We study the class of Lorentzian polynomials. The class contains homogeneous stable
polynomials as well as volume polynomials of convex bodies and projective varieties. We …

In this chapter, we survey Elias and Williamson's proof of Soergel's conjecture on the
characters of Soergel bimodules. Their work actually establishes a “Hodge theory” for …

We use the geometry of the stellahedral toric variety to study matroids. We identify the
valuative group of matroids with the cohomology ring of the stellahedral toric variety and …

We design an FPRAS to count the number of bases of any matroid given by an independent
set oracle, and to estimate the partition function of the random cluster model of any matroid …

The motivation of this work is to extend the techniques of higher order random walks on
simplicial complexes to analyze mixing times of Markov chains for combinatorial problems …

We give a deterministic polynomial time 2^ O (r)-approximation algorithm for the number of
bases of a given matroid of rank r and the number of common bases of any two matroids of …

We introduce the intersection cohomology module of a matroid and prove that it satisfies
Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. As …

We study an operation in matroid theory that allows one to transition a given matroid into
another with more bases via relaxing a stressed subset. This framework provides a new …

We introduce certain torus-equivariant classes on permutohedral varieties which we call
“tautological classes of matroids” as a new geometric framework for studying matroids …

Consider a simplicial complex that allows for an embedding into $\mathbb {R}^ d $. How
many faces of dimension $\frac {d}{2} $ or higher can it have? How dense can they be? This …