4 lectures by Simon Machado (ETH Zurich) on The Dynamics of point sets: local and global multiplicative structures.
Lecture 1: Tuesday Nov. 26th, 14:00-15:00 (B3.02)
Lecture 2-3: Wednesday Nov. 27th, 14:00-16:00 (B3.01)
Lecture 4: Thursday Nov. 28th, 09:00-10:00 (B3.01)
Title: The dynamics of point sets: local and global multiplicative structures
Abstract: The study of discrete point sets in homogeneous spaces has a long and rich history, with fascinating examples from statistical physics, aperiodic tilings, combinatorial number theory, and sphere packing. More recently, the theory of aperiodic point sets has found fruitful interactions with various branches of mathematics. This mini-course will focus on one emerging trend, exploring the relationship between global and local multiplicative structures from a dynamical perspective.
On the one hand, the concept embodying global structure will be a generalization of groups, known as approximate groups. This will lead us to approximate lattices (a.k.a. Meyer sets), a notion first explored by Yves Meyer.
On the other hand, the local structure will be embodied by notions of patterns studied dynamically. Analyzing pattern statistics dynamically is often beneficial, as evidenced by Furstenberg’s correspondence principle or the study of “tiling spaces”.
We will cover the following topics:
1. An introduction to approximate lattices, first defined by Yves Meyer and a discussion of their historical roots. We will focus on recent developments and how the global multiplicative structure arises from local patterns, such as 3-term arithmetic progressions.
2. We will discuss dynamical systems related to point sets, such as tiling spaces and Stone spaces. We will illustrate their usefulness by explaining a new proof of a recent breakthrough in approximate subgroups initially established through model theory.
3. Finally, I will briefly discuss exciting open problems that naturally fit this framework, including some related to Ulam stability and the Lorentz gas.
