The Geometry of Gerrymandering

Why math and society?

I am haunted by these existential questions:

• What is the role of mathematics in society?
• What is my role as a mathematician in society?
• What is my role as a human being in society?

I'm going to explore these questions by giving talks on the intersection of math and society at the University of Toronto, at every platform I can get. I'll highlight the interesting research of others, because I think academics need to help each other to "signal-boost" more to get important messages out, instead of just talking about their own work.

Abstract of the talk

Can geometry, graph theory and probability theory counter the manipulation of elections? A growing body of math research tackles gerrymandering, which is the practice of redrawing boundaries between voting districts to favour one's own political party, such as by splitting up communities of opposition. Let us amuse ourselves with some really weird-looking gerrymandered district maps, while I review some geometric measures of "niceness of shape" proposed to disqualify obvious cases of manipulation. I will then focus on the recent strategy of sampling the universe of possible district maps using random walks, in order to identify "outlier" maps. This approach has been discussed in the US Supreme Court, which invites us to wonder: what could closer collaboration between mathematicians and civil society look like?

This is a survey without much technical details. If you know what a subgraph is, then you will understand everything. If you don't, you can still get a lot from most of it!

(Disclaimer: This talk involves a Singaporean commenting on the politics of the United States, by speaking in Canada! I do not claim to be very familiar with the political intricacies, but by sharing what I know, I hope to spark a discussion on some very important questions.) 

Where to learn more?

 Prof. Moon Duchin's talk "Political Geometry: Voting Districts, ''Compactness," and Ideas About Fairness" is quite accessible, and she goes into deeper depth about the nuanced difficulties faced by compactness scores, what helps the judicial system to trust a mathematical metric, and the political process in the US in general. She was personally involved in the effort convince the Supreme Court of the United States to adopt the "ReCom" method presented in my talk. The Wikipedia page for the Supreme Court case in question, Rucho v. Common Cause, tells a fascinating story, and serves as a window into how democracy can be undermined by partisan manipulation (every Supreme Court justice agreed on that).

The ​Metric Geometry and Gerrymandering Group (​MGGG), founded by Prof. Duchin, has a lot of resources relating to the math behind gerrymandering, including fun interactive web applets to let you explore the concepts involved.

On the research side, the theory of counting spanning trees on lattices has surprising connections to the physics of crystals; as a result, many of the asymptotic counting formulas were derived by physicists. Unfortunately I do not have a good reference at hand to explain this connection. Please tell me if you find one so I an actually understand this stuff!

References

• Moon Duchin. Political Geometry: The Mathematics of Redistricting. Talk at the Radcliffe Institute (2018).
• Daryl DeFord, Moon Duchin, Justin Solomon. Recombination: A family of Markov Chains for redistricting. (2019)
• Moon Duchin. Political Geometry: Voting Districts, ''Compactness," and Ideas About Fairness. Talk at the Joint Mathematics Meetings (2018).
• ​MGGG, Welcome to Gridlandia! An interactive introduction to the math of redistricting.
• Glasser, M. L., and F. Y. Wu. On the entropy of spanning trees on a large triangular lattice. The Ramanujan Journal 10.2 (2005): 205-214.
• MGGG, The Mathematicians' Brief in Rucho v. Common Cause. (2019)
• Supreme Court of the United States. Rucho v. Common Cause, No. 18-422, 588 U.S. ___ (2019)

Acknowledgements

Thank you to Assaf Bar-Natan and Lorenzo Najt for connecting me to relevant research! My talk grew from a seed planted by Moon Duchin when I attend a talk by her at MIT on "compactness scores," or how to measure the messiness of voting districts. I drew inspiration from Prof. Duchin's 

​I'd also like to highlight the great work of the Metric Geometry and Gerrymandering Group (MGGG), founded by Prof. Duchin, for their excellent work not only in the mathematical theory of gerrymandering, but also in bringing together mathematicians, political scientists, researchers in law and so on to collaborate. I think the complex social issues of our time need to be understood from both the technical and humanistic viewpoints, so scientists need to learn from their counterparts in the humanities, and vice versa, in order to craft solutions that are technically sound and that can also handle the uncertainties and irrationalities of society.