## Fudan Logic Seminar 2023

### November 20

Speaker: Ehud Hrushovski

Time: 15:30 - 17:00. Location: Gu Hall, SCMS, Jiangwan Campus.

#### Model-theory and Approximate Lattices

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).

### Aug 9

Location: HGX208

Speaker: Chieu-Minh Tran

Time: 9:30 - 10:20

#### Measure growth in compact semisimple Lie groups and the Kemperman Inverse Problem

Suppose \(G\) is a compact semisimple Lie group, \(\mu\) is the normalized Haar measure on \(G\), and \(A,A^2\subseteq G\) are measurable. We show that
\(\mu(A^2)\geq \min\{1,2\mu(A)+\eta\mu(A)(1−2\mu(A))\}\)
with the absolute constant \(\eta>0\) (independent from the choice of \(G\)) quantitatively determined. This is the continuous counterpart of celebrated products theorem by Helfgott, Pyber-Szabo, and Breuillard-Green-Tao.

We also show a more general result for abstractly semisimple connected compact groups and resolve the Kemperman Inverse Problem from 1964. (Joint with Jing Yifan)

Speaker: Yifan Jing

Time: 10:35 - 11:25

#### Measure doubling for small sets in SO(3,R)

Let \(\mathrm{SO}(3,\mathbb{R})\) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure \(\mu\). Confirming a conjecture by Breuillard and Green, we show that if \(A\) is an open subset of \(\mathrm{SO}(3,\mathbb{R})\) with sufficiently small measure, then \(\mu(A^2) > 3.99 \mu(A)\). This is joint work with Chieu-Minh Tran (NUS) and Ruixiang Zhang (Berkeley).

### May 26

Speaker: 何家亮 Jialiang He

Time: 11:45 - 13:15. Location: HGW2403.

#### Structure of summable tall ideals under Katětov order

In this talk, we will present the newest progress on structure of Katetov order on summable ideals. In particular, We show that Katětov and Rudin-Blass orders on summable tall ideals coincide. We prove that Katětov order on summable tall ideals is Galois Tukey equivalent to (\(\omega^\omega, \leq^*\)). It follows that Katětov order on summable tall ideals is upwards directed which answers a question of H. Minami and H. Sakai. In addition, we prove that $l_\infty$ is Borel bireducible to an equivalence relation induced by Katětov order on summable tall ideals. This is joint work with Zuoheng Li and Shuguo Zhang.