Polytopes count (introductory talk)
Aram Bingham
Colorado School of Mines

We’ll see how many problems in combinatorics can be converted to the problem
of counting integral points inside appropriate polytopes. Not only is this
conversion a practical advantage in settings where nice, closed formulas for
combinatorial quantities aren’t available, but phrasing computations as
integral point-counting problems is often key to establishing complexity
theoretic results. We’ll take up the particular example of
Littlewood-Richardson coefficients, which have motivated many of the
conjectures on their cousins the Kronecker coefficients.

Kronecker coefficients, polytopes, and complexity
Aram Bingham
Colorado School of Mines

The “Kronecker coefficients problem” is one of the last major open questions
in the classical representa tion theory of symmetric groups. It asks for a
combinatorial rule describing the decomposition of tensor products of
irreducible symmetric group representations, and a solution is known only in
limited special cases. Kronecker coefficients have also been the subject of
much recent research motivated by the geomet ric complexity theory (GCT)
program, which hypothesized efficient computation of these numbers as part
of a strategy to separate the computational complexity classes P and NP. We
will discuss the specific conjectures coming from GCT and report some
progress on computing these coefficients via discrete volumes of polytopes
(joint work with Ernesto Vallejo).