Picture your favorite \$d\$-dimensional polytope. It has faces of every dimension below \$d\$: zero-dimensional vertices, one-dimensional edges, two-dimensional polygons, and so on. Counting the faces of each dimension yields a list of face numbers. For example, a \$3\$-cube has face numbers \$(8, 12, 6)\$: eight vertices, twelve edges, and six facets.
What integer tuples can arise as the face numbers of a polytope? Geometric methods may help us answer this question. In my talk, I'll discuss what we can learn about the face numbers of polytopes from their geometric properties--specifically, their angles. I'll present a set of tight linear inequalities on face numbers, answering a 1997 question of Bárány.
Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.
Join Zoom Meeting: https://washington.zoom.us/j/
Meeting ID: 915 4733 5974