Two permutations are conjugate if and only if they have the same cycle structure, and two complex unitary matrices are conjugate if and only if they have the same set of eigenvalues. Motivated by the large literature on cycles of random permutations and eigenvalues of random unitary matrices, we study  conjugacy classes of random elements of finite classical groups. For the case of GL(n,q), this amounts to studying rational canonical forms. This leads naturally to a probability measure on the set of all partitions of all natural numbers. We connect this measure to symmetric function theory, and give algorithms for generating partitions distributed according to this measure. We describe analogous results for the other finite classical groups (unitary, symplectic, orthogonal). We were excited to learn that (at least for GL(n,q)), exactly the same random partitions arise in the “Cohen-Lenstra heuristics” of number theory.

A classical problem in Galois theory is a strong variant of
the Inverse Galois Problem: "What finite groups arise as the Galois
group of a finite Galois extension of the rational numbers, if you
impose the additional condition that the extension can only ramify at
finite set of primes?" This question is wide open in almost every
nonabelian case, and one reason is our lack of knowledge about how to
find number fields with prescribed ramification at fixed primes. While
such fields are often constructed to answer arithmetic questions,
there is currently no known way to systematically construct such
extensions in full generality.

However, there are some inspiring programs that are gaining ground on
this front. One method, initiated by Chenevier, is to construct such
number fields using Galois representations and their associated
automorphic representations via the Langlands correspondence. We will
explain the method, show how some recent advances in these subfields
allow us to gain some additional control over the number fields
constructed, and indicate how this brings us closer to our goal. As a
application, we will show how one can use this knowledge to study the
arithmetic of curves over number fields.

Motivated from his p-adic study of the variation of the zeta function as
the variety moves through a family, Dwork conjectured that a new type of
L-function, the so-called unit root L-function, was always p-adic
meromorphic. In the late 1990s, Wan proved this using the theory of
sigma-modules, demonstrating that unit root L-functions have structure.
Little more is known.

This talk is concerned with unit root L-functions coming from families of
exponential sums. In this case, we demonstrate that Wan's theory may be
used to extend Dwork's theory -- including p-adic cohomology -- to these
L-functions. To illustrate the technique, the unit root L-function of the
Kloosterman family is studied in depth.

We study the monodromy of a certain class of semistable hyperelliptic curves over the rationals that was introduced by Shigefumi Mori forty years ago (before his Minimal Model Program). Using ideas of Chris Hall, we prove that the corresponding $\ell$-adic monodromy groups are (almost) ``as large as possible". We also discuss an explicit construction of two-dimensional families of hyperelliptic curves over an arbitrary global field with big monodromy.

Eigencurve was introduced by Coleman and Mazur to parametrize
modular forms varying p-adically. It is a rigid analytic curve such that
each point corresponds to an overconvegent eigenform. In this talk, we
discuss a result on the geometry of the eigencurve: over the boundary
annuli of the weight space, the eigencurve breaks up into infinite disjoint
union of connected components and the weight map is finite and flat on each
component. This was first observed by Buzzard and Kilford by an explicit
computation in the case of p=2 and tame level 1. We will explain a
generalization to the definite quaternion case with no restriction on p and
the tame level. This is a joint work with Ruochuan Liu and Daqing Wan,
based on an idea of Robert Coleman.