Note on the Kadison-Singer Problem and its Solution

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The Kadison-Singer problem arose from the work on quantum mechanics done by Paul Dirac in the 1930s. The problem is equivalent to fundamental problems in areas like Operator theory, Hilbert and Banach space theory, Frame theory, Harmonic Analysis, Discrepancy theory, Graph theory, Signal Processing and theoretical Computer Science. The Kadison-Singer problem had been long standing and defied the efforts of most Mathematicians until it was recently solved by Adam Wade Marcus, Daniel Alan Spielman and Nikhil Srivastava for which they were awarded the George Polya Prize in Mathematics in 2014, and very recently, the Michael and Sheila Held Prize in 2021. The proof uses an existence argument which reduces the problem to bounding the roots of the expected characteristic polynomial of certain random matrices employing tools from the theory of random polynomials.

This note was a part of my Randomized Algorithms and Probabilistic Analysis course project. The complete note can be found here.

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