Georgia Institute of Technology College of Sciences

Thomassen conjectures that every graph of sufficiently large average degree has a subgraph of average degree at least d and girth at least k, for any d and k. What if we want the subgraph to be induced? Large cliques and bicliques are the obvious obstructions; we conjecture there are no others. We survey results in this direction, and we prove that every bipartite graph of sufficiently large average degree has either K_{d,d} or an induced subgraph of average degree at least d and girth at least 6.

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.

It has been conjectured that for a sufficiently large $n$, and $p = p_n \in [\log(n)/n, 1/2)$, the probability that a $n\times n$ Bernoulli($p$) matrix $A$ is singular equals to the probability that $A$ contains of a zero row or zero column up to a negligible error.

This conjecture has been recently proved by Litvak-Tikhomirov in the regime $ C\log(n)/ n < p < 1/C$ for some universal constant $C>1$ with their new tool. While for $p = (1+o(1)) \log(n) /n$, it also holds due to a result of Basak-Rudelson. In this talk, we will discuss how to extend their results to fill the gap between these two regions. ( $1\le pn/\log(n) <\infty$ )

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

Optimization and machine learning algorithms often use real-world data that has been generated through complex socio-economic and behavioral processes. This data, however, is noisy, and naturally encodes difficult-to-quantify systemic biases. In this work, we model and address bias in the secretary problem, which has applications in hiring. We assume that utilities of candidates are scaled by unknown bias factors, perhaps depending on demographic information, and show that bias-agnostic algorithms are suboptimal in terms of utility and fairness. We propose bias-aware algorithms that achieve certain notions of fairness, while achieving order-optimal competitive ratios in several settings.

An n-lift of a graph G is a graph from which there is an n-to-1 covering map onto G. Amit, Linial, and Matousek (2002) raised the question of whether the chromatic number of a random n-lift of K_5 is concentrated on a single value. We consider a more general problem, and show that for fixed d ≥ 3 the chromatic number of a random lift of K_d is (asymptotically almost surely) either k or k+1, where k is the smallest integer satisfying d < 2k log k. Moreover, we show that, for roughly half of the values of d, the chromatic number is concentrated on k. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random n-lifts of G, for any fixed d-regular graph G. (This is joint work with JD Nir.)