The talk covered how mathematics has been quietly shaping democracy for centuries, from the design of voting systems to the drawing of political maps. Today, with classical results being repurposed in surprising ways and vast computational power at our disposal, it offers powerful new tools for understanding and improving how we govern ourselves. Volić traced the historical role of mathematics in democracy and surveyed current research at their intersection, spanning voting theory, districting, apportionment, proportional representation, and digital democracy. Far from being merely abstract or theoretical, this vibrant interdisciplinary work demonstrates how mathematical expertise speaks directly to the functioning of democratic institutions – challenging the notion that our work has little bearing on civic life. The talk highlighted how mathematical formality can be meaningfully integrated with human realities of politics, a task that feels especially urgent as democracy faces profound strain.

To watch the recording, visit the following link: https://www.youtube.com/watch?v=hm_nBJ_xpAQ 

The post IMD Co-Founder Ismar Volić Gives Talk at the JMM: “Math goes to Washington: The Mathematics of Democracy, from Theory to Practice” appeared first on Institute for Mathematics and Democracy.

See the abstract and link to the publication below:

“This article extends the analysis of Atkinson, Foley, and Ganz in “Beyond the Spoiler Effect: Can Ranked-Choice Voting Solve the Problem of Political Polarization?”. Their work uses a one-dimensional spatial model based on survey data from the Cooperative Election Survey (CES) to examine how instant-runoff voting (IRV) and Condorcet methods promote candidate moderation. Their model assumes an idealized electoral environment in which all voters possess complete information regarding candidates’ ideological positions, all voters provide complete preference rankings, etc. Under these assumptions, their results indicate that Condorcet methods tend to yield winners who are substantially more moderate than those produced by IRV. We construct new models based on CES data which take into account more realistic voter behavior,

such as the presence of partial ballots. Our general finding is that under more realistic models the differences between Condorcet methods and IRV largely disappear, implying that in real-world settings the moderating effect of Condorcet methods may not be nearly as strong as what is suggested by more theoretical models.”

Candidate Moderation Under Instant Runoff and Condorcet Voting: Evidence from the Cooperative Election Survey

The post IMD Team Releases Paper Titled “Candidate Moderation Under Instant Runoff and Condorcet Voting: Evidence from the Cooperative Election Survey” appeared first on Institute for Mathematics and Democracy.

In addition, Prof. Veomett has two articles that were recently published on gerrymandering metrics:  Bounds and Bugs: The Limits of Symmetry Metrics to Detect Partisan Gerrymandering and Don’t Trust a Single Gerrymandering Metric.  The Redistricting Data Hub recently asked her to present this work to their stakeholders (which includes staff from the ACLU, NAACP Legal Defense Fund, Southern Poverty Law Center, and other prominent civil rights organizations). 

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The post IMD Affiliate and Summer Research Students Complete Paper on “Bloc Voting on Single-Peaked Preferences” appeared first on Institute for Mathematics and Democracy.

Find out more about the program here: https://drive.google.com/file/d/162xqjHqKQMvoRQ6qEdgm2fSnneQaSeuB/view?usp=drive_link

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All students and faculty also meet once a week for guest lectures given by prominent mathematicians and democracy reformers. One session is dedicated to a discussion of the college application process and the final session consists of presentations of the projects (here are some of the presentations from the 2025 program*).

If you’re a high school student interested in participating in IMD’s summer 2026 online research program, visit this page. Linked there is a form you should fill out to apply by March 6, 2026.

If you have any questions, feel free to write to us at contact@math-democracy.org. We are looking forward to reading your application!

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Analyzing Swing States in Presidential Elections: The Case of Wisconsin – Michelle Zuo

This paper quantitatively analyzes county-level voting patterns in Wisconsin’s presidential elections from 2000 to 2024. As a pivotal swing state, Wisconsin has alternated between Democratic and Republican candidates since 2012. Using data from the Wisconsin Elections Commission, we examine vote totals across 72 counties and seven election cycles. Pearson correlations measure similarity in county voting trajectories, while choropleth maps visualize spatial shifts. Results show strong clustering of vote changes: Democratic and Republican gains between 2016 and 2020 were concentrated in southeastern urban and suburban counties, with rural areas showing little change. Correlations reveal a north-south divide, as southern counties exhibit similar trends and northern ones diverge. These findings highlight spatial heterogeneity in electoral dynamics and the decisive role of urban mobilization in statewide outcomes.

A Comparison of Precinct and District Voting Data Using Persistent Homology to Identify Gerrymandering in North Carolina – Ananya Shah

Gerrymandering is one of the biggest threats to American democracy. By manipulating district lines, politicians effectively choose their voters rather than the other way around. Current gerrymandering identification methods (namely the Polsby-Popper and Reock scores) focus on the compactness of congressional districts, making them extremely sensitive to physical geography. To address this gap, we extend Feng and Porter’s 2021 paper, which used the level-set method to turn geographic shapefiles into filtered simplicial complexes, in order to compare precinct level voting data to district level voting data. As precincts are regarded as too small to be gerrymandered, we are able to identify discrepancies between precinct and district level voting data to quantify gerrymandering in the United States. By comparing the persistent homologies of Democratic voting regions at the precinct and district levels, we detect when areas have been “cracked” (split across multiple districts) or “packed” (compressed into one district) for partisan gain. This analysis was conducted for North Carolina House of Representatives elections (2012-2024). North Carolina has been redistricted four times in the past ten years, unusually frequent as most states redistrict decennially, making it a valuable case study. By comparing persistence barcodes at the precinct and district levels (using the bottleneck distance), we show that precinct level voting patterns do not significantly fluctuate biannually, while district level patterns do, suggesting that shifts are likely a result of redistricting rather than voter behavior, providing strong evidence of gerrymandering. This research presents a novel application of topological data analysis in evaluating gerrymandering and shows persistent homology can be useful in discerning gerrymandered districts.

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Over 40 researchers will present their work in mathematical political science, focusing on recent developments in the study of alternative voting methods and the creation of district electoral maps. Over the last 20 years, many jurisdictions in Canada and the United States have adopted or have considered adopting alternative voting methods such as ranked-choice voting for electing mayors, representatives, city council members, etc. There are many questions about such methods which can be analyzed mathematically. For example, does ranked-choice voting tend to elect candidates who are more centrist than candidates elected by more commonly-used methods like plurality? How can we best achieve proportional representation in a legislative body? What makes an election fair? There are many open problems around such questions, and this workshop hopes to make some progress on them using mathematical tools.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF) and Alberta’s Technology and Innovation.

Find out more about the event here

The post IMD Affiliates Organize “Mathematics of Elections, Fairness, and Representation” Workshop at the Banff International Research Station appeared first on Institute for Mathematics and Democracy.

Read more about the paper here: https://www.siam.org/publications/siam-news/articles/how-to-win-elections-in-a-dynamically-evolving-electorate/

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