# The Ivan and Betty Niven Distinguished Lectures

## Lecture #1 (Student Lecture)

### Monodromy groups

A monodromy group is the group of symmetries of a family of objects which are "locally isomorphic". These groups frequently occur in several areas of mathematics, including algebra, topology and number theory. In many situations one would like to show that the monodromy group is "as large as possible", subject to the obvious constraints. We will consider several examples of the monodromy problem in different flavors: local versus global, p-adic versus l-adic. The first examples are from Galois theory with finite monodromy groups. In other examples from number theory the monodromy groups will be infinite.

## Lecture #2 (Number theory / Algebraic geometry seminar)

### Canonical coordinates for leaves of p-divisible groups

Let p be a prime number and g be a positive integer. Let M be the moduli space of abelian varieties of PEL type. A leaf in M is the locus corresponding to a fixed isomorphism class of polarized p-divisible group with prescribed endomorphisms. Although a leaf is defined by a "pointwise" condition, it turns out that the formal completion (or jet space) of a leaf at a point has a rigid structure: It is built up from a finite collection of p-divisible formal groups via a family of fibrations. This structural description can be regarded as a generalization of the Serre-Tate coordinates of the local deformation space of a ordinary abelian variety. We also explain a local rigidity result related to the action of the local stabilizer subgroup on the canonical coordinates.

## Lecture #3 Hecke orbits (Colloquium)

### Friday, March 10, 3:00-4:00, MATX 1100

Let p be a prime number and g be a positive integer. Let A_g be the moduli space of g-dimensional principally polarized abelian varieties over the algebraic closure of the finite field with p elements. There is a family of finite-to-one correspondences on this moduli space generated by prime-to-p isogenies between abelian varieties; these symmetries are known as Hecke correspondences. Recently F. Oort defined a family of locally closed smooth algebraic subvarieties of A_g, called "leaves"; a leaf is the locus in A_g corresponding to a fixed isomorphism class of polarized p-divisible group. Clearly every leaf is stable under all prime-to-p Hecke correspondences. Oort conjectured that every Hecke orbit is dense in the leaf containing it. We will explain methods motivated by this conjecture, and how these methods can be used to prove the conjecture.