The Mirror of Mathematics
Spring School in Quantum Structures in Physics and Computer Science, 19-22 May 2014, University of Oxford: .
Abstract: The word "duality" is tossed around in almost all areas of mathematics and physics (and even parts of computer science). Roughly speaking it allows one to view two, apparently different, mathematical universes as "mirror images of each other." In mathematics the most venerable and well-known such duality is the familiar duality between a vector space and the space of linear functionals on a vector space. More sophisticated versions of this arise in functional analysis. In physics one talks about dualities between electric and magnetic fields and more sophisticated versions of that duality arise in non-abelian gauge theories.
In this series of three lectures I will discuss the three dualities mentioned in the title. Stone duality relates Boolean algebras with certain kinds of topological spaces and is fundamental for logic. Variations of this duality arise throughout theoretical computer science. The second one on the list is more functional-analytic in nature and relates compact Hausdorff spaces to commutative unital C*-algebras. This is a stepping stone towards the much deeper dualities of interest in quantum mechanics. The third duality, Pontryagin duality, is what underlies Fourier theory and will involve group theory.
A little bit of algebra (groups, rings, algebras) and topology (compactness, separation axioms) will be useful background. The categorical version of the dualities will be given but the category theory used will be elementary, say up to the level of knowing what "adjoint functor" means.
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