Titles and Abstracts

Keynote Speakers

Sami Assaf: "Multiplication by boxes: an invitation to algebraic combinatorics"

One of the central ideas in algebraic combinatorics is to represent polynomials in many variables by combinatorial containers that can hold many values. The quintessential example is the representation of Schur polynomials, the characters of irreducible modules for the general linear group, by Young tableaux, arrays of boxes filled with positive integers. In this talk I’ll survey different polynomials with nice tableaux representations and show how the combinatorics can lead to elegant solutions to difficult problems.

Jozsef Balogh: "The method of hypergraph containers"

We will give a gentle introduction to a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. In the first half of the talk we will attempt to convey a general high-level overview of the method; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry.

Robin Pemantle: "Combinatorics, Probability, and Analysis"

Combinatorics students sometimes wonder how deep the connections are between combinatorics and other branches of mathematics. Beyond representation theory, what are the most prominent connections? In this talk I will discuss several connections that arise through generating functions.

First, the relations between where the zeros of a generating function are, or are not, and the behavior of the coefficients, yield surprising results relating to classical combinatorial properties (unimodality, log-concavity) as well as probabilistic properties such as central limit theorems and negative association.

Second, I will discuss the use of generating functions in exactly solvable models in probability and statistical mechanics. Probability limit theorems, not provable by any other means, can be established via generating functions and ACSV (analytic combinatorics in several variables).

Student Presentations

Student presentations may be found below. If you notice a mistake or need to make a correction, please let us know.