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Nonlocal Mathematics

Mukul Patel
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Abstract

This is a research announcement on what is best termed `nonlocal' methods in mathematics. (This is not to be confused with global analysis.) The nonlocal formulation of physics in \cite{principia} points to a fresh viewpoint in mathematics: Nonlocal viewpoint. It involves analyzing objects of geometry and analysis using nonlocal methods, as opposed to the classical local methods, e.g., Newton's calculus. It also involves analyzing new nonlocal geometries and nonlocal analytical objects, i.e. nonlocal fields. In geometry, we introduce and study (nonlocal) forms, differentials, integrals, connections, curvatures, holonomy, G-structures, etc. In analysis, we analyze local fields using nonlocal methods (semilocal analysis); nonlocal fields nonlocally (of course); and the connection between nonlocal linear analysis and local nonlinear analysis. Analysis and geometry are next synthesized to yield nonlocal (hence noncommutative) homology, cohomology, de Rham theory, Hodge theory, Chern-Weil theory, K-theory (called N-theory) and index theory. Applications include theorems such as nonlocal-noncommutative Riemann-Roch, Gauss-Bonnet, Hirzebruch signature, etc. The nonlocal viewpoint is also investigated in algebraic geometry, analytic geometry, and, to some extent, in arithmetic geometry, resulting in powerful bridges between the classical and nonlocal-noncommutative aspects.

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Mukul Patel. 2000-08-02. Nonlocal Mathematics. https://arxiv.org/abs/math/0008023

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