Victor Seixas Souza
Klarman Fellow at Department of Mathematics
Cornell University

505 Malott 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. In Cornell, I am one of the organisers of the Probability Seminar.

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 2026 Disjoint and nearly disjoint sums of matrix multiplication tensors and their centroids M. Kassabov, J. M. Landsberg, V. Souza, and P. Speegle arxiv This paper addresses centroids, which are fundamental invariants of tensors. Our main results are as follows: (i) The construction of explicit tensors with very large centroids, whereas previously it had been conjectured that none such exist. (ii) An upper bound on the dimension of the centroid that is essentially attained by our examples. (iii) The development of a geometric technique to write down border rank decomposition of tensors using centroids and "extended centroids". (iv) The technique is applied to tensors of this paper to prove they are of minimal border rank. The technique is versatile and enables us to geometrically derive and improve upon previous ad hoc decompositions. (v) The construction of symmetric tensors with large centroids and proof that they are wild in the sense of Buczyńska-Buczyński.

Our results also pave the way for new upper bounds on the exponent of matrix multiplication. The geometric technique also constructs new "better" tensors for Strassen's laser method from old, and we apply this to the tensors of Strassen and Schönhage to get better tensors in the sense that they give better upper bounds on the exponent than the original tensors. 
@MISC{KLSS-26-tensor-centroids,
    title = {Disjoint and nearly disjoint sums of matrix multiplication tensors and their centroids},
    author = {Martin Kassabov and J.M. Landsberg and Victor Souza and Philip Speegle},
    archivePrefix = {arXiv},
    primaryClass = {math.AG},
    eprint = {2608.27434},
    doi = {https://arxiv.org/abs/2608.27434},
}
2026 Dense sets without large sumsets G. Dahia, J. P. Marciano, and V. Souza arxiv We prove, for all fixed $0 < \delta < 1$, and all sufficiently large $n$, that there exists $S \subseteq [n]$ with $|S| \ge \delta n$ such that $A + B \not \subseteq S$ for all ${A, B \subseteq \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / \delta)}.$$ A very recent result of Hernández and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $\delta$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and that one can take $n^{-\alpha} \le \delta \le 1 - c$ where $c > 0$ is fixed and $\alpha > 0$ depends only on the $o(1)$ error, answering another question of the same authors in a strong form. 
@MISC{DMS-26-dense-sumsets,
    title = {Dense sets without large sumsets},
    author = {Gabriel Dahia and João Pedro Marciano and Victor Souza},
    archivePrefix = {arXiv},
    primaryClass = {math.CO},
    eprint = {2607.15269},
    doi = {https://arxiv.org/abs/2607.15269},
}
2025 Double-jump phase transition for the reverse Littlewood--Offord problem L. Hollom, J. Portier, and V. Souza Journal of the London Mathematical Society, 2026 arxiv journal 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-26-reverse-lo,
    title = {Double-jump phase transition for the reverse Littlewood--Offord problem},
    author = {Lawrence Hollom and Julien Portier and Victor Souza},
    journal = {Journal of the London Mathematical Society},
    volume = {113},
    number = {5},
    pages = {e70539},
    publisher = {Wiley},
    doi = {http://dx.doi.org/10.1112/jlms.70539},
}
2024 The Maker-Breaker percolation game on a random board V. Dvořák, A. Mond, and V. Souza Annals of Probability, 2026 arxiv journal 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. 
@ARTICLE{DMS-26-mb-random,
    title = {The Maker--Breaker percolation game on a random board},
    author = {Vojtěch Dvořák and Adva Mond and Victor Souza},
    journal = {Annals of Probability},
    volume = {54},
    number = {3},
    pages = {1530--1563},
    year = {2026},
    publisher = {Institute of Mathematical Statistics},
    doi = {https://doi.org/10.1214/25-AOP1796},
}
2023 On the number of monochromatic solutions to multiplicative equations L. Aragão, J. Chapman, M. Ortega, and V. Souza Combinatorica, 2025 arxiv journal 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. 
@ARTICLE{ACOS-25-multiplicative-schur,
    title = {On the number of monochromatic solutions to multiplicative equations},
    author = {Lucas Aragão and Jonathan Chapman and Miquel Ortega and Victor Souza},
    journal = {Combinatorica},
    volume = {45},
    number = {6},
    pages = {64},
    year = {2025},
    publisher = {Springer},
    doi = {https://doi.org/10.1007/s00493-025-00183-x},
}
2022 Expander graphs are globally synchronising P. Abdalla, A. Bandeira, M. Kassabov, V. Souza, S. Strogatz, and A. Townsend Advances in Mathematics, 2026 arxiv journal 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},
}
Coauthors Adva Mond4 Lucas Aragão2 Maurício Collares2 Vojtěch Dvořák2 Martin Kassabov2 Roberto Parente2 Leo Versteegen2 Pedro Abdalla1 José Alvarado1 Afonso Bandeira1 Cynthia Bortolotto1 Marcelo Campos1 Jonathan Chapman1 Lucas Colucci1 Gabriel Dahia1 Victor Falgas-Ravry1 Lawrence Hollom1 Yoshiharu Kohayakawa1 J. M. Landsberg1 Noah Lebowitz-Lockard1 David Lewis1 João Pedro Marciano1 Taísa Martins1 Rob Morris1 Natasha Morrison1 Miquel Ortega1 Julien Portier1 Rik Sarkar1 Philip Speegle1 Steven Strogatz1 Alex Townsend1