I am an assistant professor of mathematics at Worcester Polytechnic Institute, and a mathematical consultant for Google DeepMind.

My research interests are machine learning, combinatorics, graph theory.

Most of my past work has been in combinatorics and graph theory, focusing on central questions in these areas and their connections to other fields such as theoretical computer science and number theory.

In recent years the focus of my work has heavily shifted towards finding new and fun ways to use computers in mathematics research. This includes using reinforcement learning methods so programs can learn disprove open conjectures by themselves, exploring how generative ML methods can be used to attack open problems, and other ways machine learning can be used to guide our intuition.

My full CV.

## Selected talks with videos

### Finding counterexamples to conjectures via reinforcement learning

We use a simple reinforcement learning setup to find explicit constructions and counterexamples to several open conjectures in extremal combinatorics and graph theory. Amongst the conjectures we refute are a question of Brualdi and Cao about maximizing permanents of pattern avoiding matrices, and several problems related to the adjacency and distance eigenvalues of graphs. You do not need to know much about machine learning or maths to be able to follow this talk, I explain everything from the basics!

Watch Talk

### Finding counterexamples to conjectures via reinforcement learning

This is a short, 20 minute version of the talk above. If you don't have much time, this is the talk for you! I don't go into much detail of the setup at all, and you can enjoy some insights from Yann LeCun into automated reasoning right after the talk.

Watch Talk (Day 1, at 1:51:30)

### The largest projective cube-free subsets of $Z_{2^n}$

What is the largest subset of $Z_{2^n}$ that doesn’t contain a projective $d$-cube? In the Boolean lattice, Sperner’s, Erdos’s, Kleitman’s and Samotij’s theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $Z_{2^n}$ we work in $Z_{2^n}$, analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. The largest $d$-cube-free subset of $Z_{2^n}$, if $d$ is not a power of two, exhibits a much more interesting behaviour.

Watch Talk

## Tutorials with code

I have made several tutorials on using machine learning methods in mathematical research. Let me know if you have any suggested improvements, or if there are any related topics that I should do a tutorial on!

### Getting started with saliency analysis

Saliency analysis is a powerful tool to finding connections between parameters of some objects. Imagine you want to write a code that counts the number of 4-cycles in a graph. There is a simple $O(n^4)$ time algorithm by checking every possibility, but you have an inkling that this could be done better! Can we use the power of supervised learning to find a faster algorithm for computing the number of 4-cycles, and speed up our code significantly?

View Tutorial

### The basics of reinforcement learning

We work through a simple reinforcement learning setup, and use it to on the following toy problem: at most how many edges can a triangle-free graph have? Once we have settled this question, we have a brief look at how RL can be useful in real maths research: we generalize our previous theorem in a very natural way, that is "clearly" correct -- but before spending a few weeks rigorously proving it, we do a quick sanity check with RL. Luckily it finds a counterexample and we have saved ourselves quite a bit of time.

View Tutorial

Coming soon!