Burger's equation is one of the basic model equations in fluid mechanics. In this talk I'll give an introduction to Burger's equation from the perspective of infinite dimensional geometry. We'll discuss how diffeomorphism groups can be endowed with a manifold structure and how with certain Riemannian metrics these groups have Burger's equation as their geodesic equations. We'll also see how different versions of Burger's equation that are simply related from the analytic perspective can have wildly different geometric properties.

Let ${\displaystyle m>1}$ be an integer. Except for finitely many primes ${\displaystyle p}$, the equation ${\displaystyle x^m+y^m\equiv tmodp}$ has integer solutions ${\displaystyle x}$ and ${\displaystyle y}$ for all integer ${\displaystyle t}$. The problem is a classical one in Number Theory. In fact, if ${\displaystyle N_{p,t}}$ is the number of solution ${\displaystyle x(modp),y(modp))}$ of the equation, then ${\displaystyle |N_{p,t}-p|=O(\sqrt{p})}$.

The analog problem for infinite or more general fields is the following: Let ${\displaystyle G}$ be a multiplicative group of finite index in ${\displaystyle F^{*}}$, the multiplicative group of units of the field ${\displaystyle F}$, what does ${\displaystyle G+G}$ look like, in particular, when is ${\displaystyle G+G=F}$?.

The first publication on the subject appeard in 1989, when Leep and Shapiro, gave a positive answer when the index of ${\displaystyle G}$ in ${\displaystyle F^{*}}$ is 3. That is, they proved that ${\displaystyle G+G=F,}$ except for a small and short list of exceptional fields. They conjectured that the result was true in the case of index 5, for any infinite field ${\displaystyle F}$. They mentioned that they had not been succesfull even for ${\displaystyle F=\mathbb{Q}}$, the rational numbers.

In the talk we will prove that if ${\displaystyle G}$ is of finite index in ${\displaystyle {\mathbb{Q}}^{*}}$, then ${\displaystyle G-G=\mathbb{Q}}$, that is, every rational number is writen as the difference of two elements of ${\displaystyle G}$. The proof is a very nice application of the Van der Waerden Theorem on artithmetic progressions, that we will state without proof.

More general results in Ramsey Theory allow to prove the same result of any infinite field, and even for some other more general rings. We will brieffly describe these more general results and consequences.

We will also discuss the behaviour of ${\displaystyle G+G}$, of ${\displaystyle G+G+G}$, etc. In fact, we will scketch the solution of a conjecture posed in 1992 by Bergelson and Shapiro on an additive question that was published in 2011, in a joint publication with Florian Luca.

Part 2 of this series of two talks focuses on properties of twisted Hilbert ${\displaystyle C^{*}}$-modules, which are seen as representations of twisted ${\displaystyle C^{*}}$-dynamical systems. I will introduce Ralf Meyer’s bra-ket operators and use them to define the notion of a ‘square-integrable representation’. A construction of generalized fixed-point algebras for twisted ${\displaystyle C^{*}}$-dynamical systems will then be given. The highlight of the talk will be a classification (up to isomorphism) of Hilbert modules over a fixed non-commutative torus purely in terms of Hilbert spaces with certain special properties.

We study some models with microscale properties that are highly heterogeneous in space and time. The time variations are controlled by a stochastic particle dynamics described by a stochastic ODE (SDE).  Our main results include the derivation of macroscale equations and showing that the macroscale equations are deterministic.  We use the asymptotic properties of the SDE. The macro scale equations are derived through an averaging principle of the slow motion (fluid velocity) with respect to the fast motion (particle dynamics).  Our results can be extended to more general nonlinear diffusion equations with heterogeneous coefficients.

In this talk, we will discuss an infinite dimensional Riemannian geometric approach to the nonlinear wave equations and how this perspective provides additional insight into the properties of their solutions. To this end, we will first review finite dimensional Riemannian geometry and gently introduce the infinite dimensional version. Then we will reveal some concrete geometric interpretations of the Hunter-Saxton partial differential equation. It is inevitable to use vocabulary from differential geometry and functional analysis. But I will make sure to explain intuitive ideas with examples and maintain our focus on understanding the mathematical perspective rather than technicalities.

TBA

A flow ${\displaystyle (X,T)}$ is a jointly continuous action of a topological group ${\displaystyle T}$ on a compact Hausdorff space ${\displaystyle X}$. The flow is minimal if every orbit is dense.  The flow ${\displaystyle (X,T)}$ is said to be equicontinuous if the maps defined by ${\displaystyle T}$ form an equicontinuous family, in which case the action can be embedded in a compact group.

Distality is a generalization of equicontinuity, and the structure of distal minimal flows is provided by a deep theorem of Hillel Furstenberg which says that an arbitrary distal minimal flow can be obtained from the trivial one point flow by a (possibly transfinite) sequence of “equicontinuous extensions”.

Given an arbitrary flow, there are distal and equicontinuous structure relations–closed ${\displaystyle T}$ invariant equivalence relations such that the factor flow is, respectively distal or equicontinuous. These relations are generated, respectively, by the proximal and regionally proximal relations. For a minimal flow in many cases (including when ${\displaystyle T}$ is abelian) surprisingly the latter coincides with a smaller relation due to Veech.

TBA

The talk  presents a survey and  some remarks regarding the notion  of pathwise asymptotic optimality in infinite horizon stochastic optimization problems: setups, approaches, results.