# Hausdorff Kolloquium 2019/2020

**Date: **October 23, 2019 - January 29, 2020

**Venue:** Mathematikzentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, October 23

15:15 |
Jani Lukkarinen (University of Helsinki): Mathematical puzzles in kinetic theory |

16:45 |
Monica Visan (UCLA): Recent progress on well-posedness for integrable PDE |

## Wednesday, November 20

15:15 |
Mike Hopkins (Harvard University): Even spaces in algebra, geometry and topology |

16:45 |
Terry Lyons (University of Oxford): What have rough paths to do with data science? |

## Wednesday, January 29

15:15 |
Patricia Gonçalves (University of Lisbon): From random dynamics to fractional PDEs with several boundary conditions |

16:45 |
Philippe Michel (EPF Lausanne): Applied $\ell$-adic cohomology |

## Abstracts

#### Patricia Gonçalves (University of Lisbon): From random dynamics to fractional PDEs with several boundary conditions

In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diﬀusion equations given in terms of the regional fractional Laplacian with diﬀerent types of boundary conditions.

#### Mike Hopkins (Harvard University): Even spaces in algebra, geometry and topology

Spaces equipped with a decomposition into even dimensional cells arise naturally in many areas of mathematics. This talk will survey some of these spaces, some conjectures about analogues of them in equivariant and motivic homotopy theory, and some applications.

#### Jani Lukkarinen (University of Helsinki): Mathematical puzzles in kinetic theory

Kinetic theory and the associated Boltzmann transport equations are one of the few tools which allow to bridge the orders of magnitude from the scale of microscopic dynamical models to their macroscopic transport properties. Although it is often restricted to models whose dynamics are weak perturbations of constant velocity motion, many physical systems offer examples of such behaviour. One case which can be controlled mathematically, is a rarefied gas of hard spheres with elastic collisions, in the Boltzmann-Grad scaling limit. Other less obvious examples are given by weakly perturbed wave motions, including weakly interacting quantum particles and weakly nonlinear discrete wave equations.

In this talk, we discuss mathematical evidence which support the validity of these kinetic theory approximations, as well as their known limitations and modifications. One key step is starting the microscopic dynamics with initial data which is random and sufficiently "chaotic". The importance of this assumption is highlighted in the dynamical formation of a Bose-Einstein condensate: the macroscopic correlations appearing together with the condensate are not compatible with the standard assumptions, and indeed modified kinetic equations are expected to be needed after the formation. This picture is corroborated by the work of Escobedo and Velázquez who prove that the related bosonic kinetic theory has solutions which blow up in a finite (kinetic) time.

#### Philippe Michel (EPF Lausanne): Applied $\ell$-adic cohomology

$\ell$-adic cohomology has its origins in the study of congruences in the ring of integers and specifically in the problem of counting solutions of systems of polynomial equations modulo a prime number $q$. This is a complex and sophisticated theory, conjectured by A. Weil, constructed by A. Grothendieck and developed by P. Deligne, N. Katz, G. Laumon and many others. The basic objects of study are $\ell$-adic sheaves which lie above a given algebraic variety. To such sheaves are associated "trace functions" which are functions living on the set of solutions of the underlying polynomial equations; for instance when the variety is the affine line, one simply obtain functions on finite fields which can then be lifted to $q$-periodic function on the integers. In this talk we will discuss various problems from analytic number theory in which trace functions show up and explain how basic and not so basic methods from $\ell$-adic cohomology allow to make progress.

#### Monica Visan (UCLA): Recent progress on well-posedness for integrable PDE

I will introduce the Korteweg-de Vries equation and then describe the method developed with Rowan Killip for proving optimal well-posedness for this equation. Next, I will describe subsequent developments for proving optimal well-posedness results for other models such as the completely integrable Schrodinger and the modified Korteweg-de Vires equations.