Comprehensive Exam Presentation

Speaker: Jason Haradyn (Western)

"The title is Noncommutative Einstein Manifolds"

Time: 15:30

Room: MC 108

This talk will be focused on the theory of noncommutative Einstein manifolds. I will recall the idea of a spectral triple and specialize to the case that requires a real structure J to exist in this spectral triple. We can think of this real structure as a simultaneous generalization of the charge conjugation operator acting on the spinor bundle over a spin manifold, and as the Tomita J-operator of Tomita-Takesaki theory that classifies modular automorphisms of von Neumann algebras. I will then define a noncommutative Einstein manifold in terms of a spectral characterization -- more precisely, in terms of the asymptotic expansion of its heat trace. We will see that this definition is, in fact, correct, because of the equivalence to the classical notion of an Einstein manifold in the commutative case. Furthermore, I will explain the noncommutative Einstein-Hilbert action and Connes' trace theorem, which connects Dirac operators and the Einstein-Hilbert action via the Wodzicki residue. Finally, I will introduce the fundamental examples of noncommutative Einstein manifolds and explain how we classify abstract noncommutative spin geometries on manifolds via Connes' reconstruction theorem.

Comprehensive Exam Presentation

Speaker: Mitsuru Wilson (Western)

"Spectral Method of Computing Curvature of Noncommutative Spaces Arising From Deformation Quantization"

Time: 13:30

Room: MC 108

Deformation quantization or a star product ? is a way of producing a noncommutative algebra from the algebra C1(M) of smooth functions on a manifold. A deformation quantization gives rise to a Poisson structure on M : Conversely, which is a very nontrivial result, Kontsevich proved that every Poisson manifold (M; f; g) admits a deformation quantization. The existence of the star product creates an immense collection of examples of noncommu- tative spaces; moreover, they arise from the classical ones! In 2011, Fathizadeh and Khalkhali, and independently Connes and Moscovich in two coauthored papers computed using spectral methods the curavature of the noncommutative tori A . The rst incidence in which the curvature of a noncommutative space had ever been computed by this method. This is achieved by evaluating the value of the analytic continuation of the spectral zeta function. The modular automorphism group from the theory of type III factors and quantum statistical mechanics appears in the nal formula for the curvature. The main instrument here is the asymptotic expansion of the heat trace of the Laplacian in the spectral triple attached to A . In my talk, I will explain the basics in NCG and the recent development respecting my thesis project. Everyone is welcome :)

Homotopy Theory

Speaker: Daniel Schaeppi (Western)

"The thick subcategory theorem"

Time: 14:30

Room: MC 108

Index Theory Seminar

Speaker: Masoud Khalkhali (Western)

"The heat equation proof of the Atiyah-Singer index theorem I"

Time: 14:00

Room: MC 107

The starting point for the heat equation proof of the Atiyah-Singer index theorem is the celebrated McKean-Singer formula and the asymptotic expansion of the heat kernel. In this first lecture the following topics will be covered: basic Sobolev space theory, including Garding's inequality; finiteness and regularity results for elliptic PDE's on compact manifolds, Hodge decomposition theorem for elliptic complexes, existence of the heat kernel and its asymptotic expansion.