##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Yanki Lekili

#### University of Cambridge

## An arithmetic refinement of homological mirror symmetry for the 2-torus

##### Abstract:

We explore a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes over the "punctured disc" Spec Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We will survey a general strategy of proof of homological mirror symmetry while carrying it out in the specific case of the 2-torus. In contrast to the abstract statement of our main result, the focus of the talk will be a concrete computation which we will express in more familiar terms. This is joint work with Tim Perutz.

Host: Justin Roberts

### February 4, 2013

### 2:00 PM

### AP&M 6402

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