Georgia Institute of Technology College of Sciences

Markov chain Monte Carlo (MCMC) methods are state-of-the-art techniques for numerical integration. MCMC methods yield estimators that converge to integrals of interest in the limit of the number of iterations, obtained from Markov chains that converge to stationarity. This iterative asymptotic justification is not ideal. Indeed the literature offers little practical guidance on how many iterations should be performed, despite decades of research on the topic. This talk will describe a computational approach to address some of these issues. The key idea, pioneered by Glynn and Rhee in 2014, is to generate couplings of Markov chains, whereby pairs of chains contract, coalesce or even "meet" after a random number of iterations; we will see that these meeting times, which can be simulated in many practical settings, contain useful information about the finite-time marginal distributions of the chains. This talk will provide an overview of this line of research, joint work with John O'Leary, Yves Atchadé and various collaborators.
The main reference is available here: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12336

Two knots are concordant to each other if they cobound an annulus in the product of S^3. We will discuss a few basic facts about knot concordance and look at J. Levine’s classical result on the knot concordance group.

Freiman's theorem characterizes finite subsets of abelian groups that behave "approximately" like subgroups: any such set is (roughly) a sum of arithmetic progressions and a finite subgroup. Quantifying Freiman's theorem is an important area of additive combinatorics; in particular, proving a "polynomial" Freiman theorem would be extremely useful.

The "Helfgott-Lindenstrauss conjecture" describes the structure of finite subsets of non-abelian groups that behave approximately like subgroups: any such set is (roughly) a finite extension of a nilpotent group. Breuillard, Green, and Tao proved a qualitative version of this conjecture. In general, a "polynomial" version of the HL conjecture cannot hold, but we prove that a polynomial version of the HL conjecture is true for linear groups of bounded rank.

In this talk, we will see how the "sum-product phenomenon" and its generalizations play a crucial role in the proof of this result. The amount of group theory needed is minimal.

This talk's recording is available here.

The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will explain their geometric origin and discuss their algebraic and geometric combinatorics.

The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.

The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.