Karim Adiprasito (IHES)
Title: Approximation by polytopes, Betti numbers of toric varieties and graph chordality

Abstract: In a bold conjecture, Kalai proposed a relation between two
very different aspects of simplicial polytopes, namely -- their
Hausdorff distance to some smooth convex body and -- the differences
g_k between consecutive nontrivial Betti numbers of the associated
projective toric variety.  The proof of this connection takes us back
to the fundamental notion of chordality, a ubiquitous concept in graph
theory connected to colorings, matchings and many more. We will see
that chordality (and its generalizations) can naturally be interpreted
within the cohomology ring of the toric variety, as well as topology
of induced subcomplexes of the polytope. In addition to the above
connection, this allows us to give an elementary, combinatorial
interpretation of the Betti numbers of the projective toric variety
which subsumes the celebrated Generalized Lower Bound Conjecture of
McMullen and Walkup.