Georgia Institute of Technology College of Sciences

In his seminal 1991 paper, Thomas M. Cover introduced a simple and elegant mathematical model for trading on the stock market. This model, which later on came to be known as  online portfolio selection (OPS), is specified with only two integer parameters: the number of assets $d$ and time horizon $T$. In each round $t \in \{1, ..., T\}$, the trader selects a  portfolio--distribution $p_t \in R^d_+$ of the current capital over the set of $d$ assets; after this, the adversary generates a nonnegative vector $r_t \in R^d_+$ of returns (relative prices of assets), and the trader's capital is multiplied by the "aggregated return'' $\langle p_{t}, r_{t} \rangle$. Despite its apparent simplicity, this model captures the two key properties of the stock market: (i) it "plays against'' the trader; (ii) money accumulates multiplicatively. In the 30 years that followed, the OPS model has received a great deal of attention from the learning theory, information theory, and quantitative finance communities.

In the same paper, Cover also proposed an algorithm, termed Universal Portfolios, that admitted a strong performance guarantee: the regret of $O(d \log (T))$ against the best portfolio in hindsight, and without any restrictions of returns or portfolios. This guarantee was later on shown to be worst-case optimal, and no other algorithm attaining it has been found to date. Unfortunately, exact computation of a universal portfolio amounts to averaging over a log-concave distribution, which is a challenging task. Addressing this, Kalai and Vempala (2002) achieved the running time of $O(d^4 T^{14})$ per round via log-concave sampling techniques. However, with such a running time essentially prohibiting all but "toy'' problems--yet remaining state-of-the-art--the problem of finding an optimal and practical OPS algorithm was left open.

In this talk, after discussing some of the arising challenges, I shall present a fast and optimal OPS algorithm proposed in a recent work with R. Jezequel and P. Gaillard (arXiv:2209.13932). Our algorithm combines regret optimality with the runtime of $O(d^2 T)$, thus dramatically improving state of the art. As we shall see, the motivation and analysis of the proposed algorithm are closely related to establishing a sharp bound on the accuracy of the Laplace approximation for a log-concave distribution with a polyhedral support, which is a result of independent interest.

Zoom link to the talk: https://gatech.zoom.us/j/98280978183

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.