# 2024 Student Essay Contest: Undergraduate Honorable Mention

## “Random Walks for Representation”

### by: Charlotte Richards (Pitzer College)

### Interviewee: Sarah Cannon (Claremont McKenna College)

In math classrooms across the world, the same question is asked again and again, much to the chagrin of teachers; “When will I use this in real life?” Responses may vary from eyerolls to more obvious applications such as finance or accounting. However, for mathematicians like Sarah Cannon, the answer extends into an unlikely place: politics.

Political districting has been a hot topic in the news for decades, as the drawing of lines directly impacts representation and policy outcomes. However, it is extremely difficult to test a political district’s adherence to laws set to promote fairness and equal voting power due to the multitude of possible plans. How can one find a sample of what nonpartisan political districts should look like, when there are countless options and factors?

Cannon’s research aims to answer this question. She specializes in random sampling algorithms, taking a space, such as all districting maps, that is too large to describe in its entirety and establishing methods to take “snapshots” that are representative of the space. She first became interested in this topic in graduate school under the instruction of her advisor, as it aligned with her multifarious motivations; Cannon has an intrinsic appreciation for proofs, starting from a proof-based calculus class in college that solidified her desire to major in mathematics. Additionally, real world problems such as gerrymandering, which involves manipulating political districts to favor a specific party, beg theoretical questions that Cannon hopes to collaboratively solve.

One of her more recent projects, a collaborative paper entitled “Voting Rights, Markov Chains, and Optimization by Short Bursts” combines mathematics’ potential to, as Cannon describes, “help us understand what is going on in the world” through its “order and rigor,” in addition to finding new solutions. Under current redistricting laws, political districting plans should take into consideration the number of majority-minority voting districts, as racial gerrymandering caused by unequal voting power may lead to litigation under the Voting Rights Act. However, the largest number of possible districts that can simultaneously accomplish this goal is difficult to determine. Mathematicians have established the maximum number of majority-minority districts through various computational techniques; however, common strategies such as unbiased random walks that randomly “step” from one plan to another do not come close to this maximization, since the origin point greatly influences the outcome.

In this paper, Cannon and her co-authors introduce a new type of random sampling algorithm called a “short burst,” applying these unbiased walks with different starting points to create non-representative samples that prioritize majority-minority districts. After running a practical comparison test based upon real world data, they concluded that this new technique outperformed the previous standard, making it easier to discover potential plans that would enable greater voting representation. Although the researchers did not precisely know why short bursts outperformed other methods, Cannon states that this limitation “stimulates more questions” that would be “interesting to dive in,” yet does not undercut its value in practical uses. In Cannon’s opinion, these problems are not discouraging, but instead are another puzzle to solve.

Mathematics is often considered to be impartial, yet these random walk algorithms are inherently not by design, raising political skepticism seen in the recent U.S. Supreme Court case Allen v Milligan. Cannon admits that “as soon as you go from deeply mathematical to the messy real world, there are a lot of questions to answer for how you choose to merge these two together.” Nonetheless, these algorithms favoring majority-minority districts alert mathematicians and larger society to gerrymandering, leading to legal action advocating for changes in districting plans in order to allocate more representation to minority voters.

In addition to applied mathematics, half of Cannon’s research in random sampling algorithms is “highly theoretical with no clear real-world applications.” Nonetheless, Cannon believes that exploring mathematics without current connections to the real world is also crucial. Cannon highlights that number theory was a pure mathematical field for years with few applications. Yet, with the inception of cryptography and computer science, previously niche mathematical topics inform modern life. She asserts that “it is essential to not only work with applications we can see, but explore all of the possibilities of what can be done, as it might be useful in the future.”

Persistence is a dominating theme in Cannon’s mathematical journey. In her PhD program at Georgia Tech, Cannon was the only woman in a program of ten people, finding it difficult to connect with others in her program. Additionally, she faced imposter syndrome, believing that she was not worthy of the recognition she was receiving; these feelings extended to her time at Claremont McKenna College, where she is currently an assistant professor of mathematics. In spite of these challenges, Cannon continued her work by finding both research networks and teaching support through a peer woman network of computer science and mathematics professors.

These experiences with mathematics and the importance of persistence inform Cannon’s style of teaching as a lecturing professor. In her “Math of Political Districting” class, Cannon taught the basics of her research and invited students to do their own exploration through a final project combining their new political knowledge and computer algorithms. These ranged from analysis of city council districts in particular cities to examining how the location of prisons impacts districting. Through her guidance, Cannon pushed her undergraduate students to become the next generation of researchers.

Cannon plans to continue with her work in theoretical and applied mathematics, as she feels there are a multitude of questions left unanswered. Just in redistricting alone, there are currently tens of court cases in the United States of accused partisan and racial gerrymandering, many of which rely on mathematical research for evidence.

So, how can we use mathematics in real life? In the case of Sarah Cannon, mathematics reaches unexpected places, from driving efforts for equality within voting to applications of theoretical mathematics into currently unknown spheres.