Georgia Institute of Technology College of Sciences

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.

Algebraic Combinatorics originated in Algebra and Representation Theory, yet its objects and methods turned out applicable to other fields from Probability to Computer Science. Its flagship hook-length formula for the number of Standard Young Tableaux, or the beautiful Littlewood-Richardson rule have inspired large areas of study and development. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity and Asymptotics. We will also show how an 80 year old open problem on Kronecker coefficients of the Symmetric group lead to the disprove of the wishful approach towards the resolution of the algebraic P vs NP Millennium problem.

Two of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erdős in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we highlight a few important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas.