Piotr M. Hajac (IMPAN and University of Colorado Boulder) From pushouts to pullbacks: a sample of noncommutative topology
Thu, Oct. 10 2pm (MATH 350)
Functional Analysis
Mitsuru Wilson (CU Boulder) Symbolic calculus on homogeneous vector bundles (Part II)
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In topology, pushouts are formal recipes for collapsing and gluing topological spaces. For instance, shrinking the boundary circle of a disc to a point yields a sphere, shrinking the equator of a sphere to a point gives two spheres joined at the point, collapsing the boundary of a solid torus to a circle, or gluing two solid tori over their boundaries, produces a three-sphere. In noncommutative topology, such procedures are expressed in terms of pullbacks of C*-algebras. It turns out that one can visualize a pullback of C*-algebras of graphs as a pushout of these graphs thus providing much needed intuition to the abstract setting of operator algebras. The goal of this talk is to discuss how to make this visualization rigorous by conceptualizing abundant examples from noncommutative topology that lead to a new concept of morphisms of graphs. In particular, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. (Based on joint works with Alexandru Chirvasitu, Sarah Reznikoff and Mariusz Tobolski.)
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This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators.