Noncommutative Geometry Seminar

The main theme of this talk is of "emergent geometry". We think of geometry as a tool to study analysis. We argue that a linear elliptic second-order partial differential operator on a vector bundle over a smooth manifold determines the local geometry of the manifold. The essential properties of the partial differential operators are determined by their leading symbols. The operators with scalar leading symbols (so called Laplace type operators) determine the Riemannian geometry. However, Riemannian geometry is inadequate to study the operators with non-scalar leading symbols (so called non-Laplace type operators). Such operators naturally define a collection of Finsler geometries on the manifold, which can be thought of as a non-commutative deformation of Riemannian geometry when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a "non-commutative" metric. We generalize the basic concepts of Riemannian geometry to the non-commutative setting. We define the natural partial differential operators associated with it and study their spectral asymptotics. We compute the first two non-zero heat trace coefficients for manifolds without boundary. We propose a non-commutative deformation of the Einstein-Hilbert action functional as a linear combination of the first two spectral invariants. The critical points of this functional naturally define the "non-commutative" generalization of Einstein equations.