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. 

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.