Graduate Seminar

Amenability is a fundamental notion in group theory, consisting of the existence of a translation-invariant "average" functional on a group. Amenable groups were first introduced in the 1920s by John Von Neumann, while studying the so-called "Banach Tarski paradox", and nowadays are one of the most studied classes of groups in the literature, especially in the fields of geometric group theory, functional analysis and ergodic theory.

 In the course of the talk, we will give some characterizations of amenability, we will prove that abelian groups are amenable (a result due to Von Neumann, although the proof will be more modern), and we will see how the nonamenability of free groups, along with the the axiom of choice, is one of the main ingredients of the Banach-Tarski paradox.

 The talk will have minimal prerequisites, as every notion used will be properly introduced.

 For fans of: Geometry, Functional Analysis, Group Theory.

Mean field games (MFGs) study Nash equilibria in systems with a very large number of strategically interacting agents, in settings where players are symmetric and interaction is realized through the population distribution. This framework has become a standard tool for modeling large-population strategic behavior in economics and engineering.

This talk will give a game-theoretic introduction to mean field games, starting from familiar finite-player examples and progressing toward dynamic games with large populations. I will explain the mean field equilibrium concept, introduce the coupled backward–forward MFG system in a finite-state continuous-time setting, and conclude with an informal discussion of the master equation, which characterizes equilibrium values as a function of the population distribution.

This talk aims to motivate the concept of a uniform space, which holds the essentials to define uniform continuity and completion. We will begin with a brief and pictoral introduction to filters, connectors, and uniformities---set theoretic machinery necessary to define them. After demonstrating that both (semi-)metric spaces and topological groups are uniform spaces, we show how this abstraction gives intuitive proofs of some separation results for this family. We'll conclude by sketching a proof that the uniform spaces are exactly the completely regular* topological spaces