In  Ramsey  Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple
points.

In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.
 

Associativity is ubiquitous in mathematics.  Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance.  But over the past few decades, associativity has blossomed and matured, appearing in theories of particle collisions, elliptic curves, and enumerative geometry.  We start with a brief look at this history, and then explore the visualization of associativity in the forms of polytopes, manifolds, and complexes. This talk is heavily infused with imagery and concrete examples.

How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from Z^d to Z^d. The cokernel of M is Z^d/Im(M). If det(M) is nonzero, then the cokernel is a finite abelian group of order det(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of Z^d.