Georgia Institute of Technology College of Sciences

In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties. 

A matroid is modular if its lattice of flats is self-dual. Prototypical examples include the Boolean matroid $U_{n,n}$ and finite projective geometry matroid $M(\mathbb{F}_q^n)$. We discovered that the space of quotients of any modular matroids has remarkable structure. For instance, quotients of the Boolean matroid $U_{n,n}$, also known as the set of all matroids on $[n]$, are equipped with operations such as matroid duality and matroid intersection. All these operations exist canonically for quotients of arbitrary modular matroids. We will showcase some implications of these operations, including a generalization of stable intersection on the Bergman fan of any modular matroid.

In this talk I will discuss a particular convex relaxation of quadratic programs called the Shor relaxation. We study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. I will discuss conditions under which this region is invariant under the choice of generators defining the feasible set as well as how this region reflects the symmetry in the feasible region. Finally, I will discuss applications of these results to quadratic binary programs. This is joint work with Jose Rodriguez.

Deligne connects the weight-zero compactly supported cohomology of a complex variety to the combinatorics of its compactifications. In this talk, we use this to study the moduli space of n-marked hyperelliptic curves. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for its weight-0 compactly supported Euler characteristic, as a virtual representation of S_n. This is joint work with Melody Chan and Siddarth Kannan.