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## Upcoming Seminars

### Wed May 30, 2018

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.

### Thu May 31, 2018

We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over **Q**, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that E/**F** is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if **F** is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have 3-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Robert Lemke Oliver, and Ari Shnidman.

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.

### Fri Jun 1, 2018

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.