Alice Mark- Vanderbilt University

We will talk about immediate thinks like planning for next semester, but we will also talk about documenting your teaching and why you might want to.

Madeline Brandt- Brown University

Moduli spaces offer a geometric solution to geometric classification problems by parameterizing all objects of some type. Tropical versions of these spaces explain the combinatorics of their compactifications. Moreover, these tropical moduli spaces can be used to compute a piece of the cohomology of the corresponding classical moduli space. In joint work with Melody Chan and Siddarth Kannan, we study the topology of the moduli space of hyperelliptic curves using these techniques.

Patrick Heslin, Maynooth University – Ireland

Arnold’s celebrated 1966 paper illustrates that the Euler equations of ideal hydrodynamics arise naturally, from the perspective of particle trajectories, as a geodesic equation on the group of smooth volume-preserving diffeomorphisms of the fluid domain equipped with a right-invariant metric corresponding to the fluid’s kinetic energy.

In this talk we will investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation, a well-known model for the three-dimensional Euler equations.

We will see that these equations can also be realised as geodesic equations on groups of diffeomorphisms. We will then illustrate precisely when the corresponding exponential map is non-linear Fredholm of index 0. Finally, we will examine the distribution of conjugate points in these settings via a Morse theoretic argument.

Nate Tenpas – Vanderbilt University

We’ll show how recently introduced linear programming bounds can be used to show the optimality of certain lattice configurations in $\mathbb{R}^2$ among a large class of periodic configurations. Some of the lattice configurations address the important conjecture that the hexagonal lattice is universally optimal, while others provide a new, and largest of its kind, class of periodic energy minimizers which are not obtained from a universally optimal lattice.

Kevin Boucher – University of Southampton