Members Seminar

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.

The Mathematical Challenge of Large Networks

László Lovász
Eotvos Lorand University, Budapest; Institute for Advanced Study
October 24, 2011

It is becoming more and more clear that many of the most exciting structures of our world can be described as large networks. The internet is perhaps the foremost example, modeled by different networks (the physical internet, a network of devices; the world wide web, a network of webpages and hyperlinks). Various social networks, several of them created by the internet, are studied by sociologist, historians, epidemiologists, and economists. Huge networks arise in biology (from ecological networks to the brain), physics, and engineering.

First Steps in Symplectic Dynamics

Helmut Hofer
Institute for Advanced Study
September 26, 2011

The modern theory of dynamical systems, as well as symplectic geometry, have their origin with Poincare as one field with integrated Ideas. Since then these fields developed quite independently. Given the progress in these fields one can make a good argument why the time is ripe to bring them closer together around the core area of Hamiltonian dynamics