Mathematical Moments from the American Mathematical Society

Dr. Alex Best, of Harmonic discusses AI's struggles with technical and ethical limitations like uninterpretable methods and a lack of diverse datasets make facial recognition and medical software dangerously biased. If algorithms output answers without little explanation, how can one trust those results? Interpretability is a prerequisite to replace the work of human researchers. But using AI for formalizing mathematics offers some mitigation, Alex Best says. "Instead of asking for an answer, you ask for an answer and a machine-checkable proof that that answer is correct."

Dr. Kristen Lauter, director of FAIR (Fundamental AI Research) Labs, Meta, North America, discusses how cryptographers are using AI. Online, private information is secured using difficult-to-solve math problems. Researchers must test those problems thoroughly to be sure they are truly difficult - if hackers can solve the problem, they'll have access to your information.

Dr. Po-Shen Loh, a Professor of Mathematics at Carnegie Mellon University known for his work in combinatorics and educational initiatives, explores how AI is reshaping mathematics. He discusses AI's role in accelerating discoveries and enhancing collaboration through proof-checking tools like Lean. Loh highlights the potential for AI to guide human intuition and tackle unsolved conjectures, while emphasizing that creativity remains a uniquely human strength in the mathematical realm.

Dr. Abiy Tasissa of Tufts University, discusses the mathematics he and colleagues used to study particle collider data, including optimal transport and optimization. Collider physics often result in distributions referred to as jets. Dr. Tasissa and his team used "Earth Mover's Distance" and other mathematical tools to study the shape of jets. "It is interesting for me to see how mathematics can be applied to study these fundamental problems answering fundamental equations in physics, not only at the level of formulating new ideas, which is, in this particular case, a notion of distance, but also how the importance of designing fast optimization algorithms to be able to actually compute these distances," says Dr. Tasissa.

Dr. Outi Tervo of Greenland Institute for Natural Resources, shares how mathematics helps recommend speed limits for marine vessels, which benefits narwhals and Inuit culture. Narwhals "can only be found in the Arctic," said Outi Tervo, a senior scientist at GINR. "These species are going to be threatened by climate change more than other species that can live in a bigger geographical area." The collaboration has already lobbied on behalf of the narwhals to reduce the level of sea traffic in their habitat, after using mathematical analysis to identify how noise from passing boats changes the narwhals' foraging behavior.

Dr. Valentina Wheeler, a mathematician at the University of Wollongong with a focus on geometric analysis, discusses her innovative work on wildfires and red blood cells. She explains how wildfires merge to create faster V-shaped flames using the principles of mean curvature flow. Wheeler emphasizes the importance of interdisciplinary research in modeling real-world events accurately. She also explores how concepts like Hellfric energy can shed light on red blood cell dynamics, potentially improving treatment options for patients.

Dr. Stan Wagon of Macalester College discusses the mathematics behind rolling a square smoothly. In 1997, inspired by a square wheel exhibit at The Exploratorium museum in San Francsico, Dr. Stan Wagon enlisted his neighbor Loren Kellen in building a square-wheeled tricycle and accompanying catenary track. For years, you could ride the tricycle at Macalester College in St. Paul, Minnesota. The National Museum of Mathematics in New York now also has square-wheeled tricycles that can be ridden around a circular track. And more recently, the impressive Cody Dock Rolling Bridge was built using rolling square mathematics by Thomas Randall-Page in London.

Dr. Rekha Thomas from the University of Washington discusses three-dimensional image reconstructions from two-dimensional photos. The mathematics of image reconstruction is both simpler and more abstract than it seems. To reconstruct a 3D model based on photographic data, researchers and algorithms must solve a set of polynomial equations. Some solutions to these equations work mathematically, but correspond to an unrealistic scenario — for instance, a camera that took a photo backwards. Additional constraints help ensure this doesn't happen. Researchers are now investigating the mathematical structures underlying image reconstruction, and stumbling over unexpected links with geometry and algebra.

Imelda Flores Vazquez from Econometrica, Inc. explains how economists use mathematics to evaluate the efficacy of health care policies. When a hospital or government wants to adjust their health policies — for instance, by encouraging more frequent screenings for certain diseases — how do they know whether their program will work or not? If the service has already been implemented elsewhere, researchers can use that data to estimate its effects. But if the idea is brand-new, or has only been used in very different settings, then it's harder to predict how well the new program will work. Luckily, a tool called a microsimulation can help researchers make an educated guess.

Stacey Finley, an Associate Professor at USC and Director of the Center for Computational Modeling of Cancer, dives into the power of mathematical models in cancer research. She highlights how these models allow researchers to simulate tumor behavior without the ethical concerns of invasive patient studies. The discussion touches on challenges like data quality and the limitations of traditional animal models. Notably, she reveals insights into cancer cell metabolism and how modeling can enhance our understanding of tumor interaction with their environment.