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.
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This article introduces and describes the mathematical structures and frameworks needed to understand the modern fundamental theory of Relativistic Spacetime Physics. The self-referential and self-contained nature of Mathematics provides enough power to prescribe a rigorous language needed to formulate the building components of the standard Einstein’s General Theory of Relativity like Spacetime, Matter, and Gravity, along with their behaviors and interactions. In these notes, we will introduce and understand these abstract components, starting with defining the arena of smooth manifolds and then adding the necessary and suffcient differential geometric structures needed to build the primers to the General Theory of Relativity. Read more
Dual spaces lie at the core of linear algebra and allows us to formally reason about the concept of duality in mathematics. Duality shows up naturally and elegantly in measure theory, functional analysis, and mathematical optimization. In this post, I have tried to learn and explore the nature of duality via Dual spaces, its interpretation in general linear algebra, all of which was motivated by the so called convex conjugate, or the Fenchel conjugate in mathematical optimization. Read more
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