Isabella Novik: Face numbers: the upper bound side of the story

What is the largest number of i-dimensional faces that a simplicial polytope of dimension d with n vertices can have? What about the largest possible number of i-dimensional faces that a triangulation of a (d - 1)-sphere can have? What are the maximizers? How common or rare are they? How do the answers change if the object in question must be centrally symmetric (i.e., endowed with a free action of Z/2Z)? This talk will be a survey of what is known and what is still unknown in this fascinating field, with an emphasis on some recent developments.