Current students

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.