Monday, July 13, 2015
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
Spectrahedra are generalizations of (convex) polyhedra sharing many of the good algorithmic features with polyhedra but allowing for roundness in their shapes. Given two spectrahedra in form of a linear matrix inequality, it is in general hard to decide whether one contains the other. To the linear matrix inequalities one can however associate not only scalar solutions but also matrix solutions. This gives rise to so called free spectrahedra. Inclusion of free spectrahedra is in many cases easy to decide, e.g., by using a generalization of the Gram matrix method and a linear Positivstellensatz for symmetric linear matrix polynomials due to Helton, Klep and McCullough. The natural question arises how the inclusion of two free spectrahedra relates to the inclusion of the corresponding classical spectrahedra. Surprisingly, this question is related to very subtle non-trivial properties of Binomial and Beta distributions some of which are known and some of which are new.
(Joint work with Bill Helton, Igor Klep and Scott McCullough)
Colloquium - 16:00
Abstract:
The plethysm problem in representation theory of GL(n) is to
compute the irreducible decomposition of a composition of Schur
functors. We give a gentle introduction to the problem and show
that the answer can be computed by lattice point counting
techniques. This allows to study the asymptotics of plethysm,
and, under certain restrictions, to derive new explicit piecewise
quasi-polynomial formulas for multiplicities of irreducible
components in the plethysm. This is joint work with Mateusz
Michalek.