A Topological Study Of Stochastic Dynamics On CW Complexes

A TOPOLOGICAL STUDY OF STOCHASTIC DYNAMICSON CW COMPLEXES byMICHAEL J. CATANZAROMAY 2016 Co-Advisor: Dr. John R. KleinCo-Advisor: Dr. Vladimir Y. ChernyakMajor:MathematicsDegree:Doctor of Philosophy 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 complexis motivated from Ito diffusions on smooth manifolds and Langevin processes in physics.We associate a Kolmogorov equation to this process, whose solutions can be interprettedin 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 ofthe complex. We show the average current generated by such a process can be written ina physically familiar form, consisting of the solution to Kirchhoff’s network problem andthe Boltzmann distribution, suitably generalized to arbitrary dimensions. We analyze thesetwo components in detail, and discover they reveal an unexpected amount of informationabout the topology of the CW complex. The main result is a quantization result for theaverage current in the low temperature, adiabatic limit. As an application, we express theReidemeister torsion of the complex, a topological invariant, in terms of these quantities.