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Large Scale Index Theory

Haj Saeedi Sadegh, Ahmad Reza | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 47085 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Esfahanizadeh, Mostafa
  7. Abstract:
  8. Studying interrelation between geometry and topology has always been an intriguing idea for classifying manifolds; characteristic classes, signature, Euler characteristic, index of Dirac operator are classical examples of topological invariants that use additional structure to be constructed. Index Theory is seeking to develop a literature to unite these invariants. A culmination in this theory happens in Atiyah and Singer index theorem, which gives a tool to relate elliptic operators on manifolds to the topology. Using this theorem, the index of Dirac operator on an spin manifold is independent of the geometric structure and so is decided by the topology. Another tool for analyzing elliptic operators is spectral theory, which in fact is wielded by index theory. By using spectral theory and asymptotic expansion, we can study geometrical structure and extract topological invariants; however, we won’t use this method in this thesis. Main purpose of this thesis is to describe Coarse geometric approach. In this way, instead of focusing on local structure, we study overall almost geometrical structure and construct or indices in new abelian group which extend the classical indices. Thus, we expand invariants’ space and actually use more new tools. For example, C*-algebras, their K-theory and K-homology, and twisted operators. This new environment provides us powerful tools for studying positive scalar curvature problems; indeed, these problems was the idea for developing Coarse Index Theory
  9. Keywords:
  10. Differential Operators ; Clifford Algebra ; Dirac Operator ; Index Theory ; Pseudo Differential Operator ; DIFFERENTIAL OPERATORS ; Elliptic Operator ; C* Algebra ; K-Theory

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