Igor Pak, a professor of mathematics at the University of California, Los Angeles

"What is a Combinatorial Interpretation?"

In enumerative combinatorics, one is often asked to count the number of combinatorial objects.  But the inverse problem is even more interesting: given some numbers, do they have a combinatorial interpretation?  In the main part of the talk I will give a broad survey of this problem, formalize the question in the language of computational complexity, and describe some connections to deep results and open problems in algebraic and probabilistic combinatorics.  In the last part of the talk, I will discuss our recent results on Stanley's inequality for the number of certain linear extensions and the implications to the equality cases and stability of the Alexandrov-Fenchel inequality.  The talk is aimed at a general audience.