Applied Algebraic Topology
My research broadly falls under the heading of applied algebraic topology. I am also interested in concrete applications of mathematical physics, where I try to answer questions in topology and physics by using tools from the other.
Here are some projects I have been fortunate enough to work on.
As a postdoc, I began work on geometric models for studying multiparameter persistence. One of the first examples of ordinary persistence arose from studying the sub-level sets of a Morse function. Together with Peter Bubenik, we propose looking at generic families of Morse functions to study multiparameter persistent homology. Each of these families has an associated Cerf diagram, and from this we attempt to build a decomposition of the associated multiparameter persistence module, analogous to the barcode.
Persistent homology for task modulation
Certain types of imaging data, like fMRI for example, naturally lend themselves to analysis using topological methods. The acquisition of an fMRI signal naturally breaks a region of interest up into three-dimensional voxels. This voxel structure can be used to model the region by a cubical complex, a topological object which is amenable to efficient homological calculations. This space acquires a natural filtration via the fMRI signal, and so the methods of topological data analysis are readily available. Using this set-up, we are currently working on understanding how task modulation in the ACC can be understood using fMRI through the lens of persistent homology. This project is joint with Vaibhav Diwadkar, Sam Rizzo, and Peter Bubenik.
Stochastic Currents on CW complexes
My thesis is based on a generalization of electrical current to higher dimensions. Specifically, I’m interested in quantization results for the `current’ generated by sub-objects under stochastic motion. This breaks into two cases: discrete (CW complexes with Markov processes) or continuous (smooth manifolds with stochastic vector fields), and the tools involved vary for each. One such tool in the discrete case are `higher spanning trees’, and we’ve enumerated them using Reidemeister torsion, an invariant in Algebraic K-theory. We’re still investigating the smooth case, using tools from functional analysis. All of this work generalizes the 1-dimensional graph results of my advisors, Vladimir Chernyak and John Klein. The interplay between these two settings also provides a nice framework to compare invariants in one to those in the other, e.g. Reidemeister to Ray-Singer torsion, using Witten-style deformations.
Electronic Excitations using Cohomology
One such problem involves counting electronic excitations, or excitons. Excitons are naturally occurring quasi-particles associated with the conversion of light to energy (e.g. photosynthesis). The number of such excitations in certain systems can be computed via a topological winding number using an index-like theorem.