Victor Seixas Souza
Klarman Fellow at Department of Mathematics
Cornell University

505 Mallot Hall
Ithaca, 14853 NY
United States

Research Fellow at Sidney Sussex College
Cambridge, CB2 3HU
United Kingdom

vsouza cornell edu
vss28 cam ac uk

I am a Klarman Fellow in the Department of Mathematics at Cornell University since July of 2025. I am delighted to be hosted by Martin Kassabov and Steven Strogatz.

I am also a Research Fellow at Sidney Sussex College, Cambridge. I previously held the position of Research Associate in the Algorithms and Complexity group of the University of Cambridge, under the brilliant mentorship of Tom Gur.

I hold a PhD from the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, where I was very fortunate of being supervised by Professor Béla Bollobás. Prior to that, I had the pleasure of being a master student of Rob Morris at IMPA surounded by the forests of Rio de Janeiro.

Research Interests

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.

Selected Publications 2025 Double-jump phase transition for the reverse Littlewood--Offord problem L. Hollom, J. Portier, and V. Souza arxiv Erdős conjectured in 1945 that for any unit vectors $v_1, \dotsc, v_n$ in $\mathbb{R}^2$ and signs $\varepsilon_1, \dotsc, \varepsilon_n$ taken independently and uniformly in $\{-1,1\}$, the random Rademacher sum $\sigma = \varepsilon_1 v_1 + \dotsb + \varepsilon_n v_n$ satisfies $\|\sigma\|_2 \leq 1$ with probability $\Omega(1/n)$. While this conjecture is false for even $n$, Beck has proved that $\|\sigma\|_2 \leq \sqrt{2}$ always holds with probability $\Omega(1/n)$. Recently, He, Juškevičius, Narayanan, and Spiro conjectured that the Erdős' conjecture holds when $n$ is odd. We disprove this conjecture by exhibiting vectors $v_1, \dotsc, v_n$ for which $\|\sigma\|_2 \leq 1$ occurs with probability $O(1/n^{3/2})$. On the other hand, an approximated version of their conjecture holds: we show that we always have $\|\sigma\|_2 \leq 1 + \delta$ with probability $\Omega_\delta(1/n)$, for all $\delta > 0$. This shows that when $n$ is odd, the minimum probability that $\|\sigma\|_2 \leq r$ exhibits a double-jump phase transition at $r = 1$, as we can also show that $\|\sigma\|_2 \leq 1$ occurs with probability at least $\Omega((1/2+\mu)^n)$ for some $\mu > 0$. Additionally, and using a different construction, we give a negative answer to a question of Beck and two other questions of He, Juškevičius, Narayanan, and Spiro, concerning the optimal constructions minimising the probability that $\|\sigma\|_2 \leq \sqrt{2}$. We also make some progress on the higher dimensional versions of these questions. 
@MISC{HPS-25-reverse-lo,
    title = {Double-jump phase transition for the reverse Littlewood--Offord problem},
    author = {Lawrence Hollom and Julien Portier and Victor Souza},
    archivePrefix = {arXiv},
    primaryClass = {math.CO},
    eprint = {2503.24202},
    doi = {https://arxiv.org/abs/2503.24202},
}
2024 The Maker-Breaker percolation game on a random board V. Dvořák, A. Mond, and V. Souza To appear in the Annals of Probability arxiv The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After that, Maker chooses a vertex $v_0$ to protect. Then, in each round Maker and Breaker claim respectively $m$ and $b$ unclaimed edges of $G$. Breaker wins if after the removal of the edges claimed by him the component of $v_0$ becomes finite, and Maker wins if she can indefinitely prevent Breaker from winning.

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. 
@MISC{DMS-24-mb-random,
    title = {The Maker-Breaker percolation game on a random board},
    author = {Vojtěch Dvořák and Adva Mond and Victor Souza},
    archivePrefix = {arXiv},
    primaryClass = {math.PR},
    eprint = {2402.17547},
    doi = {https://arxiv.org/abs/2402.17547},
}
2023 On the number of monochromatic solutions to multiplicative equations L. Aragão, J. Chapman, M. Ortega, and V. Souza To appear in Combinatorica arxiv Given an $r$-colouring of the interval $\{2, \dotsc, N \}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always asymptotically at least $(1/2\sqrt{2}) N^{1/2} \log N$ monochromatic solutions, and that the leading constant is sharp. We also establish a stability version of this result. For general $r$, we show that there are at least $C_r N^{1/S(r-1)}$ monochromatic solutions, where $S(r)$ is the Schur number for $r$ colours and $C_r$ is a constant. This bound is sharp up to logarithmic factors when $r \leq 4$. We also obtain results for more general multiplicative equations of the form $x_1^{a_1} \dotsb x_k^{a_k} = y$, where $a_1, \dotsc, a_k$ are positive integers, at least one of which equals $1$. Our corresponding upper bounds are given in terms of certain 'interval Rado numbers' for additive equations. We pose a number of open problems concerning these numbers. 
@MISC{ACOS-23-multiplicative-schur,
    title = {On the number of monochromatic solutions to multiplicative equations},
    author = {Lucas Aragão and Jonathan Chapman and Miquel Ortega and Victor Souza},
    archivePrefix = {arXiv},
    primaryClass = {math.CO},
    eprint = {2311.18742},
    doi = {https://arxiv.org/abs/2311.18742},
}
2022 Expander graphs are globally synchronising P. Abdalla, A. Bandeira, M. Kassabov, V. Souza, S. Strogatz, and A. Townsend arxiv quanta The Kuramoto model is a prototypical model used for rigorous mathematical analysis in the field of synchronisation and nonlinear dynamics. A realisation of this model consists of a collection of identical oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that the Kuramoto model on such a graph will converge to the fully synchronised state with all the oscillators with same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon)(\log n)/n$, the Kuramoto model on the Erdős-Rényi graph $G(n,p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz and Townsend and solves a conjecture of Ling, Xu and Bandeira. We also show that the Kuramoto model is globally synchronizing on any $d$-regular Ramanujan graph with $d \geq 600$ and that, for the same range of degrees, a $d$-regular random graph is typically globally synchronizing. 
@MISC{ABKSST-22-expander-sync,
    title = {Expander graphs are globally synchronizing},
    author = {Pedro Abdalla and Afonso Bandeira and Martin Kassabov and Victor Souza and Steven Strogatz},
    archivePrefix = {arXiv},
    primaryClass = {math.CO},
    eprint = {2210.12788},
    doi = {https://arxiv.org/abs/2210.12788},
}
2019 The typical structure of sets with small sumset M. Campos, M. Collares, R. Morris, N. Morrison, and V. Souza International Mathematics Research Notices, 2022 arxiv journal In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $\lambda > 2$ and every $k \geq (\log n)^4$: if $\omega \to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A+A| \leq \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$. 
@ARTICLE{CCMMS-22-small-sumset,
    title = {The typical structure of sets with small sumset},
    author = {Marcelo Campos and Maurício Collares and Robert Morris and Natasha Morrison and Victor Souza},
    journal = {International Mathematics Research Notices},
    volume = {2022},
    number = {14},
    pages = {11011--11055},
    year = {2022},
    publisher = {Oxford University Press},
    doi = {https://doi.org/10.1093/imrn/rnab021},
}
Coauthors Adva Mond4 Lucas Aragão2 Maurício Collares2 Vojtěch Dvořák2 Roberto Parente2 Leo Versteegen2 Pedro Abdalla1 José Alvarado1 Afonso Bandeira1 Cynthia Bortolotto1 Marcelo Campos1 Jonathan Chapman1 Lucas Colucci1 Victor Falgas-Ravry1 Lawrence Hollom1 Martin Kassabov1 Yoshiharu Kohayakawa1 Noah Lebowitz-Lockard1 David Lewis1 Taísa Martins1 Rob Morris1 Natasha Morrison1 Miquel Ortega1 Julien Portier1 Rik Sarkar1 Steven Strogatz1 Alex Townsend1