Interlacing Families and Kadison–Singer
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This talk will focus on the resolution of two conjectures — Weaver's so-called "kS_2" and the Anderson Paving Conjecture. These conjectures have been shown to be equivalent to open problems in a number of fields, including C^* algebras, Banach space theory, harmonic analysis, and frame theory. In particular, they were known to imply an answer to a fundamental open question in mathematical physics due to Kadison and Singer.
I will discuss the connections and ramifications of these problems and then walk through the ideas and techniques that contribute to the proofs. This will include introducing a new technique for establishing the existence of certain combinatorial objects that we call the "method of interlacing polynomials." This technique seems to be interesting in its own right (it was also the main tool for resolving the existence of Ramanujan graphs of arbitrary degree).
This represents join work with Dan Spielman of Yale University and Nikhil Srivastava of Microsoft Research, India.