Members Seminar

Parallel Repetition of Two Prover Games: A Survey

Ran Raz
Weizmann Institute; Member, School of Mathematics
October 8, 2012

 

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

Andras Stipsicz
Renyi Institute of Mathematics, Hungarian Academy of Sciences
April 9, 2012

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

Andras Stipsicz
Renyi Institute of Mathematics, Hungarian Academy of Sciences
April 9, 2012

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.

Local Correction of Codes and Euclidean Incidence Geometry

Avi Wigderson
Institute for Advanced Study
March 5, 2012

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

Toward Enumerative Symplectic Topology

Aleksey Zinger
SUNY, Stony Brook;Institute for Advanced Study
February 6, 2012
Enumerative geometry is a classical subject often concerned with enumeration of complex curves of various types in projective manifolds under suitable regularity conditions. However, these conditions rarely hold. On the other hand, Gromov-Witten invariants of a compact symplectic manifold are certain virtual counts of J-holomorphic curves. These rational numbers are rarely integer, but are generally believed to be related to some integer counts.

Members Seminar: The Role of Symmetry in Phase Transitions

Tom Spencer
Professor, School of Mathematics, Institute for Advanced Study
January 23, 2012

This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.

Strong and Weak Epsilon Nets and Their Applications

Noga Alon
Tel Aviv University; Institute for Advanced Study
November 7, 2011

I will describe the notions of strong and weak epsilon nets in range spaces, and explain briefly some of their many applications in Discrete Geometry and Combinatorics, focusing on several recent results in the investigation of the extremal questions that arise in the area, and mentioning some of the remaining open problems.