Members Seminar

Self-Avoiding Walk and Branched Polymers

John Imbrie
University of Virginia; Member, School of Mathematics
March 7, 2011

I will introduce two basic problems in random geometry. A self-avoiding walk is a sequence of steps in a d-dimensional lattice with no self-intersections. If branching is allowed, it is called a branched polymer. Using supersymmetry, one can map these problems to more tractable ones in statistical mechanics. In many cases this allows for the determination of exponents governing the relationship between the diameter and the number of steps.

Does Infinite Cardinal Arithmetic Resemble Number Theory?

Menachem Kojman
Ben-Gurion University of the Negev; Member, School of Mathematics
February 28, 2011

I will survey the development of modern infinite cardinal arithmetic, focusing mainly on S. Shelah's algebraic pcf theory, which was developed in the 1990s to provide upper bounds in infinite cardinal arithmetic and turned out to have applications in other fields.

This modern phase of the theory is marked by absolute theorems and rigid asymptotic structure, in contrast to the era following P. Cohen's discovery of forcing in 1963, during which infinite cardinal arithmetic was almost entirely composed of independence results.

Some Equations and Games in Evolutionary Biology

Christine Taylor
Harvard University; Member, School of Mathematics
February 14, 2011

The basic ingredients of Darwinian evolution, selection and mutation, are very well described by simple mathematical models. In 1973, John Maynard Smith linked game theory with evolutionary processes through the concept of evolutionarily stable strategy. Since then, cooperation has become the third fundamental pillar of evolution. I will discuss, with examples from evolutionary biology and ecology, the roles played by replicator equations (deterministic and stochastic) and cooperative dilemma games in our understanding of evolution.

Members Seminar: Linear Equations in Primes and Nilpotent Groups

Tamar Ziegler
Technion--Israel Institute of Technology
January 30, 2011
A classical theorem of Dirichlet establishes the existence of infinitely many primes in arithmetic progressions, so long as there are no local obstructions. In 2006 Green and Tao set up a program for proving a vast generalization of this theorem. They conjectured a relation between the existence of linear patterns in primes and dynamics on nilmanifolds. In recent joint work with Green and Tao we completed the final step of this program.

Moment-Angle Complexes, Spaces of Hard-Disks and Their Associated Stable Decompositions

Fred Cohen
University of Rochester; Member, School of Mathematics
January 10, 2011

Topological spaces given by either (1) complements of coordinate planes in Euclidean space or (2) spaces of non-overlapping hard-disks in a fixed disk have several features in common. The main results, in joint work with many people, give decompositions for the so-called "stable structure" of these spaces as well as consequences of these decompositions.

This talk will present definitions as well as basic properties.