Mathematical Moments from the American Mathematical Society

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 from University of Southern California discusses how mathematical models support the research of cancer biology. Cancer research is a crucial job, but a difficult one. Tumors growing inside the human body are affected by all kinds of factors. These conditions are difficult (if not impossible) to recreate in the lab, and using real patients as subjects can be painful and invasive. Mathematical models give cancer researchers the ability to run experiments virtually, testing the effects of any number of factors on tumor growth and other processes — all with far less money and time than an experiment on human subjects or in the lab would use.

Rodney Kizito from U.S. Department of Energy discusses solar energy, mathematics, and microgrids. When you flip a switch to turn on a light, where does that energy come from? In a traditional power grid, electricity is generated at large power plants and then transmitted long distances. But now, individual homes and businesses with solar panels can generate some or all of their own power and even send energy into the rest of the grid. Modifying the grid so that power can flow in both directions depends on mathematics. With linear programming and operations research, engineers design efficient and reliable systems that account for constraints like the electricity demand at each location, the costs of solar installation and distribution, and the energy produced under different weather conditions. Similar mathematics helps create "microgrids" — small, local systems that can operate independent of the main grid.

Karen Rios Soto explains how mathematics illuminates the link between air pollution from motor vehicle emissions and asthma. Air pollution causes the premature deaths of an estimated seven million people each year, and it makes life worse for all of us. People with asthma can experience chest tightness, coughing or wheezing, and difficulty breathing when triggered by air pollution. One major source is gas- and diesel-powered cars and trucks, which emit "ultrafine" particles less than 0.1 micrometers across. That's about the width of the virus that causes COVID-19, so tiny that these particles are not currently regulated by the US Environmental Protection Agency. Yet ultrafine particles can easily enter your lungs and be absorbed into your bloodstream, causing health issues such as an asthma attack or even neurodegenerative diseases. Mathematics can help us understand the extent of the problem and how to solve it.

Malena Espanol explains how she and others use linear algebra to correct blurry images. Imagine snapping a quick picture of a flying bird. The image is likely to come out blurry. But thanks to mathematics, you might be able to use software to improve the photo. Scientists often deal with blurry pictures, too. Linear algebra and clever numerical methods allow researchers to fix imperfect photos in medical imaging, astronomy, and more. In a computer, the pixels that make up an image can be represented as a column of numbers called a vector. Blurring happens when the light meant for each pixel spills into the adjacent pixels, changing the numbers in a way that can be mathematically represented as an enormous matrix. But knowing that matrix is not enough if you want to reconstruct the original (non-blurry) image.

Tim Chumley explains the connections between random billiards and the science of heat and energy transfer. If you've ever played billiards or pool, you've used your intuition and some mental geometry to plan your shots. Mathematicians have gone a step further, using these games as inspiration for new mathematical problems. Starting from the simple theoretical setup of a single ball bouncing around in an enclosed region, the possibilities are endless. For instance, if the region is shaped like a stadium (a rectangle with semicircles on opposite sides), and several balls start moving with nearly the same velocity and position, their paths in the region soon differ wildly: chaos. Mathematical billiards even have connections to thermodynamics, the branch of physics dealing with heat, temperature, and energy transfer.

Lorin Crawford explains how he uses math to analyze interactions between genes. Your DNA (the biological instruction manual in all of your cells) contains a mind-boggling amount of information represented in roughly 20,000 genes that encode proteins, plus a similar number of genes with other functions. As the cost of analyzing an individual's DNA has plummeted, it has become possible to search the entire human genome for genetic variants that are associated with traits such as height or susceptibility to certain diseases. Sometimes, one gene has a straightforward impact on the trait. But in many cases, the effect of one gene variant depends on which variants of other genes are present, a phenomenon called "epistasis." Studying such interactions involves huge datasets encompassing the DNA of hundreds of thousands of people. Mathematically, that requires time-intensive calculations with massive matrices and a good working knowledge of statistics.

Angela Robinson explains the math behind the next generation of cryptographic algorithms. Whenever you log in to a website, send an email, or make an online purchase, you're counting on your data being sent securely, without hackers being able to crack the code. Our standard cryptographic systems hinge on mathematical problems that stump present-day computers, like finding the prime factors of a very large number. But in the coming decades, powerful quantum computers are expected to be able to rapidly solve some such problems, threatening the security of our online communications. To develop new methods that can withstand even the most sophisticated quantum computer, cryptographers are using a wide range of mathematical tools, many of which were originally developed without any real-life applications in mind.

Ricardo Bermudez-Otero and Tobias Galla discuss the mathematics describing the evolution of human languages. The sounds and structures of the world's approximately 7,000 languages never stop changing. Just compare the English in Romeo and Juliet or the Spanish in Don Quixote to the modern forms. But historical records give an incomplete view of language evolution. Increasingly, linguists draw upon mathematical models to figure out which features of a language change often and which ones change more rarely over the course of thousands of years. A new model inspired by physics assigns a "temperature" to many sounds and grammatical structures. Features with higher temperatures are less stable, so they change more often as time goes on. The linguistic thermometer will help researchers reconstruct how our languages came to be, and how they might change in future generations.

Math may sometimes seem as if it's comprised of countless meaningless unconnected exercises, but in reality, it's much more. It's figuring out how to do something, and, even better, why something works the way it does. The math you're doing now can open doors for you so that you can answer deep questions yourself about a subject or idea that you're interested in. Give those questions a shot and perhaps someday also help others solve their problems. Five mathematicians (Alexander Diaz-Lopez, Trachette Jackson, Francis Su, Erika Tatiana Camacho, and Deanna Haunsperger) talk about what mathematics means to them.

We've seen that the availability of hospital beds is important during a pandemic, and it's important during normal times as well. Whether it's for emergency medical help or for a scheduled procedure (for example, chemotherapy), access to hospital space, staff, and equipment can be a matter of life and death. Mathematics helps medical center staff manage their resources more efficiently so that they are available when needed. An optimization technique called integer programming is used along with tools from statistics, probability, and machine learning to create better schedules for operating rooms, treatment centers, and the people who staff them. David Scheinker talks about the mathematics involved in hospital operations.

In many places, fire seasons keep getting longer with larger and ever more destructive wildfires. Teams of mathematicians, computer scientists, meteorologists, and firefighters are working to reduce the number of large fires before they happen and to contain those that do occur. Mark Finney talks about the math involved in modeling and fighting wildfires.

Math is often described as the science of patterns, which makes it a natural subject to help in the study of the underlying causes of patterns found in nature, for example, bands of vegetation that often occur on gently sloped terrains in certain near-desert ecosystems worldwide. We are starting to learn more about these bands' common properties by using mathematical models built on data, such as rainfall totals and the curvature of the terrain. Mary Silber talks about these mathematical models of vegetation bands.

Math's connection with cooking extends beyond the mathematical constant that sounds like a dessert. For example, using differential equations to model fluid flow and heat transfer, research teams have found how spaghetti curls as it's cooked, how to rotate a pan to make the perfect crepe (thin pancake), and the temperature setting to get the perfect steak. Mathematics helps understand cooking, and parallels it in that following a recipe can lead to good results, but asking questions like "What if we tried this?" can lead to a masterpiece. Eugenia Cheng talks about the mathematics of cooking and baking.

Algorithms can be very useful, but lately, with so much data being created and shared, and with the increase in their use in critical areas such as hiring, credit, and health care, algorithms are under intense scrutiny about their fairness. People experience the effects of an algorithm's conclusion, but the data and steps that form the basis for that conclusion are frequently hidden from them (as if inside a black box). Cathy O'Neil talks about the unfairness of most predictive algorithms.

What causes wine legs (tears)? Andrea Bertozzi explains and describes how to generate legs.

Fumie Tazaki talks about creating the first image of a black hole and its shadow, which relied on Fourier transforms. About the work to make the image, she says, "Our collaboration has 200 members and we did it with all of our efforts."

Hany Farid talks about fighting fake videos: "Mathematically, there's a lot of linear algebra, multivariate calculus, probability and statistics, and then a lot of techniques from pattern recognition, signal processing, and image processing."

Rob Schneiderman talks about the metaphorical connections between math and music

Steven Strogatz and Mary Bushman talk about math's role in controlling HIV and understanding malaria, respectively. Mary Bushman says, "It's really cool to try and use math to nail down some questions that have gone unanswered for a really long time."

Steven Strogatz and Mary Bushman talk about math's role in controlling HIV and understanding malaria, respectively. Mary Bushman says, "It's really cool to try and use math to nail down some questions that have gone unanswered for a really long time."

Tom Patterson and Bojan Savric discuss the Equal Earth projection map that they created with Bernhard Jenny.

Researcher: Jordan Hashemi, Duke University Description: Jordan Hashemi talks about an easy-to-use app to screen for autism.

Researchers: Vikash V. Gayah and S. Ilgin Guler, Pennsylvania State University Description: Gayah and Guler talk about mitigating the clustering of buses on a route.

Researcher: Christine Darden, NASA (retired) Description: Christine Darden on working at NASA.

Researchers: Christopher Brinton, Zoomi, Inc. and Princeton University, and Mung Chiang, Purdue University Moment: http://www.ams.org/samplings/mathmoments/mm139-netflix.pdf Description: Christopher Brinton and Mung Chiang talk about the Netflix Prize competition.

Researcher: Derek Moulton, University of Oxford Moment: http://www.ams.org/samplings/mathmoments/mm138-shells.pdf Description: Derek Moulton explains the math behind the shapes of seashells.

Researcher: Stefan Siegmund, TU-Dresden Moment: http://www.ams.org/samplings/mathmoments/mm137-hurricane.pdf Description: Stefan Siegmund talks about his an invention to protect homes during hurricanes.

Researcher: Andy Andres, Boston University Moment: http://www.ams.org/samplings/mathmoments/mm136-baseball.pdf Andy Andres on baseball analytics.

Researcher: Michel C. Molinkovitch, University of Geneva Description: Michel C. Milinkovitch used math, physics, and biology for an amazing discovery about the patterns on a lizard's skin.

Researcher: Michel C. Molinkovitch, University of Geneva Description: Michel C. Milinkovitch used math, physics, and biology for an amazing discovery about the patterns on a lizard's skin.

Researcher: Konstantin Batygin, Caltech Description: Konstantin Batygin talks about using math to investigate the existence of Planet Nine.

Researcher: Jim Papadopoulos, Northeastern University Description: Jim Papadopoulos talks about his years of research analyzing bicycles.

Researchers: Eleanor Jenkins, Clemson University and Kathleen (Fowler) Kavanagh, Clarkson University. Lea Jenkins and Katie Kavanagh talk about their work making farming more efficient while using water wisely.

Researcher: Daniel Rothman, MIT. Dan Rothman talks about how math helped understand a mass extinction.

Researcher: Daniel Rothman, MIT. Dan Rothman talks about how math helped understand a mass extinction.

Researcher: Hamsa Balakrishnan, MIT. Hamsa Balakrishnan talks about her work to shorten airport runway queues.

Researcher: Annalisa Crannell, Franklin & Marshall College. Annalisa Crannell on perspective in art.

Researcher: John A. Adam, Old Dominion University. John A. Adam explains the math and physics behind rainbows.

Researchers: Eleanor Jenkins, Clemson University, and Katie Kavanagh, Clarkson University. Eleanor Jenkins and Katie Kavanagh talk about their interdisciplinary team's work helping farmers.

Researcher: Andrew Beveridge, Macalester College Moment Title: Dis-playing the Game of Thrones Description: Andrew Beveridge uses math to analyze Game of Thrones.

Researcher: Andrew Beveridge, Macalester College Moment Title: Dis-playing the Game of Thrones Description: Andrew Beveridge uses math to analyze Game of Thrones.

Researcher: Thomas Snitch, University of Maryland Description: Thomas Snitch talks about nabbing poachers with math. Audio files: podcast-mom-poaching-1.mp3 and podcast-mom-poaching-2.mp3

Researcher: Thomas Snitch, University of Maryland Description: Thomas Snitch talks about nabbing poachers with math. Audio files: podcast-mom-poaching-1.mp3 and podcast-mom-poaching-2.mp3

Researcher: Cristina Stoica, Wilfrid Laurier University Description: Cristina Stoica talks about celestial mechanics.

Researcher: Cristina Stoica, Wilfrid Laurier University Description: Cristina Stoica talks about celestial mechanics.

Researcher: Jackson Cothren, University of Arkansas Moment Title: Scanning Ancient Sites Description: Jackson Cothren talks about creating three-dimensional scans of ancient sites.

Researcher: Wesley Pegden, Carnegie Mellon University Moment Title: Piling On and on and on! Description: Wesley Pegden talks about simulating sandpiles

Researcher: Norbert Stoop, MIT Title: Adding a New Wrinkle Description: Norbert Stoop talks about new research on the formation of wrinkles.

Researcher: Sidney Redner, Santa Fe Institute Moment: Moment Title: Holding the Lead Description: Sidney Redner talks about how random walks relate to leads in basketball.

Researcher: Meredith Greer, Bates College. Going Over the Top Description: Meredith Greer talks about math and roller coasters.