## Seminar:

Spring 2013, Thursdays, Milner 216, 4:00-4:50 PM

Date: February 7

"  Mini-Talks

* Unitary Yang-Baxter Operators

* Can you hear the density of a taut string "

* Prof. Eric Rowell

* Prof. Willaim Rundell

Abstract

* The Yang-Baxter equation (YBE) goes back at least to the late 1960s appearing in the papers of physicists working in statistical mechanics. In the 1980s the YBE fell into the hands of algebraists, who built machines for producing solutions to the YBE, namely quantum groups. Now the YBE is of interest to physicists again, with potential applications to quantum information. I will briefly describe this recent connection and a very simple stated conjecture that would have important information-theoretic ramifications.

* The title could also have been "can you determine the interior density of the sun from vibrational models". I admit these are a bit cryptic from a mathematical perspective, so here is a more concrete problem: If $A$ is an $n \times n$ matrix with unknown entries can I determine these if I know the eigenvalues $\{\mu_k\}_1^n$ of $A$? Clearly, the answer is no, certainly no in a unique way: we have $n^2$ unknowns but only $n$ pieces of information. What if now $A$ is completely known but I add an unknown diagonal matrix $D$ to it and give you the eigenvalues $\{\lambda_k\}_1^n$ of $A+D$? Can I determine $D$? Now the known/unknown count is correct. Good, but certainly not enough to give a firm answer. I actually want to pose this problem but in a pde setting and the answers will give insight into the two original questions.