Hausdorff Kolloquium 2018/2019

Date: October 31, 2018 - January 23, 2019

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

Abstracts

Giada Basile (Sapienza Universit├Ąt Rom): A gradient flow approach to kinetic equations

In the seminal paper of Jordan, Kinderlehrer and Otto the heat equation has been interpreted as the gradient flow, or deepest descent, of the entropy with respect to the  Wasserstein L^2 metric. In the same spirit the evolution associated to some partial differential equations has been characterized  in terms of an entropy dissipation inequality. Recently there have been some attempts to formulate the Fokker-Planck equation associated to continuous time reversible Markov chains, equivalently homogeneous linear kinetic equations, as gradient flow, and the approach has been extended to the case of homogeneous Boltzmann equations.
I will present some results on gradient flow formulation of linear, non-homogeneous kinetic equations. In particular, I will discuss  how the functional giving rise to the entropy dissipation inequality is related to large deviations and  I will derive the diffusive limit. I will finally discuss how the above approach could be extended to  non-linear Boltzmann equations.

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Nicolas Bergeron (ENS Paris): Linking in torus bundles and Hecke L functions

Torus bundles over the circle are among the simplest and cutest examples of 3-dimensional manifolds. After presenting some of these examples, using in particular animations realized by Jos Leys, I will consider periodic orbits in these fiber bundles over the circle. We will see that their linking numbers --- that are rational numbers by definition --- can be computed as certain special values of Hecke L-functions.  Properly generalized this viewpoint makes it possible to give new topological proof of now classical rationality or integrality theorems of Klingen-Siegel and Deligne-Ribet. It also leads to interesting new "arithmetic lifts" that I will briefly explain. All this is extracted from an on going joint work with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

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Jean Michel Coron (UPMC Paris): Stabilization of control systems: From the clepsydras to the regulation of rivers

A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: Is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think to the classical experiment of an upturned broomstick on the tip of one's finger.) On this problem, we present some pioneer devices and works (Ctesibius, Watt, Maxwell, Lyapunov,...), some more recent results, and an application to the regulation of the rivers La Sambre and La Meuse in Belgium. A special emphasize is put on positive or negative effects of the nonlinearities.

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Jean Christophe Mourrat (ENS Paris): Quantitative stochastic homogenization

Over large scales, many disordered systems behave similarly to an equivalent "homogenized" system of simpler nature. A fundamental example of this phenomenon is that of reversible diffusion operators with random coefficients. The homogenization of these operators has been well-known since the late 70's. I will present recent results that go much beyond this qualitative statement, reaching optimal rates of convergence and a precise description of the next-order fluctuations. The approach is based on a rigorous renormalization argument and the idea of linearizing around the homogenized limit.

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Pablo Shmerkin (UTDT Buenos Aires): Multiplying by 2 and by 3: old conjectures and new results

In the 1960s, H. Furstenberg proposed a series of conjectures which, in different ways, aim to capture the heuristic principle that "expansions in different bases have no common structure". I will discuss a sample of these conjectures, some which remain open and some which have been established in recent years. These problems lie at the intersection of Ergodic Theory and Fractal Geometry, but no previous background on either area will be assumed.

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Ulrike Tillmann (University of Oxford): Moduli spaces of manifolds and quantum field theories

There has been a rich and fruitful interplay between quantum field theory and the study of topological moduli spaces of manifolds including the proof of Mumford's conjecture and the cobordism hypothesis. We will explore some of these interactions emphasising the role of homological stability.

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