Activities This Week

Let $V$ be a representation of a connected reductive group $G$. A representation is cofree if $k[V]$ is a free $k[V]^G$ module. There is a long history of work studying and classifying cofree representations of reductive groups. In this talk I present a simple conjectural characterization of cofree representations in terms of geometric invariant theory. Matt Satriano and I have proved the conjecture for irreducible representations of SL_n as well as for torus actions. I will give motiviation for the conjecture and explain the techniques which can be used for its verification. This talk based on joint work with Matt Satriano.

We will consider some basic optimization problems and how they relate to optics. We then define an index of refraction to any given distribution. We conjecture an estimate for the index and explain how its related to some natural operations research questions. We also consider lenses and ask questions about the probabilistic behavior of discrete geodesics in a lens setting.

Several functional analogs of fundamental geometric inequalities have appeared in recent decades, beginning with the works of Prekopa and Leindler in the 1970’s. In this talk I will, after discussing the method of functionalization of geometry, present new functional extensions of the Brunn Minkowski inequality and their consequences.