Abstract: Understanding face numbers of convex polytopes is one of the oldest branches of mathematics. The celebrated g-theorem, conjectured by McMullen '70 and proved by Billere-Lee '80 and Stanley '80, gives a complete characterization of the face numbers of simplicial polytopes, namely polytopes all whose proper faces are simplices. It is conveniently phrased is terms of the g-vector, obtained by a linear transformation of the face numbers. The g-vector encodes primitive Betti numbers of the toric varieties of these polytopes. The tight lower bounds are given by the nonnegativity of the g-vector, componentwise.

Kalai '94 conjectured that g-vectors of simplicial polytopes which well approximate a convex body are unbounded, componentwise. We resolve this conjecture by relating the underling geometry of the polytope with the geometry of its associated toric variety. Further, for C² -convex bodies we give asymptotically tight lower bounds on the g-numbers of the approximating polytopes, in terms of their Hausdorff distance from the convex body. This sharpen's results of Boroczky '00 on lower bounds for individual face numbers.

In the talk I'll give the needed background and sketch the proof - moving from geomerty, to algebra, to algebraic topology. Time permitting, I'll mention the second part of Kalai's conjecture, on upper bounds on the g-vector, and its resolution for approximation by random polytopes.

Joint work with Karim Adiprasito and Jose Samper.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006