A higher-dimensional generalization of electrical current, studied via algebraic topology and stochastic processes.
In a graph, an electrical current is a flow along edges that satisfies Kirchhoff’s laws. Think of an electron moving through a wire. The flow can be deterministic or stochastic, and so random walks on graphs generate currents. In this case, the current encodes topological properties of the graph and can be studied under various dynamical limits. We think of current as a statistical object encoding how much net flow passes through each edge. This relationship between probability and topology is classical and well understood in dimension one.
My thesis generalizes this to higher dimensions(Catanzaro, 2016). Instead of measuring flow along edges (1-dimensional objects) in a graph, we consider flow along higher-dimensional sub-objects, like embedded surfaces or volumes. We study these questions in both CW complexes and smooth manifolds. In either case, A stochastic process can be defined, whether it is a Markov chain or a stochastic differential equation. Instead of thinking about points moving around the graph, we again generalize to moving higher-dimensional or extended objects around. These objects moving similarly generate a higher current, and the central question is: what topological information does this current carry?
The project splits into two parallel settings and the tools involved vary for each:
Discrete case (CW complexes with Markov processes): Under dynamical limits on the Markov process, the motion of the embedded object tends to be supported along higher spanning trees, the analog of a spanning tree on a graph. These higher spanning trees can be enumerated using Reidemeister torsion, an invariant from Algebraic K-theory. The combinatorial complexity here is much richer than the graph case and is analyzed in (Catanzaro et al., 2015).
Continuous case (smooth manifolds with stochastic vector fields): The analogous invariant is Ray-Singer torsion, an analytic object defined via zeta-regularized determinants of Laplacians. The two settings are connected by Witten-style Laplacian deformations, which relate the discrete and analytic torsions.
There is an interactive version of this higher dimensional analogue of random walks on a CW decomposition of a torus here: random torus walk
Beyond topology, there are also statistical mechanics implications of these processes. Quantization of currents and their relationship to nonequilibrium steady states are explored in (Catanzaro et al., 2016) and (Catanzaro et al., 2016). A mathematical generalization to a broader class of hypercurrents is pursued in (Catanzaro et al., 2020).
An initial cycle (shown on the right side) within a CW decomposition of a torus. The evolution of this initial condition is also shown later during evolution (back left). While the evolved cycle may become more complicated, it is always homologous to the initial cycle throughout the process.
We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. If the space is a one-dimensional manifold, then a protocol determines a continuous time Markov chain. When a homological gap condition is present, we associate to each protocol a ’characteristic’ cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the low temperature limit. We also exhibit examples of protocols having nontrivial hypercurrent.
2016
A Topological Study Of Stochastic Dynamics On CW Complexes
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.
Stochastic dynamics of extended objects in driven systems: I. Higher-dimensional currents in the continuous setting
The probability distributions, as well as the mean values of stochastic currents and fluxes, associated with a driven Langevin process, provide a good and topologically protected measure of how far a stochastic system is driven out of equilibrium. By viewing a Langevin process on a compact oriented manifold of arbitrary dimension m as a theory of a random vector field associated with the environment, we are able to consider stochastic motion of higher-dimensional objects, which allow new observables, called higher-dimensional currents, to be introduced. These higher dimensional currents arise by counting intersections of a k -dimensional trajectory, produced by a evolving ( k - 1 ) -dimensional cycle, with a reference cross section, represented by a cycle of complimentary dimension ( m - k ) . We further express the mean fluxes in terms of the solutions of the Supersymmetric Fokker–Planck (SFP), thus generalizing the corresponding well-known expressions for the conventional currents.
Stochastic dynamics of extended objects in driven systems II: Current quantization in the low-temperature limit
Driven Langevin processes have appeared in a variety of fields due to the relevance of natural phenomena having both deterministic and stochastic effects. The stochastic currents and fluxes in these systems provide a convenient set of observables to describe their non-equilibrium steady states. Here we consider stochastic motion of a ( k - 1 ) -dimensional object, which sweeps out a k-dimensional trajectory, and gives rise to a higher k-dimensional current. By employing the low-temperature (low-noise) limit, we reduce the problem to a discrete Markov chain model on a CW complex, a topological construction which generalizes the notion of a graph. This reduction allows the mean fluxes and currents of the process to be expressed in terms of solutions to the discrete Supersymmetric Fokker–Planck (SFP) equation. Taking the adiabatic limit, we show that generic driving leads to rational quantization of the generated higher dimensional current. The latter is achieved by implementing the recently developed tools, coined the higher-dimensional Kirchhoff tree and co-tree theorems. This extends the study of motion of extended objects in the continuous setting performed in the prequel (Catanzaro et al.) to this manuscript.
2015
Kirchhoff’s theorems in higher dimensions and Reidemeister torsion
Kirchhoff’s theorems in higher dimensions and Reidemeister torsionHomology, Homotopy and Applications, Dec 2015