Georgia Institute of Technology College of Sciences

I will give a hopefully accessible introduction to some work on
tropical moduli spaces of curves and abelian varieties. I will report
on joint work with Madeline Brandt, Juliette Bruce, Margarida Melo,
Gwyneth Moreland, and Corey Wolfe, in which we find new rational
cohomology classes in the moduli space A_g of abelian varieties using
tropical techniques. And I will try to touch on a new point of view on
this topic, namely that of differential forms on tropical moduli
spaces, following the work of Francis Brown.

A meromorphic inner function is a bounded analytic function on the upper half plane with unit modulus almost everywhere on the real line and a meromorphic continuation to the complex plane. Meromorphic inner functions and equivalently meromorphic Herglotz functions play a central role in inverse spectral theory of differential operators. In this talk, I will discuss some uniqueness problems for meromorphic inner functions and their applications to inverse spectral theory of canonical Hamiltonian systems as Borg-Marchenko type results.

A \emph{list assignment} $L$ gives to each vertex $v$ in a graph $G$ a
list $L(v)$ of
allowable colors.  An \emph{$L$-coloring} is a proper coloring $\varphi$ such that
$\varphi(v)\in L(v)$ for all $v\in V(G)$.  An \emph{$L$-recoloring move} transforms
one $L$-coloring to another by changing the color of a single vertex.  An
\emph{$L$-recoloring sequence} is a sequence of $L$-recoloring moves.  We study
the problem of which hypotheses on $G$ and $L$ imply that for that every pair
$\varphi_1$ and $\varphi_2$ of $L$-colorings of $G$ there exists an $L$-recoloring
sequence that transforms $\varphi_1$ into $\varphi_2$.  Further, we study bounds on
the length of a shortest such $L$-recoloring sequence.

We will begin with a survey of recoloring and list recoloring problems (no prior
background is assumed) and end with some recent results and compelling
conjectures.  This is joint work with Stijn Cambie and Wouter Cames van
Batenburg.

We study the problem of designing a robust parameter-independent feedback control input that steers, with minimum energy, the average of a linear system submitted to parameter perturbations where the states are initialized and finalized according to a given initial and final distribution. We formulate this problem as an optimal transport problem, where the transport cost of an initial and final state is the minimum energy of the ensemble of linear systems that have started from the initial state and the average of the ensemble of states at the final time is the final state. The by-product of this formulation is that using tools from optimal transport, we are able to design a robust parameter-independent feedback control with minimum energy for the ensemble of uncertain linear systems. This relies on the existence of a transport map which we characterize as the gradient of a convex function.