Course Number:

Credit Hours:

4

Course Title:

Von Neumann Algebras and Applications

Course Description:

Introduces von Neumann algebras and their applications to analysis, geometry, and topology. Topics include algebras of operators in a Hilbert space; uniform, strong, and weak topology in the algebra of bounded linear operators in a Hilbert space; von Neumann algebras, traces, and von Neumann dimensions; ideals in von Neumann algebras; factors and their classification; von Neumann algebras and traces associated with actions of discrete groups on manifolds; trace class operators and Hilbert-Schmidt operators; tensor products of von Neumann algebras and traces; analytic expression of traces; elliptic operators; pseudodifferential operators and their Schwartz kernels; uniform Sobolev spaces; index theory in von Neumann algebras and Atiyah L2 index theorem on covering manifolds; von Neumann Betti numbers and Euler characteristics; heat kernel invariants and spectra-near-zero invariants, their interpretation as near-cohomology and homotopy invariance; Witten deformation and semiclassical asymptotics on covering manifolds with applications to L2 Morse inequalities; and L2 Riemann-Roch theorem for elliptic operators.