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 on the Traveling Salesman
problem that has produced an interesting anomaly in computer science.
This talk will be directed at a general mathematics audience and
represents joint work with Dan Spielman and Nikhil Srivastava.
|