Georgia Institute of Technology College of Sciences

Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer $s$, there exists a positive integer $n$ such that for every partition of the edges of the complete graph on $n$ vertices into two classes, one of the classes must contain a complete subgraph on $s$ vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest $n$ that satisfies this property.

For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that "pseudo-randomness" and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

Graph burning is a simplified model for the spread of influence in a network. Associated with the process is the burning number, which quantifies the speed at which the influence spreads to every vertex. The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil$. While the conjecture remains open, we prove the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) \le \sqrt{4n/3} + 1$, improving on the previously known $\sqrt{3n/2}+O(1)$ bound.

The famous Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will briefly sketch a proof of this conjecture for every large n. If time permits, I will also talk about our solution to a problem of Erdős from 1977 about chromatic index of hypergraphs with bounded codegree. Joint work with D. Kang, T. Kelly, D.Kuhn and D. Osthus.

Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the "game" of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.