Mon, 29 Nov 2021

16:00 - 17:00
Virtual

Qualitative properties on a Fokker Planck equation model on neural network

Delphine Salort
(Sorbonne Université)
Abstract

The aim of this talk is to understand the qualitative properties that emerge from a PDE model inspired from neurosciences, in order to understand what are the key processes that lead to mathematical complex patterns for the solutions of this equation.

Tue, 01 Jun 2021
14:00
Virtual

Why are numerical algorithms accurate at large scale and low precisions?

Theo Mary
(Sorbonne Université)
Abstract

Standard worst-case rounding error bounds of most numerical linear algebra algorithms grow linearly with the problem size and the machine precision. These bounds suggest that numerical algorithms could be inaccurate at large scale and/or at low precisions, but fortunately they are pessimistic. We will review recent advances in probabilistic rounding error analyses, which have attracted renewed interest due to the emergence of low precisions on modern hardware as well as the rise of stochastic rounding.

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Mon, 08 Mar 2021

16:00 - 17:00
Virtual

Singular solutions of the binormal flow

Valeria Banica
(Sorbonne Université)
Abstract

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. This flow is also related to the classical continuous Heisenberg model in ferromagnetism and to the 1-D cubic Schrödinger equation. In this lecture I will first talk about the state of the art of the binormal flow conjecture, as well as about mathematical methods and results for the binormal flow. Then I will introduce a class of solutions at the critical level of regularity that generate singularities in finite time and describe some of their properties. These results are joint work with Luis Vega.

Mon, 01 Feb 2021

16:00 - 17:00

Extremal distance and conformal radius of a CLE_4 loop.

TITUS LUPU
(Sorbonne Université)
Abstract

The CLE_4 Conformal Loop Ensemble in a 2D simply connected domain is a random countable collection of fractal Jordan curves that satisfies a statistical conformal invariance and appears, or is conjectured to appear, as a scaling limit of interfaces in various statistical physics models in 2D, for instance in the double dimer model. The CLE_4   is also related to the 2D Gaussian free field. Given a simply connected domain D and a point z in D, we consider the CLE_4 loop that surrounds z and study the extremal distance between the loop and the boundary of the domain, and the conformal radius of the interior surrounded by the loop seen from z. Because of the confomal invariance, the joint law of this two quantities does not depend (up to a scale factor) on the choice of the domain D and the point z in D. The law of the conformal radius alone has been known since the works of Schramm, Sheffield and Wilson. We complement their result by deriving the joint law of (extremal distance, conformal radius). Both quantities can be read on the same 1D Brownian path, by tacking a last passage time and a first hitting time. This joint law, together with some distortion bounds, provides some exponents related to the CLE_4. This is a joint work with Juhan Aru and Avelio Sepulveda.

Mon, 26 Oct 2020

16:00 - 17:00
Virtual

The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary

Grigorios Fournodavlos
(Sorbonne Université)
Abstract

Unlike the classical Cauchy problem in general relativity, which has been well-understood since the pioneering work of Y. Choquet-Bruhat (1952), the initial boundary value problem for the Einstein equations still lacks a comprehensive treatment. In particular, there is no geometric description of the boundary data yet known, which makes the problem well-posed for general timelike boundaries. Various gauge-dependent results have been established. Timelike boundaries naturally arise in the study of massive bodies, numerics, AdS spacetimes. I will give an overview of the problem and then present recent joint work with Jacques Smulevici that treates the special case of a totally geodesic boundary.

Thu, 30 Apr 2020

16:00 - 16:45
Virtual

Learning with Signatures: embedding and truncation order selection

(Sorbonne Université)
Further Information

Abstract

Abstract: Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. We will be concerned with a novel approach for sequential learning, called the signature method, and rooted in rough path theory. Its basic principle is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. On the one hand, this approach relies critically on an embedding principle, which consists in representing discretely sampled data as paths, i.e., functions from [0,1] to R^d. We investigate the influence of embeddings on prediction accuracy with an in-depth study of three recent and challenging datasets. We show that a specific embedding, called lead-lag, is systematically better, whatever the dataset or algorithm used. On the other hand, in order to combine signatures with machine learning algorithms, it is necessary to truncate these infinite series. Therefore, we define an estimator of the truncation order and prove its convergence in the expected signature model.

Thu, 27 Feb 2020
11:30
C4

Non-archimedean parametrizations and some bialgebraicity results

François Loeser
(Sorbonne Université)
Abstract

We will provide a general overview on some recent work on non-archimedean parametrizations and their applications. We will start by presenting our work with Cluckers and Comte on the existence of good Yomdin-Gromov parametrizations in the non-archimedean context and a $p$-adic Pila-Wilkie theorem.   We will then explain how this is used in our work with Chambert-Loir to prove bialgebraicity results in products of Mumford curves.

Tue, 15 Oct 2019

15:30 - 16:30
L6

On random waves in Seba's billiard

Henrik Ueberschär
(Sorbonne Université)
Abstract

In this talk I will give an overview of Seba's billiard as a popular model in the field of Quantum Chaos. Consider a rectangular billiard with a Dirac mass placed in its interior. Whereas this mass has essentially no effect on the classical dynamics, it does have an effect on the quantum dynamics, because quantum wave packets experience diffraction at the point obstacle. Numerical investigations of this model by Petr Seba suggested that the spectrum and the eigenfunctions of the Seba billiard resemble the spectra and eigenfunctions of billiards which are classically chaotic.

I will give an introduction to this model and discuss recent results on quantum ergodicity, superscars and the validity of Berry's random wave conjecture. This talk is based on joint work with Par Kurlberg and Zeev Rudnick.

Tue, 15 May 2018

16:00 - 17:00
L5

Non-archimedean integrals as limits of complex integrals.

Antoine Ducros
(Sorbonne Université)
Abstract

Several works (by Kontsevich, Soibelman, Berkovich, Nicaise, Boucksom, Jonsson...) have shown that the limit behavior of a one-parameter family $(X_t)$ of complex algebraic varieties can often be described using the associated Berkovich t-adic analytic space $X^b$. In a work in progress with E. Hrushovski and F. Loeser, we provide a new instance of this general phenomenon. Suppose we are given for every t an  $(n,n)$-form $ω_t$ on $X_t$ (for n= dim X). Then under some assumptions on the formula that describes $ω_t$, the family $(ω_t)$ has a "limit" ω, which is a real valued  (n,n)-form in the sense of Chambert-Loir and myself on the Berkovich space $X^b$, and the integral of $ω_t$ on $X_t$ tends to the integral of ω on $X^b$.
In this talk I will first make some reminders about Berkovich spaces and (n,n)-forms in this setting, and then discuss the above result.
In fact, as I will explain, it is more convenient to formulate it with  $(X_t)$ seen as a single algebraic variety over a non-standard model *C of C and (ω_t) as a (n,n) differential form on this variety. The field *C also carries a t-adic real valuation which makes it a model of ACVF (and enables to do Berkovich geometry on it), and our proof uses repeatedly RCF and ACVF theories.

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