Logic Week with Ehud Hrushovski

November 20 to October 22, 2023

Model-theory and Approximate Lattices

Ehud Hrushovski

Time: 15:30 - 17:00, November. 20, 2023. Location:

Gu Hall, SCMS, Jiangwan Campus.

An approximate subgroup is a subset $X$ of a group $G$, such that $1\in X$, $X = X^{-1}$, and such that the set of products $X\cdot X = \{x\cdot y : x, y \in X\}$ is ‘almost’ equal to $X$, more precisely it is contained in $X\cdot F$ for some finite $F \subset G$. Approximate subgroups arise in many areas of analysis, combinatorics and geometry, as well as in model theory. The finite ones were classified by Breuillard, Green and Tao; they essentially arise in nilpotent groups.

An approximate lattice in $G = \mathbb{R}^n$, or in the matrix group $\textrm{GL}_n(\mathbb{R})$, is a discrete approximate subgroup $X$ that has finite covolume; i.e. there exists a subset $D \subset G$ of finite measure, with $XD = G$. Approximate lattices in $\mathbb{R}^n$ were classified by Meyer in the 1970's, and eventually became the mathematical model for quasicrystals. I will present a generalization to semisimple groups; in effect all irreducible approximate lattices have arithmetic origin. They arise from number fields via a classical construction of Borel-Harish-Chandra; the approximate setting allows greater flexibility in putting archimedean and non-archimedean places on the same footing.

The proof uses a construction arising naturally from basic questions in model theory (amalgamation, the Lascar group).

This talk is one of Fudan Logic Seminar, SCMS Gu Lecture, and Fudan University Mathematical Sciences Academy Lecture, and it is jointly organized by School of Philosophy, School of Mathematical Sciences at Fudan University, and Shanghai Center for Mathematical Sciences.