Bachir Bekka (Université de Rennes)
Alessandra Iozzi (ETH Zürich)
Yves Benoist (Université de Rennes)
Serge Cantat (CNRS et Université de Renne)
09:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Gigliola Staffilani (Massachusetts Institute of Technology)
Dispersive equations and wave turbulence theory abstract
Abstract:
In this course we will investigate questions of weak turbulence theory by using as explicit example of wave interactions the solutions to periodic and nonlinear Schrödinger equations. We will start with Strichartz estimates on periodic setting, then we will move to well-posedness.We will then present two different ways of introducing the evolution of the energy spectrum. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then, we will give some ideas of how to derive rigorously the effective dynamics of the energy spectrum itself (wave kinetic equation), when one considers weakly nonlinear dispersive equations.
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Daniel Ruprecht (Technische Universität Hamburg)
Recent results on spectral deferred corrections, parallel and otherwise abstract
Abstract:
Spectral deferred corrections (SDC) remain an underutilized approach for solving initial value problems. SDC can be considered a preconditioned, iterative solver for the collocation problem arising in a fully implicit Runge-Kutta method with a dense Butcher table. The key trick in SDC is that each iteration, or sweep, only requires stepping through the nodes with a simple, low order method like implicit Euler. In that way, SDC provide a framework to build methods of arbitrary high order out of simple, low order methods.SDC-based methods have many advantages. They typically come with large stability domains, easily tunable order of accuracy and are often more accurate than RKM of similar order when using comparable time step sizes. However, they require more right hand side evaluations per step than RKM (sometimes a lot more) and can only be computationally efficient if they allow a user to take much larger step sizes than RKM. In the talk, I will present recent work aiming to increase the computational efficiency of SDC. This will include approaches to parallelize SDC "across-the-method" to allow the use of small-scale parallelism to reduce the wallclock time of iterations. I will also show how the iterative structure of SDC can easily be exploited to construct a form of embedded method that can be used for step size control. Finally, I will comment one ongoing work regarding the use of reduced asymptotic coarse models to speed up SDC convergence as well as attempts to use physics-informed neural operators (PINO) to generate accurate starting values.This is joint work with J. Fregin, T. Lunet, I. Akramov, P. Freese, S. Götschel (TUHH), G. Čaklovic (KIT), T. Baumann (FZ Jülich) and M. Schreiber (Grenoble).
14:00 • Université de Genève, Conseil Général 7-9, Room 1-07
Vanessa Piccolo (ENS Lyon, FR)
Dynamics of optimization in high dimensions for multi-spiked tensor PCA abstract
Abstract:
We will study the high-dimensional statistical problem of multi-spiked tensor PCA, where the goal is to infer a finite number of unknown, orthogonal signal vectors (or spikes) from noisy observations of a p-tensor. I will present our recent results on the sample complexity required for stochastic gradient descent to efficiently recover the signal vectors from natural initializations. In particular, we will show that it is possible to recover a permutation of all spikes provided a number of sample scaling as N^{p-2}, aligning with the computational threshold identified in the rank-one tensor PCA problem. The recovery process is governed by a sequential elimination phenomenon. As one correlation exceeds an explicit critical threshold, all correlations that share a row or column index become sufficiently small to be negligible, allowing the subsequent correlation to grow and become macroscopic. The order in which correlations become macroscopic is determined by their initial values and the associated signal-to-noise ratios. Based on recent joint work with Gérard Ben Arous (NYU) and Cédric Gerbelot (NYU).
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Anshul Adve
Density criteria for Fourier uniqueness phenomena in R^d abstract
Abstract:
A remarkable byproduct of the work of Viazovska et al. on sphere packing is the notion of a "Fourier uniqueness set": a discrete subset A of R^d such that any Schwarz function on R^d is determined by its restriction and the restriction of its Fourier transform to A. Once one knows such sets exist, it is natural to ask how dense they must be / how sparse they can be. This was determined in dimension 1 by Kulikov, Nazarov, and Sodin, but the higher dimensional story was much less well understood. We will give an answer in all dimensions using new methods.
14:15 • EPF Lausanne, CM 1 517
Dr. Jaume De Dios Pont (ETH Zürich, Switzerland)
Log-concave measures can have interior hot spots abstract
Abstract:
Let u(x,t) be the temperature distribution of a d-dimensional convex domain at time t with given initial temperature u(x,0) and insulating boundary. The hot-spots conjecture of Rauch asserts that for large times, the maximum of the function x -> u(x,t) is taken near the boundary of the domain.Equivalently, the conjecture asserts that the first nontrivial Neumann Laplace eigenfunction of a convex domain takes its maximum (and minimum) in the boundary.A general philosophy in convex analysis is that dimension free statements about convex sets imply an analogous, dimension-free statement about log-concave measures.In this talk I will construct the log-concave analogue to the hot spots conjecture, and construct a counterexample for it in high dimensions.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Serge Cantat (CNRS et Université de Rennes)
Croissance du groupe de Cremona abstract
Abstract:
Le groupe de Cremona est formé de toutes les transformations birationnelles du plan, c’est-à-dire les transformations inversibles qui s’expriment - ainsi que leur inverse - à l\'aide de fractions rationnelles en les coordonnées cartésiennes. Je décrirai ce groupe en me concentrant sur la notion de croissance.
Ce colloque fait partie du Séminaire Groupes et Géométrie qui tiendra une session spéciale à l’occasion des 80 ans de de Pierre de la Harpe.
De plus, que pour des raisons logistiques, l\'inscription est nécessaire avant le 20 septembre.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-05
Rupert Frank (Universität München )
The Wehrl entropy problem: mathematical physics meets complex analysis and representation theory abstract
Abstract:
The coherent state transform, under various names, appears in many fields of mathematics and physics. It is associated with representations of a group. In this talk we are concerned with the problem of minimizing the entropy of the coherent state transform and we explain how complex analysis can be used to achieve this in certain settings. We discuss various open questions
16:30 • UZH Zentrum, Building KO2, Room F 150