Participants with different backgrounds and trajectories will discuss their experiences navigating the joys and struggles of graduate school and finding ways to thrive, personally and professionally.

The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will briefly explain their geometric origin and discuss their algebraic and geometric combinatorics.

The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-vector, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-vector, which gives the dimensions of the intersection cohomology of the associated toric variety, is log-concave.

The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges.

These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.

Tropical algebraic geometry studies varieties by `tropicalizing' them into polyhedral complexes. The hope, which is realized surprisingly often, is that difficult algebro-geometric questions will reduce to simpler polyhedral questions. The resulting questions are usually not easy, and that’s good news for us: they have led to several new and interesting research directions in geometric combinatorics. Matroids often play a fundamental role because they are equivalent to tropical linear spaces. This talk will survey a few central aspects of the interplay between matroids and tropical geometry, and discuss some recent developments.