Currently, I am a pursuing PhD in mathematics at the University of Oxford
under the supervision of Prof. Oliver Riordan. Prior to that, I earned
a master's degree in mathematics under the supervision of Prof. Krzysztof Oleszkiewicz,
and a bachelor's degree under the supervision of Marcin Kotowski,
both at the University of Warsaw.
My research interests lie broadly in probabilistic combinatorics, with a
particular appreciation for its interactions with geometry and analysis. I
study topics such as threshold phenomena, the convergence of random walks, percolation,
Boolean functions, and related areas.
During my school years, I was fortunate to be part of the vibrant Polish
mathematical society clustered together thanks to mathematical olympiads.
One of the first people
to ignite my passion for mathematics was a fantastic educator, Henryk Pawłowski,
from whom I learnt a lot. Perhaps these entities are a reason why I enjoy
tackling interesting and elegant problems, and prefer the problem-solving
side of mathematics.
Publications
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KKL theorem for the influence of a set of variables
by T. Przybyłowski
arXiv abstractWe study the notion of the influence of a set of variables on a Boolean function, which was recently introduced by Tal. We show that for an arbitrary fixed \(d\), every Boolean function \(f\) on \(n\) variables admits a \(d\)-set of influence at least \(10 \textbf{W}^{\geq d}(f)(\frac{\log n}{n})^d\), where \(\textbf{W}^{\geq d}(f) = \sum_{A \subset [n]\colon |A| \geq d} \hat{f}(A)^2\).
It is a direct generalisation of the Kahn-Kalai-Linial theorem. We give an example demonstrating essential sharpness of this result. Further, we generalise a related theorem of Oleszkiewicz regarding influences of pairs of variables. -
Random walks on Coxeter interchange graphs
by M. Buckland, B. Kolesnik, T. Przybyłowski, R. Mitchell
Electronic Journal of Probability, 2025
arXiv journal abstract videoA tournament is an orientation of a graph. Vertices are players and edges are games, directed away from the winner. Kannan, Tetali and Vempala and McShine showed that tournaments with given score sequence can be rapidly sampled, via simple random walks on the interchange graphs of Brualdi and Li. These graphs are generated by the cyclically directed triangle, in the sense that traversing an edge corresponds to the reversal of such a triangle in a tournament. We study Coxeter tournaments on Zaslavsky's signed graphs. These tournaments involve collaborative and solitaire games, as well as the usual competitive games. The interchange graphs are richer in complexity, as a variety of other generators are involved. We prove rapid mixing by an intricate application of Bubley and Dyer's method of path coupling, using a delicate re-weighting of the graph metric. Geometric connections with the Coxeter permutahedra introduced by Ardila, Castillo, Eur and Postnikov are discussed.
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Coxeter interchange graphs
by B. Kolesnik, T. Przybyłowski, R. Mitchell
arXiv abstract videoBrualdi and Li introduced tournament interchange graphs. In such a graph, each vertex represents a tournament. Traversing an edge corresponds to reversing a cyclically directed triangle. Such a triangle is neutral, in that its reversal does not affect the score sequence. An interchange graph encodes the combinatorics of the set of tournaments with a given score sequence, or equivalently, of a given fiber of the classical permutahedron from discrete geometry. Coxeter tournaments were introduced by the first author and Sanchez, in relation to the Coxeter permutahedra in Ardila, Castillo, Eur and Postnikov. Coxeter tournaments have collaborative and solitaire games, in addition to the usual competitive games in classical tournaments. We introduce Coxeter interchange graphs. These graphs are more intricate, as there are multiple neutral structures at play, which interact with one another. Our main result shows that the Coxeter interchange graphs are regular, and we describe the degree geometrically, in terms of distances in the Coxeter permutahedra. We also characterize the set of score sequences of Coxeter tournaments, generalizing a classical result of Landau.
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Thresholds, expectation thresholds and cloning
by T. Przybyłowski, O. Riordan
Electronic Journal of Combinatorics, 2024
arXiv journal abstractLet \(p_{\textrm{c}}\) and \(q_{\textrm{c}}\) be the threshold and the expectation threshold, respectively, of an increasing family \(\mathcal{F}\) of subsets of a finite set \(X\), and let \(l\) be the size of a largest minimal element of \(\mathcal{F}\). Recently, Park and Pham proved the Kahn-Kalai conjecture, which says that \(p_{\textrm{c}} \leqslant K q_{\textrm{c}} \log_2 l\) for some universal constant \(K\). Here, we slightly strengthen their result by showing that \(p_{\textrm{c}} \leqslant 1 - \textrm{exp}(-K q_{\textrm{c}} \log_2 l)\). The idea is to apply the Park-Pham Theorem to an appropriate `cloned' family \(\mathcal{F}_k\), reducing the general case (of this and related results) to the case where the individual element probability \(p\) is small.