math.princeton.edu/~rvan/tuniv220113.pdf
14:45 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Christopher Garban (Université de Lyon & Genève)
Title T.B.A.
15:00 • EPF Lausanne, Salle CM 0 11
Andras Szenes (Université de Genève)
Intersection cohomology of the moduli spaces of stable bundles abstract
Abstract:
Intersection cohomology is a topological notion adapted to the description of singular topological spaces. The Decomposition Theorem for maps of algebraic varieties is a key tool in the subject, generalizing the Leray-Hirsch theorem.The study of the intersection cohomology of the moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Camilla Felisetti and Olga Trapeznikova, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In these talk, I will give an introduction to the subject, and describe our results.
15:15 • Université de Genève, Conseil Général 7-9, Room 1-07
Hannah Markwig (University of Tübingen)
Counting bitangents of quartic curves - arithmetic, real, tropical abstract
Abstract:
We showcase tropical geometry as a tool for geometric counting problems. A nice feature of tropical geometry is that many techniques can be applied simultaneously over various ground fields, e.g. for complex or real counting problems. Our prime example will be the count of bitangent liens to a smooth plane quartic.Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold\'s trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work with Sam Payne and Kris Shaw.
16:30 • UZH Zentrum, Building KO2, Room F 150
Martin Leguil (Université de Picardie, Amiens)
Some topics of smooth ergodic theory abstract
Abstract:
One of the main goals of ergodic theory is to describe the statistical behavior of most orbits of a given dynamical system. For "chaotic systems" (more precisely, those exhibiting some form of hyperbolicity), certain dynamical foliations (stable/unstable foliations) prove to be very useful in the study of statistical properties of such systems. In this talk I will recall basic notions of these foliations, and how these geometric objects interact with some invariant measures of particular interest, the so-called SRB measures, which describe the statistical behavior of a "large" set of points. I will also mention the special case of conservative systems, where a famous argument due to Hopf shows that a certain property of these foliations, called accessibility, can be employed to show that the system is ergodic. For partially hyperbolic systems, I will introduce a certain class of measures, the so-called u-Gibbs measures, explain how such measures can be constructed, and how they are related to the SRB measures.
17:00 • Université de Neuchâtel, Rue Emile Argand 11, Auditoire B013