Interlacing families: a new technique for controlling eigenvalues

Job Candidate Talk
Thursday, January 15, 2015 - 1:00pm for 1 hour (actually 50 minutes)
Adam Marcus – Yale University
Christopher Heil
Matrices are one of the most fundamental structures in mathematics, and it is well known that the behavior of a matrix is dictated by its eigenvalues. Eigenvalues, however, are notoriously hard to control, due in part to the lack of techniques available. In this talk, I will present a new technique that we call the "method of interlacing polynomials" which has been used recently to give unprecedented bounds on eigenvalues, and as a result, new insight into a number of old problems. I will discuss some of these recent breakthroughs, which include the existence of Ramanujan graphs of all degrees, a resolution to the famous Kadison-Singer problem, and most recently an incredible result of Anari and Gharan that has led to an interesting new anomaly in computer science. This talk will be directed at a general mathematics audience and represents joint work with Dan Spielman and Nikhil Srivastava.