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

- Redistricting Reform in Virginia: Districting Criteria in Context, with M. Duchin, Virginia Policy Review, 12(2), 120-146, (2019).
- Total Variation Isoperimetric Profiles, with H. Lavenant, Z. Schutzman, and J. Solomon, SIAM Journal on Applied Algebra and Geometry, (2019).
- Complexity and Geometry of Sampling Connected Graph Partitions, with L. Najt and J. Solomon, arXiv:1908.08881, (2019).
- Mathematics of Nested Districts: The Case of Alaska, with S. Caldera, M. Duchin, S. Gutenkust, and C. Nix, Preprint, (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).

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, arXiv:1903.07134, (2019).
- 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).

#### Entropy in Time Series

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

#### 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.

- Computing Stirling numbers for families of graphs and graph products
- Finding a module-theoretic decomposition of LHCCRR space and high-order Lucas Bases.
- Creating an efficient algorithm for finding minimal division chains
- Investigating the splitting behaviour of theta series lifted via spherical polynomials
- Applying SFC methods to develop efficient implementations for scientific computing