\indent We use toric symmetry and degeneration to study the GW theory of $X = {\bf P^1} \times {\bf P^1} \times {\bf P^1}$ . There exists a toric blowup $Y$ of $X$ whose polytope is the permutohedron. The permutohedron admits a symmetry which is manifest in the GW theory of $Y$. We use degeneration to show this symmetry descends to $X$ itself. This also shows a subtle relationship between the GW theories of ${\bf P^1} \times {\bf P^1} \times {\bf P^1}$ and ${\bf P^3}$. All this is joint work with Dhruv
Ranganathan.

\indent Starting with $n$ particles at the origin in $Z^d$, let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a ball. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the ball, answering a question posed by Lawler in 1995 and confirming a prediction made by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.

\indent I will first review the notion of Galois averages of Rankin-Selberg $L$-functions, in particular those of Rankin-Selberg $L$-functions of weight-two cusp forms times theta series associated to Hecke characters of imaginary quadratic fields. I will then present a conjecture about the behaviour of these averages with the conductor of the character, of which the nonvanishing theorems of Rohrlich, Vatsal and Cornut-Vatsal are special cases. Finally, I will explain a strategy of proof, at least in the setting where the class number is equal to one.

\indent This talk discusses the derivation and analysis of mathematical models motivated by the experimental induction of contour phosphenes in the retina. First, a spatially discrete chain of periodically forced coupled oscillators is considered via reduction to a chain of scalar phase equations. Each isolated oscillator locks in a 1-2 manner with the forcing, so there is intrinsic bistability, with activity peaking on either the odd or even cycles of the forcing. If half the chain is started on the odd cycle and half on the even cycle ("split state"), then with sufficiently strong coupling a wave can be produced which can travel in either direction due to symmetry. Numerical and analytic methods are employed to determine the size of coupling necessary for the split state solution to destabilize such that waves appear. Next we take a continuum limit, reducing the chain to a partial differential equation. We use a Melnikov function to compute, to leading order, the speed of the traveling wave solution to the partial differential equation as a function of the form of coupling and the forcing parameters and compare our result to numerically computed discrete and continuum wave speeds. This is joint work with Bard Ermentrout and Jonathan Rubin, published in Physica D volume 240, issue 7 as a paper under the same name.

\indent A tropical curve is a vertex-weighted metric graph. It is hyperelliptic if it admits an involution whose quotient is a tree. Assuming no prior knowledge of tropical geometry, I will develop the theory of tropical hyperelliptic curves and discuss the relationship with classical algebraic curves and their Berkovich skeletons. Along the way, we will see some nice combinatorics, including an analogue for graphs of holomorphic maps of Riemann surfaces.

\indent We discuss the challenge of understanding the WILD representation theory of Lie algebras over fields of positive characteristic. Even very explicit examples lead to difficult, if not impossible, problems. One can make some computations, but how does one give structure to these computations? Recent joint work with Julia Pevtsova introduces algebro-geometric invariants for such representations, an approach which leads to algebraic vector bundles on familiar (and not so familiar) algebraic varieties.