School of Mathematics

Geometric and Numerical Approaches to KAM Theory

Rafael de la Llave
Georgia Institute of Technology
February 8, 2012
We review some recent developments in KAM theory. By exploiting some identities of a geometric nature, one can obtain iterative steps which lead to numerical algorithms and which can follow the tori till breakdown.

We present theoretical results in several contexts:
A) Persistence of non-twist tori (these are tori whose frequency map is degenerate).
B) Conformally symplectic systems (systems with friction proportional to the velocity)
C) Pre-symplectic systems
D) Some ill-posed equations such as Boussinesq equation for water waves.

Randomness Extraction: A Survey

David Zuckerman
University of Texas at Austin; Institute for Advanced Study
February 7, 2012
A randomness extractor is an efficient algorithm which extracts high-quality randomness from a low-quality random source. Randomness extractors have important applications in a wide variety of areas, including pseudorandomness, cryptography, expander graphs, coding theory, and inapproximability. In this talk, we survey the field of randomness extraction and discuss connections with other areas.

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.

Graphlets: A Spectral Perspective for Graph Limits

Fan Chung
University of California at San Diego
February 6, 2012
To examine the limiting behavior of graph sequences, many discrete methods meet their continuous counterparts, leading to numerous theoretical and applicable advancements. For dense graph sequences, the graph limits have recently been well developed by many researchers, mostly based on Szemeredi's regularity partition and the algebra of graph homomorphisms. For sparse graphs with a linear number of edges, the graph limits have very different behavior and are much less well understood.

Primes and Equations

Richard Taylor
Institute for Advanced Study
February 1, 2012

One of the oldest subjects in mathematics is the study of Diophantine equations, i.e., the study of whole number (or fractional) solutions to polynomial equations. It remains one of the most active areas of mathematics today. Perhaps the most basic tool is the simple idea of “congruences,” particularly congruences modulo a prime number. In this talk, Richard Taylor, Professor in the School of Mathematics, introduces prime numbers and congruences and illustrates their connection to Diophantine equations. He also describes recent progress in this area, an application, and reciprocity laws, which lie at the heart of much recent progress on Diophantine equations, including Wiles’s proof of Fermat’s last theorem.

CSDM: A Survey of Lower Bounds for the Resolution Proof System

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics, Institute for Advanced Study
January 31, 2012
The Resolution proof system is among the most basic and popular for proving propositional tautologies, and underlies many of the automated theorem proving systems in use today. I'll start by defining the Resolution system, and its place in the proof-complexity picture.

Symplectic Dynamics Seminar: Symplectic Structures and Dynamics on Vortex Membranes

Boris Khesin
University of Toronto; Member, School of Mathematics, Institute for Advanced Study
January 25, 2012
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on higher vortex filaments of codimension 2 in any any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D.