Several Questions in Noncommutative Geometry

Vanderbilt University, Nashville TN

Investigators

Abstract

Abstract Yao The investigator proposes to study several different topics in noncommutative geometry (NCG). The first topic concerns one class of deformations of algebras arising from the theory of modular forms. These deformations have been related to the Hopf algebras which appear naturally as local symmetry for noncommutative spaces, such as those of leaves of a foliation. The PI plans to use methods and techniques of harmonic analysis (on the group of affine transformations of a real line) to search for a corresponding deformation theory in the framework of C* algebras, which can be considered as the set of continuous functions over noncommutative spaces. Other topics in this project concern cyclic cohomology, a noncommutative analogue of the classical De Rham cohomology, which encodes some topological properties of the algebras in question. The PI intends to apply some ideas of complex geometry to study some noncommutative manifolds. The PI plans also to investigate the cyclic cohomology of Hölder functions on the circle and other manifolds, and find applications on problems in harmonic analysis. Noncommutative geometry is a domain which is originated in the study of operator algebras(which, historically, has its origin in the mathematical foundation of quantum mechanics). Since late 70?s, this area interacts to a great extent with differential geometry, group theory, topology, etc.; and later with number theory or quantum field theory. By addressing both internal questions of this domain and its interactions with other research areas, one could expect further applications in physics once the primary goals of the project achieved.