Activities This Week

A polytope is called simplicial if all its proper faces are simplices. The celebrated g-theorem gives a complete characterization of the possible face numbers (a.k.a. f-vector) of simplicial polytopes, conjectured by McMullen ’70 and proved by Billera-Lee (sufficiency) and by Stanley (necessity) ’80. The latter uses deep relations with commutative algebra and algebraic geometry. Moving to general polytopes, a finer information than the f-vector is given by the flag-f-vector, counting chains of faces according to their dimensions. Here much less is known, or even conjectured.

I will discuss what works and what breaks, at least conjecturally, when passing from simplicial to general polytopes, or subfamilies of interest.

I will describe recent threads in the study of number theory in function fields, the different techniques that are used, the challenges, and if time permits the applications of the theory to other subjects such as probabilistic Galois theory.