Polynomial Techniques in Quantitative Linear Algebra

Job Candidate Talk
Wednesday, March 7, 2018 - 11:00am for 1 hour (actually 50 minutes)
Skiles 005
Adam Marcus – Princeton University – amarcus@math.princeton.eduhttps://web.math.princeton.edu/~amarcus/
Galyna Livshyts
I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the "method of interlacing polynomials") and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.