Office C0.03

Centre for Mathematical Sciences

Cambridge CB3-0WA

United Kingdom

I am a third year PhD student in the Department of Pure Mathematics and Mathematical Statistics at Cambridge, supervised by Professor Béla Bollobás.

I am broadly interested in combinatorics, number theory, probability theory and related areas in statistical physics and theoretical computer science.

Recently, I have been involved with the study of synchronisation phenomena. My work on synchronisation has been recently featured in Quanta magazine.

We show that for any $p < 1$, Breaker almost surely has a winning strategy for the $(1,1)$ game on $(\mathbb{Z}^2)_p$. This fully answers a question of Day and Falgas-Ravry, who showed that for $p = 1$ Maker has a winning strategy for the $(1,1)$ game. Further, we show that in the $(2,1)$ game on $(\mathbb{Z}^2)_p$ Maker almost surely has a winning strategy whenever $p > 0.9402$, while Breaker almost surely has a winning strategy whenever $p < 0.5278$. This shows that the threshold value of $p$ above which Maker has a winning strategy for the $(2,1)$ game on $\mathbb{Z}^2$ is non-trivial. In fact, we prove similar results in various settings, including other lattices and biases $(m,b)$.

These results extend also to the most general case, which we introduce, where each edge is given to Maker with probability $\alpha$ and to Breaker with probability $\beta$ before the game starts.

Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the $(m,2m)$ game even if he allows Maker to claim $c$ edges before the game starts, for any integer $c$, and that he can moreover win rather quickly as a function of $c$.

We also consider the game played on the so-called polluted board, obtained after performing Bernoulli bond percolation on $\mathbb{Z}^2$ with parameter $p$. We show that for the $(1,1)$ game on the polluted board, Breaker almost surely has a winning strategy whenever $p \leq 0.6298$.

- 2023 University of Cambridge - Supervisor Part II Graph Theory.
- 2022 University of Cambridge - Supervisor Part II Number Theory.
- 2021 University of Cambridge - Supervisor Part II Number Theory.
- 2021 University of Cambridge - Supervisor Part IA Analysis I.
- 2020 University of Cambridge - Supervisor Part II Number Theory.
- 2017 IMPA - Teacher Assistant in Probability I.