Two weeks on Global Analysis

course: C-algebras and elliptic theory

speaker: Alexandr Mishchenko (Moscow State University)

abstract: Elements of vector bundles theory and topological K-theory. Vector bundles. Homotopy theory of vector bundles. K-groups. Bott periodicity in K-theory. The Thom isomorphism. Characteristic classes and the Chern character. Calculus in K-theory. Elements of operator algebras and modules. Elements of C-algebras theory. The 1st and 2nd Gelfand-Neimark theorems. Basic example of C-algebras (the group C-algebras of locally compact groups, the Calkin algebra, the Cuntz algebra, von Neumann algebras, factors, C-algebras of foliations, C-algebra of gruppoid). Hilbert modules and operators. The Kasparov theorem on stabilization. Operator algebras on the Hilbert modules. Multipliers. Homotopy theory in the category of C-algebras. Operator K-theory. Fredholm representations and KK-functor by Kasparov. Product-intersection. Asymptotical homomorphisms of C-algebras. The Voiculescu example. E-thoery by Connes-Higson. Relation to K- and KK-theory. Elliptic operators and index theory. Jets, differential operators. Sobolev spaces, the inclusion theorems. Pseudo-differential operators. Axiomatic approach. Examples: the Laplace, Hirzebruch, Dirac operators. The index theorem. C-theorem about the index. K-homology and the index theorem. Application to topology and geometry. Fundamental group and invariants of manifolds. Algebraic Poincaré complexes. Higher structures. Canonical bundle. The Novikov conjecture. Amenable and hyperbolic groups. The Connes-Moscovici theorem. Curvature and the Gromov-Lawson-Rosenberg conjecture. The Baum-Connes conjecture. Extension of C-algebras.

timetable:
Mon 21 Feb, 9:00 - 9:55, Sala Conferenze Centro De Giorgi
Tue 22 Feb, 9:00 - 9:55, Sala Conferenze Centro De Giorgi
Wed 23 Feb, 9:00 - 9:55, Sala Conferenze Centro De Giorgi
Thu 24 Feb, 9:00 - 9:55, Sala Conferenze Centro De Giorgi
Fri 25 Feb, 9:00 - 9:55, Sala Conferenze Centro De Giorgi
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