Access Type
Open Access Dissertation
Date of Award
1-1-2016
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
Vladimir Y. Chernyak
Second Advisor
John R. Klein
Abstract
In this dissertation, we consider stochastic motion of subcomplexes of a CW complex, and explore the implications on the underlying space. The random process on the complex is motivated from Ito diffusions on smooth manifolds and Langevin processes in physics. We associate a Kolmogorov equation to this process, whose solutions can be interpretted in terms of generalizations of electrical, as well as stochastic, current to higher dimensions. These currents also serve a key function in relating the random process to the topology of the complex. We show the average current generated by such a process can be written in a physically familiar form, consisting of the solution to Kirchhoff’s network problem and the Boltzmann distribution, suitably generalized to arbitrary dimensions. We analyze these two components in detail, and discover they reveal an unexpected amount of information about the topology of the CW complex. The main result is a quantization result for the average current in the low temperature, adiabatic limit. As an application, we express the Reidemeister torsion of the complex, a topological invariant, in terms of these quantities.
Recommended Citation
Catanzaro, Michael Joseph, "A Topological Study Of Stochastic Dynamics On CW Complexes" (2016). Wayne State University Dissertations. 1433.
https://digitalcommons.wayne.edu/oa_dissertations/1433