非局部数学
Nonlocal Mathematics
摘要：
这是一个研究公告什么是最好的所谓的“非本地”的方法在数学。(This不应与全局分析混淆）。\cite{principia}中物理学的非定域表述指向了数学中的一个新观点：非定域观点。它涉及使用非局部方法分析几何对象和分析，而不是经典的局部方法，例如，牛顿的微积分。它还涉及分析新的非局部几何和非局部分析对象，即非局部场。在几何中，我们介绍和研究（非局部）形式，微分，积分，连接，曲率，完整性，G-结构等在分析中，我们分析局部域使用非局部方法（半局部分析）;非局部域非局部（当然）;和非局部线性分析和局部非线性分析之间的联系。分析和几何接下来综合产生非局部（因此非交换）的同调，上同调，德拉姆理论，霍奇理论，陈-韦伊理论，K-理论（称为N-理论）和指数理论。应用程序包括定理，如非本地非交换黎曼-罗克，高斯-邦纳，Hirzebruch签名等非本地的观点也调查了代数几何，解析几何，并在一定程度上，在算术几何，从而在强大的桥梁之间的经典和非本地非交换方面。
英文摘要：
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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