Chern–Weil Theory Extended to a Class of Infinite Dimensional Bundles

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Hesari, Abdolaziz | 2014

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 45726 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahani Zadeh, Mostafa
Abstract:
Given a principal bundle a characteristic class is an element of the cohomology al- gebra of the classifying space of structure group of the bundle with coeﬃcients in a commutative ring with unit ,which have a functorial property .When the structure groupe of the bundle is a Lie group and the coeﬃcient ring is the real or complex num- bers , the Chern–Weil approach provides a geometric construction of char-acteristic classes. Classical Chern–Weil formalism relates geometry to topology, assigning to the cur- vature of a connection, de Rham cohomology classes of the underlying manifold.This theory developped in the 40’s by Shiing-ShenChern and Andre Weil which can be seen as a generalisation of the Chern-Gauss-Bonnet Theorem .In this thesis we extended the classical Chern–Weil formalism to an inﬁnite n-sional setup using regularised traces on classical pseudodifferential operators
Keywords:
Pseudo Differential Operator ; Infinite Dimensional Manifolds ; Wadzicki Trace ; Leading Symbol Theory ; Chern-Weil Theory