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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 in 2013. Read more
A note on Conformal Symplectic and Relativistic Optimization
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This note on a spotlight paper at NeurIPS 2020, has been made while I had been reading the literature on the principle connections between continuous and discrete optimization. The motivation is to understand and create accelerated discrete large scale optimization algorithms from first principles via considering the geometry of phase spaces and numerical integration, specifically symplectic integration. Recent works successfully have been able to throw sufficient light on the two and therefore has attracted my attention. Read more
Geometry of Relativistic Spacetime Physics
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This article introduces the mathematical structures needed to understand relativistic spacetime physics. The self-referential and self-contained nature of mathematics provides a rigorous language for formulating the components of Einstein’s general theory of relativity—spacetime, matter, and gravity—along with their behavior and interactions. These notes begin with smooth manifolds and then add the necessary and sufficient differential-geometric structures. Read more
Dual spaces and the Fenchel conjugate
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Dual spaces lie at the core of linear algebra and allow us to reason formally about duality in mathematics. Duality appears naturally in measure theory, functional analysis, and mathematical optimization. In this post, I explore dual spaces and their interpretation in linear algebra, motivated by the so-called convex conjugate, or Fenchel conjugate, in mathematical optimization. Read more
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