H.B. Keller Colloquium

We consider the problem of characterizing polynomial inequalities satisfied by traces of powers of real symmetric matrices. I will focus on inequalities that are satisfied by symmetric matrices of all sizes. The problem of determining whether a given polynomial inequality in traces of matrix powers is valid for all symmetric matrices of all sizes is undecidable. However, if we replace trace by normalized trace, i.e. trace divided by the size of the matrix, then the corresponding problem of determining validity of polynomial inequalities is decidable. I will describe the beautiful geometry that leads to decidability and undecidability. Time permitting, I will discuss related undecidability results in graph theory.

Based on joint work with Jose Acevedo, Sebastian Debus, Annie Raymond, Cordian Riener and Fan Wei.