Georgia Institute of Technology College of Sciences

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will first introduce the Kahn-Kalai Conjecture with some motivating examples and then talk about the recent resolution of a fractional version of the Kahn-Kalai Conjecture due to Frankston, Kahn, Narayanan, and myself. Some follow-up work, along with open questions, will also be discussed.

Szemerédi's regularity lemma is a game-changer in extremal combinatorics and provides a global perspective to study large combinatorial objects. It has connections to number theory, discrete geometry, and theoretical computer science. One of its classical applications, the removal lemma, is the essence for many property testing problems, an active field in theoretical computer science. Unfortunately, the bound on the sample size from the regularity method typically is either not explicit or enormous. For testing natural permutation properties, we show one can avoid the regularity proof and yield a tester with polynomial sample size. For graphs, we prove a stronger, "L_\infty'' version of the graph removal lemma, where we conjecture that the essence of this new removal lemma for cliques is indeed the regularity-type proof. The analytic interpretation of the regularity lemma also plays an important role in graph limits, a recently developed powerful theory in studying graphs from a continuous perspective. Based on graph limits, we developed a method combining with both analytic and spectral methods, to answer and make advances towards some famous conjectures on a common theme in extremal combinatorics: when does randomness give nearly optimal bounds? 

These works are based on joint works with Jacob Fox, Dan Kral',  Jonathan Noel, Sergey Norin, and Jan Volec.

Graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This problem can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For correlated Erdős-Rényi random graphs, we will give insights on the fundamental limits for the planted formulation of this problem, establishing statistical thresholds for partial recovery. From the computational point of view, we are interested in designing and analyzing efficient (polynomial-time) algorithms to recover efficiently the underlying alignment: in a sparse regime, we exhibit an local rephrasing of the planted alignment problem as the correlation detection problem in trees. Analyzing this related problem enables to derive a message-passing algorithm for our initial task and gives insights on the existence of a hard phase.

Based on joint works with Laurent Massoulié and Marc Lelarge: 

https://arxiv.org/abs/2002.01258

https://arxiv.org/abs/2102.02685

https://arxiv.org/abs/2107.07623

Given a multigraph $G=(V,E)$, the chromatic index $\chi'(G)$ is the minimum number of colors needed to color the edges of $G$ such that no two incident edges receive the same color. Let $\Delta(G)$ be the maximum degree of $G$ and let  $\Gamma(G):=\max \big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm
{\rm and \hskip 2mm odd} \big\}$. $\Gamma(G)$ is called the density of $G$. Clearly, the density is a lower bound for the chromatic index $\chi'(G)$. Moreover, this value can be computed in polynomial time. Goldberg and Seymour in the 1970s conjectured that $\chi'(G)=\lceil\Gamma(G)\rceil$ for any multigraph $G$ with $\chi'(G)\geq\Delta(G)+2$, known as the Goldberg-Seymour conjecture. In this talk we will discuss this conjecture and some related open problems. This is joint work with Guantao Chen and Wenan Zang.