Beatrice Bernasconi (ETH Zürich)
Abstract:
abstract We define the Cremona group over the projective plane and some length functions and the respective dynamical invariants on its elements; we concentrate in particular on the Sarkisov length.
12:15 • Universität Basel, Lecture Room 117, Kollegienhaus
Prof. Dr. Omri Sarig (Weizmann Institute of Science)
Stochastic Properties for Measures of Maximal Entropy for Smooth Surface Diffeomorphisms abstract
Abstract:
<div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">Suppose \\(f\\) is a general infinitely differentiable topologically mixing surface diffeomorphism with positive topological entropy.</span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">We show that the measure of maximal entropy is Bernoulli, and that all smooth observables exhibit exponential decay of correlations, the strong almost sure invariance principle, and (consequently) the central limit theorem and the law of the iterated logarithm.</span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">The proof, joint with Crovisier and Buzzi, is based on a new dynamical property called "strong positive recurrence," which guarantees the existence of a symbolic model with spectral gap for the associated transfer operator. </span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">(Joint with S. Crovisier & J. Buzzi)</span></span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div>
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Prof. Dr. Shahar Mendelson (Texas A&M University, USA )
On the structure of marginals in high dimensions abstract
Abstract:
Let X be a centred random vector in \\R^d and consider the random matrix \\Gamma=\\sum_{i=1}^N <X_i,->e_i, whose rows are independent copies of X. Given T \\subset S^{d-1}, how much of T\'s structure is captured by \\Gamma T? Our focus here is on fine notions of similarity - specifically, whether the (empirical) distributions of (<X_i,\\theta>)_{i=1}^N are close in an appropriate sense to the distributions of <X,\\theta>, uniformly in \\theta \\in T. The answer to this question was not known even when X=G, the standard gaussian. In this talk I will present the current state-of-the-art - which holds under minimal assumptions, and in particular leads to an (almost) optimal answer in the gaussian case. If time permits, I will present some of the questions that are still open. Joint work with D. Bartl.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room F 26.1
Alonso Beaumont (Rennes)
The Margulis-Soifer property for the planar Cremona group abstract
Abstract:
In 1981, Margulis and Soifer established a remarkable property of finitely generated linear groups: such a group contains uncountably many maximal subgroups if and only if it is not virtually solvable. In particular, this yields the existence of maximal subgroups of infinite index in SLn(Z). We will show how their methods can be adapted to establish an analogous result for groups of birational transformations of the plane. This is based on joint work with Serge Cantat, Julie Déserti and Seung Uk Jang.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Christian Lubich (University Tuebingen)
Time integration of tree tensor networks abstract
Abstract:
The talk first presents some numerical experiments with time-dependent tree tensor network algorithms for the approximation of quantum spin system dynamics. It continues with the basics in the design of time integration methods that are robust to the typical presence of small singular values, that have good structure-preserving properties (norm, energy conservation or dissipation), and that allow for rank (= bond dimension) adaptivity and for parallelism. This discussion of basic concepts forms the main part of the talk and will be done for the smallest possible typeof tensor network differential equations, namely low-rank matrix differential equations, which are of interest in their own right. Once this nontrivial technically simplest case is understood, there is a systematic path to the extension of the integrators and their favourable properties to general tree tensor networks.This talk is based on joint work with many colleagues and former andpresent students, among which I wish to single out Othmar Koch for the first mathematical work on dynamical low-rank approximation (DLRA) in 2007, Ivan Oseledets for jointly discovering the first robust DLRA integrator in 2014(the projector-splitting integrator), Gianluca Ceruti and Jonas Kusch for jointly developing the Basis Update & Galerkin (BUG) integrators since 2022, and Hanna Walach, Gianluca Ceruti, Dominik Sulz and Charlotte Verhoeven forthe recent systematic extension from low-rank matrices to generaltree tensor networks both in theory and in increasingly efficientimplementations.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Prof. Dr. Giovanni Conforti (University of Padova)
Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations abstract
Abstract:
A classical consequence of the Prékopa–Leindler inequality is that log-concavity is preserved along the heat flow. Beyond the classical heat semigroup, however, this phenomenon typically breaks down. In this talk, I will introduce a weaker notion of log-concavity that remains stable under generalized heat semigroups. I will show how this framework yields new log-semiconcavity estimates for ground states of Schrödinger operators with non-convex potentials, as well as propagation results for functional inequalities along generalized heat flows.I will also discuss how these weak log-concavity properties behave under conditioning and marginalization, extending classical ideas of Brascamp and Lieb beyond the log-concave setting. The main ingredient is a stochastic control perspective on quadratic Hamilton–Jacobi–Bellman equations, combined with a second-order analysis of reflection couplings along HJB characteristics.
17:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H12