Research Projects

To view my full CV and published papers please click the Research link at the top of the page.

Main Research Interests

The focus of my research is on applications of algebraic and combinatorial methods to problems that arise in data analysis. Some of the specific topics that I am currently studying are sketched below. For software and data related to these projects, see this page.

Political Redistricting

Research Papers

Total Variation Isoperimetric Profiles, with H. Lavenant, Z. Schutzman, and J. Solomon, SIAM J. Appl. Algebra Geometry, 3(4), 585-613, (2019).
Redistricting Reform in Virginia: Districting Criteria in Context, with M. Duchin, Virginia Policy Review, 12(2), 120-146, (2019).
Complexity and Geometry of Sampling Connected Graph Partitions, with L. Najt and J. Solomon, arXiv:1908.08881, (2019).
A Computational Approach to Measuring Vote Elasticity and Competitiveness, with M. Duchin and J. Solomon, submitted, (2019).
ReCombination: A family of Markov chains for redistricting, , with M. Duchin and J. Solomon, arXiv:1911.05725, (2019).
Partisan Dislocation: A Precinct-Level Measure of Representation and Gerrymandering, with N. Eubank and J. Rodden, submitted, (2020).
Mathematics of Nested Districts: The Case of Alaska, with S. Caldera, M. Duchin, S. Gutenkust, and C. Nix, Statistics and Public Policy (to appear), (2020).

Expository Papers and Empirical Projects

Aftermath: The Ensemble Approach to Political Redistricting (with J. Clelland and M. Duchin), MAA Math Horizons, 28(1), 34-35, (2020).
Introduction to Discrete MCMC for Redistricting (with Scrabble), Teaching Materials (2019).
Building Ensembles of Graph Partitions, Technical Documentation, (2019).
Comparison of Districting Plans for the Virginia House of Delegates, with M. Duchin and J. Solomon, MGGG Technical Report, (2019).
Mathematician's amicus brief to the Supreme Court, with M. Duchin and G. Charles et al., Rucho v. Common Cause, (2019).
Study of Reform Proposals for Chicago City Council, with M. Duchin et al., MGGG Technical Report, (2019).

Political redistricting can be abstracted as a graph partitioning problem subject to a variety of legal constraints. My main research in this area focuses on developing methods for efficient sampling of graph partitions usiing Markov chain techniques. I am also interested in shape based analysis of districting plans and metrics defined on the space of permissible graph partitions. For more details, see the description and notes from my 2019 MIT IAP course here.

There are many interesting, open problems in this domain that are well suited for student research projects. I also maintain an active list of open mathematical problems related to redistricting here, please feel free to email me for more details if any of these catch your interest.

Multiplex Networks

Spectral Clustering Methods for Multiplex Networks, with S. Pauls, Physica A, 121949, (2019).
A new framework for dynamical models on multiplex networks, with S. Pauls, Journal of Complex Networks, 6(3), 353-381, (2018).
Multiplex Dynamics on the World Trade Web, Proc. 6th International Conference on Complex Networks and Applications, Studies in Computational Intelligence, Springer, 1111-1123, (2018).
A Random Dot Product Model for Weighted Networks, with D. Rockmore, arXiv:1611.02530, (2016).

While standard network models consider a set of nodes an a binary relation between them, multiplex networks allow for many different types of connections between the nodes. For example, we might consider a social multiplex whose nodes are people and where different types of edges represent familial ties, coworkers, and friendship relations.

I am particularly interested in dynamical models on these networks, such as information flow through our social network example, and the properties of their associated operators. Current research includes developing clustering algorithms that respect multiplex structure and better understanding the effects of modeling choices in application domains, as well as machine learning approaches for inferring inter-layer edge weights.

Representations and Eigenvalues

On the Spectrum of Finite Rooted Homogeneous Trees, with D. Rockmore, Linear Algebra and Applications, 598, 165-185, (2020).
Fourier transforms on \(SL_2(\mathbb{Z}/p^n\mathbb{Z})\) and related numerical experiments, with B. Breen, J. Linehan, and D. Rockmore, arxiv: 1710.02687, (2017).

The connection between the Fourier transform on finite groups and eigenvalues of Cayley graphs leads to many natural questions, both in graph theory and representation theory. I am particularly interested in questions relating to the convergence of the spectral measures for sequences of these graphs. One way to formalize this quesion is to ask for a combinatorial characterization of increasing graph sequences that satisy for all \(\varepsilon > 0\) there exists a finite set \(\Lambda \subseteq \mathbb{R}\) and an integer \(N\) such that for all \(n > N\) we have \(\dfrac{|\{ \lambda \in \operatorname{spec}(G_n): \lambda \notin \Lambda\}|}{|\operatorname{spec}(G_n)|} \). The future work sections of the papers linked above provide sketches of additional current research questions. Addtionally, various bases for the dihedral groups lead to families of interger spectra graphs, while others define a periodic spectral structure that appears to obey a well-defined limit law. Explaining these phenomena is an ongoing research project, see here for related software and figures.

Entropy in Time Series

Random Walk Null Models for Time Series Data, with K. Moore, Entropy, 19(11):615, (2017).

Time series data consists of real valued samples, indexed by a time parameter, such as daily stock returns or temperature values. I am particularly interested in probabilistic generative models for this data and their relation to entropy measures. Current work with Kate Moore focuses on clustering and detecting distributional changes using information-theoretic metrics and Markov based null models. These null models represent a given time series by a Markov chain defined on permutations, whose transition probabilities are inferred from the data. This model allows for both theoretical results for known distributions as well as an empirical test for divergence between the steady state of the Markov chain and the observed distribution.

Other Projects:

This section contains links to brief descriptions of various research projects, outside of my main areas, that have also captured my interest. Each .pdf contains some background material, thoughts on possible approaches, and a bibliography.