Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
In this talk we will discuss how ideas from noncommutative geometry put forward by Kontsevich can be used as a framework to study matrix models. We will explore the case of the Gaussian Unitary Ensemble from this perspective and determine some of its large N behavior.We will discuss perturbative path integral quantization of scalar field theory with a polynomial interaction potential on two-dimensional compact Riemannian manifolds with boundary. The perturbative partition function is defined in terms of configuration space integrals on the surface. Moreover, partition functions can be organized into a functor, in the sense of Atiyah-Segal axiomatics, from the Riemannian cobordism category to the category of Hilbert spaces. A crucial role in the result is played by non-trivial behavior of tadpoles (short loops) under gluing. This is based on joint work with Pavel Mnev and Konstantin Wernli.