If G is a group acting on a vector space V by linear transformations, then the invariant polynomial functions on V form a ring. In this talk we will discuss upper bounds for the degrees of generators of this invariant ring. An example of particular interest is the action of the group $SL_n x SL_n$ on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. In this case, Visu Makam and the speaker recently proved that invariants of degree at most $mn^4$ generate the invariant ring. We will explore an interesting connection between this result and the notion of noncommutative rank.

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a 'cute' eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories.

The hook length formula for d-complete posets states that the P-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using q-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the P-partition generating function for each case as a q-integral and then we evaluate the q-integrals. Several q-integrals are evaluated using partial fraction expansion identities and others are verified by computer. This is joint work with Meesue Yoo.

Schur Q-functions are a family of symmetric functions introduced by Schur in his study of projective representations of symmetric groups. They are obtained by putting t = -1 in the Hall-Littlewood functions associated to the root system of type A. (Schur functions are the t = 0 specialization.) This talk concerns symplectic Q-functions, which are obtained by putting t = -1 in the Hall-Littlewood functions associated to the root system of type C. We present several Pfaffian formulas for symplectic Q-functions similar to those for Schur Q-functions and give a tableau description. Also we discuss some conjectures including the positivity conjecture of structure constants.

We study tensor powers of rank 1 Drinfeld A-modules, where A is
the affine coordinate ring of an elliptic curve. Using the theory of
A-motives, we find explicit formulas for the A-action of these modules.
Then, by developing the theory of vector valued Anderson generating
functions, we give formulas for the coefficients of the logarithm and
exponential functions associated to these A-modules, as well as formulas
for the fundamental period. This allows us to relate function field zeta
values to evaluations of the logarithm function and prove transcendence
facts about these zeta values.

In the 1970's Stark made precise conjectures about the leading
term of the Taylor series expansion at $s=0$ of Artin $L$-functions,
refining Dirichlet's class number formula. Around the same time Barsky,
Cassou-Nogu\`{e}s, and Deligne and Ribet for totally real fields, along
with Katz for CM fields defined $p$-adic $L$-functions of ray class
characters. Since then Stark-type conjectures for these $p$-adic
$L$-functions have been formulated, and progress has been made in some
cases.

The goal of this talk is to discuss a new definition of a $p$-adic
$L$-function and Stark conjecture for a mixed signature character of a real
quadratic field. After stating the definition and conjecture, theoretical
and numerical evidence will be discussed.

When studying supersymmetric equations from physics, Tseng and Yau introduced several Laplacians on symplectic manifolds in 2012. These Laplacians behave different from usual ones in Rimannian case and Complex case. They contain both 2nd and 4th order operators. In this talk, we will discuss these ``symplectic Laplacians'' and their relations with cohomologies on compact symplectic manifolds with boundary. For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds. Their properties and importance will be discussed.

I'll survey a circle of ideas around the question of measuring ``how irrational'' are various classes of non-rational varieties.

A drawing of a graph on a surface is independently even if every pair of independent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most $2$ common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on surfaces.