Future students

Title: Nearly-linear stable sets

Abstract: The Gyárfás-Sumner conjecture says that for every forest 𝐻 and complete graph 𝐾, there exists 𝑐 such that every {𝐻,𝐾}-free graph (that is, containing neither of 𝐻,𝐾 as an induced subgraph) has chromatic number at most 𝑐. This is still open, but we have proved that every {𝐻,𝐾}-free graph 𝐺 has chromatic number at most |𝐺|^o(1).

Title: Contracts for Functions Based on Graphs

Abstract: We study contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a contract with several agents, whose actions the principal is unable to observe. The principal is able to see only the outcome of the agents' collective actions; and the outcome is either a success or failure. All agents that decided to exert effort incur costs, and so naturally all agents expect a fraction of the principal's reward as a compensation. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized.

Title: Combinatorial models in enumerative geometry

Abstract: How many circles are tangent to three given circles in the plane?  How many lines lie on a smooth cubic surface? How many lines intersect four given lines in 3-space? These are classical questions in enumerative geometry, a field at least as old as Apollonius of Perga, to whom the question about tangent circles is attributed in the third century BCE.  It is not hard to see that the answers to these questions may depend on the choice of circles, surface, or lines, respectively. It is harder to see that, in each case, there is a "typical" answer. 

Title: On the use of senders in Ramsey Theory

Abstract: In this talk I will introduce and investigate some parameters in Graph Ramsey theory, beyond the traditional Ramsey numbers. A crucial ingredient for their analysis is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős and Lovász in 1976. I will explain their origin, properties, and try to convey their surprising strength. Using probabilistic methods, we will see how to build such gadgets, and how to use them to prove some theorems, previously out of reach without these tools.

Title: Bipartite Perfect Matching is in Quasi-NC, Part II

Abstract: Mulmuley, Vazirani, and Vazirani gave a randomized parallel algorithm for checking whether a perfect matching exists in a graph. In doing so, they came up with the infamous isolation lemma, which found several uses in other areas of computer science. The isolation lemma is inherently randomized, and it has been a long-standing open problem to derandomize the lemma. In this talk, I will go over the breakthrough work of Fenner, Gurjar, and Thierauf where they almost completely derandomize the isolation lemma in the special case when applied to the bipartite perfect matching problem. In doing so, they give a deterministic parallel algorithm for perfect matching that uses a quasi-polynomial number of processors.

Title: Supersingular isogeny graphs, modular curves and Galois Representations.

Abstract: In this talk we will discuss the remarkable interconnection among supersingular elliptic curves, modular curves and Galois representations with a focus on cryptographic applications.

Title: A relaxation of Woodall’s conjecture

Abstract: In a directed graph, a directed cut (dicut for short) is a cut where all arcs are directed from one shore to the other; a directed join (dijoin for short) is a set of arcs whose contraction makes the digraph strongly connected. The celebrated Lucchesi–Younger theorem states that for any directed graph the size of the smallest dijoin equals the maximum number of pairwise disjoint dicuts. Woodall’s conjecture posits that the size of the smallest dicut equals the maximum number of pairwise disjoint dijoins. 

Title: Bipartite Perfect Matching is in Quasi-NC

Abstract: Mulmuley, Vazirani, and Vazirani gave a randomized parallel algorithm for checking whether a perfect matching exists in a graph. In doing so, they came up with the infamous isolation lemma, which found several uses in other areas of computer science. The isolation lemma is inherently randomized, and it has been a long-standing open problem to derandomize the lemma. In this talk, I will go over the breakthrough work of Fenner, Gurjar, and Thierauf where they almost completely derandomize the isolation lemma in the special case when applied to the bipartite perfect matching problem. In doing so, they give a deterministic parallel algorithm for perfect matching that uses a quasi-polynomial number of processors.

Title: Shellability of complexes over lattices

Abstract: In this work, we introduce the notion of power lattices, which are a more general class of ranked lattices with additional properties. Then we generalize the concept of shellable simplicial complexes in the lattice of subsets to P-shellable P-complexes in a power lattice P. We show that when the P-complex is P-shellable, its order complex is a shellable simplicial complex. We demonstrate that these P-complexes can be constructed by generalizing the concept of matroids to matroids in a power lattice P. This provides various constructions of posets with desirable topological and algebraic properties. In the particular class of lattices of multiset subsets, we show how to construct shellable 'multicomplexes' from weighted graphs. Finally, we illustrate how shellable multicomplexes give rise to rings that are sequentially Cohen-Macaulay.

Title: Combinatorial proof of a Non-Renormalization Theorem

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Abstract: In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that controls certain deformations in quantum field theory. These integrals themselves are differential forms, and the authors conjectured that one class of them squares to zero. This phenomenon can be interpreted as absence of quantum corrections in topological quantum field theories with more than one topological direction, or as an analogue of Kontsevich's formality theorem. In my talk, I will present a purely combinatorial proof of the conjecture for arbitrary graphs. It is based on graph matrices and graph polynomials, and a careful analysis of the involved signs and multiplicities. No knowledge or intution of the underlying physics is required.

In the preseminar, I will review the necessary definitions and properties of graph polynomials, and how they are typically applied in Feynman integrals. If time permits, I might also comment on the physical background.