Georgia Institute of Technology College of Sciences

How complicated can successive manifolds get in a tower of covering
spaces? Specifically, how large can the dimension of the first
cohomology get? We will begin with a tour of possible behaviors for
low-dimensional spaces, and then focus on arithmetic manifolds.
Specifically, for towers of complex-hyperbolic manifolds, I will
describe how to bound the rates of growth using known instances of
Langlands functoriality.

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

The theory of graph quasirandomness studies graphs that "look like" samples of the Erdős--Rényi
random graph $G_{n,p}$. The upshot of the theory is that several ways of comparing a sequence with
the random graph turn out to be equivalent. For example, two equivalent characterizations of
quasirandom graph sequences is as those that are uniquely colorable or uniquely orderable, that is,
all colorings (orderings, respectively) of the graphs "look approximately the same". Since then,
generalizations of the theory of quasirandomness have been obtained in an ad hoc way for several
different combinatorial objects, such as digraphs, tournaments, hypergraphs, permutations, etc.

The theory of graph quasirandomness was one of the main motivations for the development of the
theory of limits of graph sequences, graphons. Similarly to quasirandomness, generalizations of
graphons were obtained in an ad hoc way for several combinatorial objects. However, differently from
quasirandomness, for the theory of limits of combinatorial objects (continuous combinatorics), the
theories of flag algebras and theons developed limits of arbitrary combinatorial objects in a
uniform and general framework.

In this talk, I will present the theory of natural quasirandomness, which provides a uniform and
general treatment of quasirandomness in the same setting as continuous combinatorics. The talk will
focus on the first main result of natural quasirandomness: the equivalence of unique colorability
and unique orderability for arbitrary combinatorial objects. Although the theory heavily uses the
language and techniques of continuous combinatorics from both flag algebras and theons, no
familiarity with the topic is required as I will also briefly cover all definitions and theorems
necessary.

This talk is based on joint work with Alexander A. Razborov.

A central question in the field of optimal transport studies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimal transport maps on a Euclidean metric space are Lipschitz. Lipschitz regularity is a desirable property because it allows for the transfer of analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimal transport.

In this talk, we will further explore the Lipschitz properties of transport maps. Our main observation is that, when one seeks Lipschitz mappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction of transport maps based on interpolation of measures, and introduce a set of techniques to analyze the Lipschitz constant of this construction. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces.

Some applications, such as functional inequalities, normal approximations, and generative diffusion models will also be discussed.