To view my full CV and published papers please click the Research link at the top of the page. If you are an undergraduate or graduate student at WSU interested in working on one of these projects, please send me an email.
Main Research Projects
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
- Empirical Sampling of Connected Graph Partitions for Redistricting with L. Najt and J. Solomon, Physical Review E, 104, 064130, (2021).
- ReCombination: A family of Markov chains for redistricting, with M. Duchin and J. Solomon, Harvard Data Science Review, 3(1), (2021).
- Colorado in Context: Congressional Redistricting and Competing Fairness Criteria in Colorado, with J. Clelland, H. Colgate, B. Malmskog, and F. Sancier-Barbosa, Journal of Computational Social Science, Online First, (2021).
- Implementing partisan symmetry: Problems and paradoxes, with N. Dhamankar, M. Duchin, V. Gupta, M. McPike, G. Schoenbach, K. W. Sim, Political Analysis, (to appear 2021).
- Partisan Dislocation: A Precinct-Level Measure of Representation and Gerrymandering, with N. Eubank and J. Rodden, Political Analysis, (to appear 2021).
- Medial Axis Isoperimetric Profiles, with P. Zhang and J. Solomon, SGP'20 Computer Graphics Forum, 39(5), 1-13, (2020).
- A Computational Approach to Measuring Vote Elasticity and Competitiveness, with M. Duchin and J. Solomon, Statistics and Public Policy, 7(1), 69-86, (2020).
- Mathematics of Nested Districts: The Case of Alaska, with S. Caldera, M. Duchin, S. Gutenkust, and C. Nix, Statistics and Public Policy, 7(1), 39-51, (2020).
- 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 J. Appl. Algebra Geometry, 3(4), 585-613, (2019).
- 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).
- Expository paper explaining the basics of the ensemble method for analyzing political districts using MCMC.
- Expert Report (and rebuttal report) for Pennsylvania (2022)
- Analysis of proposed Congressional redistricting plans for Pennsylvania on behalf of Citizen Mathematicians and Scientists in litigation before the Commonwealth Court.
- Expert Report (and Rebuttal Report) for Wisconsin (2021 and 2022)
- Analysis of proposed Congressional and State Legislative redistricting plans for Wisconsin on behalf of Citizen Mathematicians and Scientists in litigation before the Supreme Court of Wisconsin.
- Analysis of Prospective Districts in Colorado (2021)
- Reports comparing Colorado's Staff maps for Legislative and Congressional districts to a large ensemble of randomly generated maps. Uses 2020 precinct data and election data from 2016-2020.
- Applying GerryChain:
A User's Guide for Redistricting Problems (2021)
- Description of modeling methodology for applying the ensemble method using GerryChain to analyze political redistricting problems. This guide was created by a team of research fellows that I supervised through the 2021 UW Data Science for Social Good program.
- Random Walks and the Universe of Districting Plans, with M. Duchin, Political Geography, Birkhauser, (to appear 2022).
- Book chapter describing the main discrete MCMC approaches employed in the analysis of districting plans. Includes numerous examples and background material intended to assist students with the underlying mathematical ideas.
- Mathematician's amicus brief to the Supreme Court, with M. Duchin and G. Charles et al., Rucho v. Common Cause, (2019).
- Brief to the US Supreme Court discussing the ensemble method in the context of North Carolina's Congressional districts.
- Study of Reform Proposals for Chicago City Council, with M. Duchin et al., MGGG Technical Report, (2019).
- Analysis of multimember ranked choice proposals for Chicago City Council districts.
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
- 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).
- Spectral Clustering Methods for Multiplex Networks, with S. Pauls, Physica A, 121949, (2019).
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.
Random Dot Product Graphs
- A Random Dot Product Model for Weighted Networks, with D. Rockmore, arXiv:1611.02530, (2016).
- Maximum a Posteriori Inference of Random Dot Product Graphs via Conic Programming with D. Wu and D. Palmer, SIOPT, (2022).
The Random Dot Product Graph (RDPG) model is a latent space model for complex networks, where each node is associated to a vector in \(\mathbb{R}^d\) and nodes are connected with probability equal to the dot product of their respective vectors. The geometry of this relationship provides natural connections to notions of comunity assignment and centrality as well as allowing for the construction of multiresolution models with community structure. This model has proven useful theoretically, for proving consistency results about spectral embeddings, as well as practically, with recent applications to many machine learning problems on graphs. I am particularly interested in methods for learning RDPG representations from empirical networks as well as proving results relating the geometric properties of the embeddings to the combinatorial structure of the associated graphs.
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).
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