# Members Seminar

## Algebraic K-Theory Via Binary Complexes

## How to Find Periodic Orbits and Exotic Symplectic Manifolds

## Parallel Repetition of Two Prover Games: A Survey

I will give an introduction to the problem of parallel repetition of two-prover games and its applications and related results in theoretical computer science (the PCP theorem, hardness of approximation), mathematics (the geometry of foams, tiling the space R^n) and, if time allows, physics (Bell inequalities, the EPR paradox).

## Computations of Heegaard Floer Homologies

Heegaard Floer homology groups were recently introduced by Ozsvath and Szabo to study properties of 3-manifolds and knots in them. The definition of the invariants rests on delicate holomorphic geometry, making the actual computations cumbersome. In the lecture we will recall the basic definitions and theorems of the theory, and show how to define the simplest version in a purely combinatorial manner. For a special class of 3-manifolds the more general version will be presented by simple combinatorial ideas through lattice homology of Nemethi.

## Computations of Heegaard Floer Homologies

Heegaard Floer homology groups were recently introduced by Ozsvath and Szabo to study properties of 3-manifolds and knots in them. The definition of the invariants rests on delicate holomorphic geometry, making the actual computations cumbersome. In the lecture we will recall the basic definitions and theorems of the theory, and show how to define the simplest version in a purely combinatorial manner. For a special class of 3-manifolds the more general version will be presented by simple combinatorial ideas through lattice homology of Nemethi.

## The Heisenberg Algebra in Symplectic Algebraic Geometry

## Polynomial Methods in Learning and Statistics

My goal in this talk is to survey some of the emerging applications of polynomial methods in both learning and in statistics. I will give two examples from my own work in which the solution to well-studied problems in learning and statistics can be best understood through the language of algebraic geometry.

## Local Correction of Codes and Euclidean Incidence Geometry

A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and related ones), which are motivated from questions about locally correctable codes and matrix rigidity. The proofs use an interesting combination of combinatorial, algebraic and analytic tools.

Joint work with Boaz Barak, Zeev Dvir and Amir Yehudayoff