Events

Abstract:  In this talk, we demonstrate a connection between log-concavity statements and sampling algorithms via high-dimensional expanders and Lorentzian polynomials. To do this, we first discuss two conjectures which were resolved about 5-10 years ago: one on the log-concavity of independent sets of matroids (due to Brändén-Huh and Anari-Liu-Oveis Gharan-Vinzant), and one on efficiently sampling bases of matroids (due to Anari-Liu-Oveis Gharan-Vinzant). From there we will present some new results on generalized graph colorings which extend these and other previous results. In particular, we will discuss how this can be used to obtain log-concavity statements and sampling algorithms for linear extensions of posets. Joint work with Kasper Lindberg and Shayan Oveis Gharan.

Abstract: In the space of type A quiver representations, putting rank conditions on the maps cuts out subvarieties called "open quiver loci." These subvarieties are closed under the group action that changes bases in the vector spaces, so their closures define classes in equivariant cohomology, called "quiver polynomials." Knutson, Miller, and Shimozono found a pipe dream formula to compute these polynomials in 2006. To study the geometry of the open quiver loci themselves, we might instead compute "equivariant Chern-Schwartz-MacPherson classes," which interpolate between cohomology classes and Euler characteristic. I will introduce objects called "chained generic pipe dreams" that allow us to compute these CSM classes combinatorially, and along the way give streamlined formulas for quiver polynomials.

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