##### Department of Mathematics,

University of California San Diego

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### RTG Colloquium

## Alireza Salehi Golsefidy

#### UCSD

## Super-approximation and its applications

##### Abstract:

Let G be a finitely generated subgroup of GL(n,Q). Under certain algebraic conditions, strong approximation describes the closure of G with respect to its congruence topology. Super-approximation essentially tells us how dense G is in its closure! Here is my plan for this talk: 1. I will start with the precise formulation of this property. 2. Some of the main results on this subject will be mentioned. 3. Some of the (unexpected) applications of super-approximation will be mentioned, e.g. Banach-Ruziewicz problem, orbit equivalence rigidity, variation of Galois representations. 4. Some of the auxiliary results that were needed in the proof of super-approximation will be mentioned: sum-product phenomena, existence of small solutions

### March 2, 2016

### 2:30 PM

### AP&M 6402

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