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.

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.

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.

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.

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.

High-Confidence Predictions under Adversarial Uncertainty

Andrew Drucker
Massachusetts Institute of Technology
February 13, 2012
We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x . Our main focus is the problem of successfully predicting a single 0 from among the bits of x . In our model we get just one chance to make a prediction, at a time of our choosing. This models a variety of situations in which we need to perform an action of fixed duration, and must predict a "safe" time-interval to perform it.

On the Colored Tverberg Problem

Benjamin Matschke
Institute for Advanced Study
February 14, 2012
In this talk I will present a colored version of Tverberg's theorem about partitioning finite point sets in R^d into rainbow groups whose convex hulls intersect. This settles the famous Bárány-Larman conjecture (1992) for primes minus one, and asymptotically in general. It implies some complexity estimates in computational geometry. I will also give some generalizations that connect to mass partition and center point theorems. The proofs use equivariant topology, which I will try to keep as elementary as possible. This is joint work with Pavle V.M.