Karim Adiprasito: New upper bound theorems

The classical Upper Bound Theorem of McMullen-Stanley bounds the number of faces of a simplicial sphere on a given number of vertices. Stanley's method stems from commutative algebra: he uses the fact that the f-vector can be bounded by estimating the Hilbert series of the Stanley--Reisner ring associated to the simplicial polytope.

Since then, there have been several questions similar to the Upper Bound Problem, some of which notoriously resisted a treatment based on algebraic methods. Among them is the problem of providing a sharp upper bound for the number of faces of the Minkowski sum of simplicial polytopes. 

In my talk, I will sketch some new methods, and illustrate them by examples. In particular, I will present a complete resolution to the Upper Bound Problem for Minkowski sums.

This reflects joint work with Raman Sanyal.