Events

Title: A Proof of Nash-Williams’ Conjecture

Abstract: A central open question in extremal design theory is Nash-Williams’ Conjecture from 1970, namely that every triangle-divisible graph on n vertices (for n large enough) with minimum degree at least 0.75n has a triangle decomposition. In this talk, we discuss the history of the problem and our recent resolution of this conjecture, as well as other applications in design theory. We also overview the proof, highlighting the new techniques we developed to resolve the fractional version as well as the full conjecture. Joint work with Michelle Delcourt.

Abstract: The mirror of a smooth Fano toric variety is a Laurent polynomial, and this Laurent polynomial encodes interesting enumerative data of the toric variety.  For example, the Jacobian ring of the Laurent polynomial is isomorphic to the small quantum cohomology ring of the toric variety. In this talk, I’ll describe this aspect of the (now basically classical) toric mirror theorem, and explain how this picture is expected to extend to other smooth Fano varieties. The Grassmannian’s strong combinatorial structure  makes it one of the nicest examples of non-toric Fano varieties to study in this context. I’ll present several mirror constructions for the Grassmannian that each reflect a different perspective on its geometry. Finally, I’ll discuss some extensions of these constructions to generalizations of the Grassmannian. 

For this term's reading group we will be hosting a study group on quantum tools in cryptography. For the first week, we start with a motivation on quantum computing including the basic definitions and discussion on complexity classes. This will serve as a elementary foundation for the rest of the reading group. A week-by-week plan outline.

Abstract: On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with n distinguished boundary vertices. Mathematical investigation of on-shell forms has largely focused on the case of planar graphs, where one can use tools developed by Postnikov in the study of the totally nonnegative Grassmannian. In this talk we develop the mathematics of nonplanar on-shell forms, and explain how they arise in physics and math. For certain forms on the Grassmannian Gr(2,n), we prove a determinantal formula appearing in the physics literature, and give a connection to the hypertree divisors of Castravet and Tevelev.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.

Abstract: Various notions of negative dependence arise naturally and/or are desirable in various random processes and randomized algorithms. We survey how to generate and utilize a few such notions of negative dependence, and sketch applications to concentration inequalities, fairness, and approximation algorithms.

Abstract: Much like matroids, oriented matroids can be defined on finite ground sets using a set of circuit axioms. Oriented matroid duality is an important property which is difficult to maintain in the infinite setting. One way to define infinite oriented matroids in a way that preserves duality is using signed set orthogonality. This allows for the notion of finitary and cofinitary oriented matroids. An oriented matroid is called finitary if all of its circuits have finite support, and cofinitary if it is the dual of a finitary oriented matroid. It has recently been shown that cofinitary oriented matroids do not necessarily satisfy the circuit axioms. In the finite setting, there are many ways to define oriented matroids which are equivalent to the circuit axioms. Here, we prove that some of these definitions still hold for cofinitary oriented matroids, even when the circuit axioms fail.

Abstract: For positive integers $s$ and $n$, let $f_{s}(n)$ denote the maximum $k$ such that every $n$-vertex $K_{s+1}$-free graph contains an induced $K_s$-free subgraph with $k$ vertices. These are the Erd\H{o}s-Rogers functions, introduced by Erd\H{o}s and Rogers in 1962, and are generalizations of Ramsey numbers.

In this talk we survey recent progress, culminating with the recent result of Morris, Saharasbudhe which finally determines the order of magnitude of the Erd\H{o}s-Rogers functions.

Joint work with Rob Morris and Julian Saharasbudhe.