Introduction to Interlacing Polynomials, Barrier Functions, and Kadison-Singer

I will introduce the method of interlacing polynomials and analysis by barrier functions through a proof of the Restricted Inevitability Principle.  I will state Weaver's discrepancy-theoretic version of the Kadison-Singer Conjecture, and sketch the role that hyperbolic polynomials play in its proof.  If time permits, I will connect these with Ramanujan graphs and the matchings polynomials of graphs.

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