RUI: Versal Deformations, Deformation Quantization, Moduli Spaces and Graph Complexes

University Of Wisconsin-Eau Claire, Eau Claire WI

Investigators

Abstract

This project has three parts: (1) To study the versal deformations of A_\infty , L_\infty algebras, which are generalizations of Lie and associative algebras; (2) To study the problem of deformation quantization of polynomial Poisson algebras; (3) To address some problems in the study of the moduli space of Riemann surfaces with marked points that arise from the combinatorial equivalence of this space and the orbifold of metric Ribbon graphs. The first part continues a program being carried out jointly with Alice Fialowski. The second part is about generalizing a purely cohomological construction of the PI and Pol Vanhaecke of the unique generic deformation quantizationof order three which extends to a fourth order deformation. The third part, which is joint work with Motohico Mulase, is more open-ended. Reasonable goals might be: to establish a canonical orbifold diffeomorphism between the moduli space of Riemann surfaces and the Ribbon graph complex; to give concrete characterizations of the Strebel differential in some interesting cases; to study solutions to the KP system determined by the correspondence between ribbon graphs and Riemann surfaces over the algebraic closure of the rationals; to describe in detail the correspondence between the homology of the Ribbon graph comples, and the moduli space of Riemann surfaces with marked points. This project consists of three problems in mathematics related to or motivated by physics. The first problem is to study what happens when a very general kind of algebraic structure, one that arises in both mathematics and physics, is "deformed," for example by incorporating some special little twisting process into the usual algebraic operation of multiplication. This investigation continues collaborative work with a colleague in Hungary. The second problem involves a different kind of deformation process applied to a certain class of polynomial algebras. The third, and most open-ended problem, is about investigating a remarkable and unexpected correspondence between an algebraically defined geometric space and a differentially (as in calculus) defined geometric object. This third part also provides numerous opportunities for undergraduate involvement in research.