Hi! Since September 2018, I am Szegő Assistant Professor in the mathematics department of Stanford University. Before coming to Stanford, my doctoral studies were at ETH Zurich under the guidance of Benny Sudakov, and my undergraduate studies were at UNSW (Sydney), where my adviser was Catherine Greenhill. You can find my CV here.
Sometimes, it is possible to represent a complicated polytope as a “shadow” of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which P can be obtained as a linear projection. It is an important question to understand the extent to which the extension complexity of a polytope is controlled by its dimension, and in this paper we prove three different results along these lines. First, we show that there exists an no(1)-dimensional polytope with at most n facets and extension complexity n1−o(1). Second, we obtain optimal bounds for the extension complexity of random d-polytopes, and third, we obtain an optimal upper bound for the extension complexity of cyclic polygons (all of whose vertices lie on a common circle).
Here is a recording of a talk I gave on the results in this paper.
Almost all Steiner triple systems are almost resolvable (with Asaf Ferber). Forum of Mathematics, Sigma 8:E39 (2020).
Almost all Steiner triple systems have perfect matchings. Proceedings of the London Mathematical Society 121.6 (2020), 1468–1495.
In the theory of combinatorial design, a Steiner triple system of order n is a system of 3-element subsets (“triples”) of a ground set of size n, such that every pair of elements in the ground set is contained in exactly one triple. We prove a general theorem comparing a uniformly random Steiner triple system to the outcome of the so-called triangle removal process. In a first paper, we use this theorem to show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (in fact, we show that almost all such Steiner triple systems have essentially the maximum possible number of perfect matchings). In a second paper, we show that almost all such Steiner triple systems in fact admit a decomposition of almost all their triples into disjoint perfect matchings. That is, almost all Steiner triple systems are almost resolvable.
Here is a recording of one of my talks that featured these results.
Given n bases B1,...,Bn in an n-dimensional vector space V, a transversal basis is a basis of V containing exactly one element from each Bi. Rota's basis conjecture posits that it is always possible to find n disjoint transversal bases. In this paper we prove the partial result that one can always find (1/2 − o(1)) n disjoint transversal bases, improving on the previous record of Ω(n/log n). Our result generalises to the setting of matroids.
Here is a recording of a talk in which I gave a fairly complete outline of the proof of this theorem.
Proof of a conjecture on induced subgraphs of Ramsey graphs (with Benny Sudakov). Transactions of the American Mathematical Society 372 (2019), 5571–5594.
Ramsey graphs induce subgraphs of quadratically many sizes (with Benny Sudakov). International Mathematics Research Notices 2020.6 (2020), 1621–1638.
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is a line of research towards showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. In these papers we prove two conjectures along these lines. First, we prove that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of Θ(n2) different sizes. This resolves a conjecture of Narayanan, Sahasrabudhe and Tomon, motivated by an old problem of Erdős and McKay. Second, we prove a conjecture of Erdős, Faudree and Sós that in any n-vertex C-Ramsey graph, there are Ω(n5/2) induced subgraphs, no pair of which have the same numbers of edges and vertices.
Consider integers k, l such that 0 ≤ l ≤ (k choose ). Given a large graph G, what is the fraction of k-vertex subsets of G which span exactly l edges? When l is zero or (k choose ), this fraction can be exactly 1. On the other hand, with Ramsey's theorem in mind, if l is far from these extreme values we might expect that this fraction must always be substantially smaller than 1. We prove an almost-best-possible theorem to this effect, improving on results of Alon, Hefetz, Krivelevich and Tyomkyn. We also make some first steps towards some analogous questions for hypergraphs. Our proofs take a probabilistic point of view, and involve polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a “slice” of the Boolean hypercube.
Fix a sequence of nonzero real numbers a = (a1,...,an), consider a random ±1 sequence ξ = (ξ1,...,ξn), and let X = a1ξ1+...+anξn. The Erdős–Littlewood–Offord theorem shows that, regardless of a, for any x the event X = x is unlikely (that is, X is anti-concentrated). In this paper, motivated by some questions about random matrices, we study the “resilience” of this anti-concentration. For a given x, how many coordinates of ξ can we allow an adversary to change before they can force X = x? The answer is quite surprising, and its proof involves an interesting connection to combinatorial number theory.