Mathematical Moments from the American Mathematical Society

Dr. Valentina Wheeler of University of Wollongong, Australia, shares how her work influences efforts to understand wildfires and red blood cells. In Australia, where bushfires are a concern year-round, researchers have long tried to model these wildfires, hoping to learn information that can help with firefighting policy. Mathematician Valentina Wheeler and colleagues began studying a particularly dangerous phenomenon: When two wildfires meet, they create a new, V-shaped fire whose pointed tip races along to catch up with the two branches of the V, moving faster than either of the fires alone. This is exactly what happens in a mathematical process known as mean curvature flow. Mean curvature flow is a process in which a shape smooths out its boundaries over time. Just as with wildfires, pointed corners and sharp bumps will change the fastest.

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 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: 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.

Van Wedeen talks about the geometry of the brain's communication pathways.

Muthu Alagappan explains how topology and analytics are bringing a new look to basketball.

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 Moment Title: Keeping the Roof On Description: Stefan Siegmund talks about his an invention to protect homes during hurricanes. Podcast page: http://www.ams.org/samplings/mathmoments/mm137-hurricane-podcast

Researcher: Andy Andres, Boston University Moment: http://www.ams.org/samplings/mathmoments/mm136-baseball.pdf Moment Title: Scoring with New Thinking Description: Andy Andres on baseball analytics. Podcast page: http://www.ams.org/samplings/mathmoments/mm136-baseball-podcast

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. Jim Papadopoulos talks about his years of research analyzing bicycles.

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

Researcher: Thomas Snitch, University of Maryland

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 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.

Researcher: Christopher Butson, Scientific Computing and Imaging Institute, University of Utah. Christopher Butson talks about work he's done to help treat Parkinson's disease.

Researcher: Christopher Butson, Scientific Computing and Imaging Institute, University of Utah. Christopher Butson talks about work he's done to help treat Parkinson's disease.

Edward Witten talks about math and physics.

Researcher: Michael C. Ferris, University of Wisconsin-Madison. Moment Title: Providing Power. Description: Michael C. Ferris talks about power grids

Lydia Bourouiba talks about surface tension and the transmission of disease

Lydia Bourouiba talks about surface tension and the transmission of disease

Colin Adams talks about knot theory

Colin Adams talks about knot theory

Michael Trick talks about creating schedules for leagues.

Despite the considerable variety among cities, researchers have identified common mathematical properties that hold around the world, regardless of a city.s population, location or even time.

Despite the considerable variety among cities, researchers have identified common mathematical properties that hold around the world, regardless of a city.s population, location or even time.

Nazareth Bedrossian talks about using math to reposition the International Space Station.

Nazareth Bedrossian explains more about math's role in maneuvering spacecraft and why he's a consumer of mathematical results.

Van Wedeen talks about the geometry of the brain's communication pathways.

Muthu Alagappan explains how topology and analytics are bringing a new look to basketball.

James Sethian and Frank Morgan talk about their research investigating bubbles.

James Sethian and Frank Morgan talk about their research investigating bubbles.

James Sethian and Frank Morgan talk about their research investigating bubbles.

James Sethian and Frank Morgan talk about their research investigating bubbles.

Many of today.s most striking buildings are nontraditional freeform shapes. A new field of mathematics, discrete differential geometry, makes it possible to construct these complex shapes that begin as designers. digital creations. Since it.s impossible to fashion a large structure out of a single piece of glass or metal, the design is realized using smaller pieces that best fit the original smooth surface. Triangles would appear to be a natural choice to represent a shape, but it turns out that using quadrilaterals.which would seem to be more difficult.saves material and money and makes the structure easier to build.

Many of today.s most striking buildings are nontraditional freeform shapes. A new field of mathematics, discrete differential geometry, makes it possible to construct these complex shapes that begin as designers. digital creations. Since it.s impossible to fashion a large structure out of a single piece of glass or metal, the design is realized using smaller pieces that best fit the original smooth surface. Triangles would appear to be a natural choice to represent a shape, but it turns out that using quadrilaterals.which would seem to be more difficult.saves material and money and makes the structure easier to build.

Facebook has over 700 million users with almost 70 billion connections. The hard part isn.t people making friends; rather it.s Facebook.s computers storing and accessing relevant data, including information about friends of friends. The latter is important for recommendations to users (People You May Know). Much of this work involves computer science, but mathematics also plays a significant role. Subjects such as linear programming and graph theory help cut in half the time needed to determine a person.s friends of friends and reduce network traffic on Facebook.s machines by about two-thirds. What.s not to like? The probability of people being friends tends to decrease as the distance between them increases. This makes sense in the physical world, but it.s true in the digital world as well. Yet, despite this, the enormous network of Facebook users is an example of a small-world network. The average distance between Facebook users the number of friend-links to connect people is less than five. And even though the collection of users and their connections may look chaotic, the network actually has a good deal of structure. For example, it.s searchable. That is, two people who are, say, five friend-links away, could likely navigate from one person to the other by knowing only the friends at each point (but not knowing anyone.s friends of friends). For More Information: Networks, Crowds, and Markets: Reasoning about a Highly Connected World, David Easley and Jon Kleinberg, 2010.

Facebook has over 700 million users with almost 70 billion connections. The hard part isn.t people making friends; rather it.s Facebook.s computers storing and accessing relevant data, including information about friends of friends. The latter is important for recommendations to users (People You May Know). Much of this work involves computer science, but mathematics also plays a significant role. Subjects such as linear programming and graph theory help cut in half the time needed to determine a person.s friends of friends and reduce network traffic on Facebook.s machines by about two-thirds. What.s not to like? The probability of people being friends tends to decrease as the distance between them increases. This makes sense in the physical world, but it.s true in the digital world as well. Yet, despite this, the enormous network of Facebook users is an example of a small-world network. The average distance between Facebook users the number of friend-links to connect people is less than five. And even though the collection of users and their connections may look chaotic, the network actually has a good deal of structure. For example, it.s searchable. That is, two people who are, say, five friend-links away, could likely navigate from one person to the other by knowing only the friends at each point (but not knowing anyone.s friends of friends). For More Information: Networks, Crowds, and Markets: Reasoning about a Highly Connected World, David Easley and Jon Kleinberg, 2010.

There.s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three. Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, A juggler, like a mathematician, is never finished: there is always another great unsolved problem.

There.s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three. Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, A juggler, like a mathematician, is never finished: there is always another great unsolved problem.

Imagine trying to describe the circulation and temperatures across the vast expanse of our oceans. Good models of our oceans not only benefit fishermen on our coasts but farmers inland as well. Until recently, there were neither adequate tools nor enough data to construct models. Now with new data and new mathematics, short-range climate forecasting for example, of an upcoming El Nino is possible.There is still much work to be done in long-term climate forecasting, however, and we only barely understand the oceans. Existing equations describe ocean dynamics, but solutions to the equations are currently out of reach. No computer can accommodate the data required to approximate a good solution to these equations. Researchers therefore make simplifying assumptions in order to solve the equations. New data are used to test the accuracy of models derived from these assumptions. This research is essential because we cannot understand our climate until we understand the oceans. For More Information: What.s Happening in the Mathematical Sciences, Vol 1, Barry Cipra.

It may be hard to accept but it.s likely that we.d all be much safer in autonomous vehicles driven by computers, not humans. Annually more than 30,000 Americans die in car crashes, almost all due to human error. Autonomous vehicles will communicate position and speed to each other and avoid potential collisions-without the possibility of dozing off or road rage. There are still many legal (and insurance) issues to resolve, but researchers who are revving up the development of autonomous vehicles are relying on geometry for recognizing and tracking objects, probability to assess risk, and logic to prove that systems will perform as required. The advent of autonomous vehicles will bring in new systems to manage traffic as well, for example, at automated intersections. Cars will communicate to intersection-managing computers and secure reservations to pass through. In a matter of milliseconds, the computers will use trigonometry and differential equations to simulate vehicles. paths through the intersection and grant entry as long as there is no conflict with other vehicles. paths. Waiting won.t be completely eliminated but will be substantially reduced, as will the fuel--and patience--currently wasted. Although the intersection at the left might look wild, experiments indicate that because vehicles would follow precise paths, such intersections will be much safer and more efficient than the ones we drive through now.

No one can predict who will commit a crime but in some cities math is helping detect areas where crimes have the greatest chance of occurring. Police then increase patrols in these "hot spots" in order to prevent crime. This innovative practice, called predictive policing, is based on large amounts of data collected from previous crimes, but it involves more than just maps and push pins. Predictive policing identifies hot spots by using algorithms similar to those used to predict aftershocks after major earthquakes. Just as aftershocks are more likely near a recent earthquake.s epicenter, so too are crimes, as criminals do indeed return to, or very close to, the scene of a crime. Cities employing this approach have seen crime rates drop and studies are underway to measure predictive policing.s part in that drop. One fact that has been determined concerns the nature of hot spots. Researchers using partial differential equations and bifurcation theory have discovered two types of hot spots, which respond quite differently to increased patrols. One type will shift to another area of the city while the other will disappear entirely. Unfortunately the two appear the same on the surface, so mathematicians and others are working to help police find ways to differentiate between the two so as to best allocate their resources.

No one can predict who will commit a crime but in some cities math is helping detect areas where crimes have the greatest chance of occurring. Police then increase patrols in these "hot spots" in order to prevent crime. This innovative practice, used to predict aftershocks after major earthquakes. Just as aftershocks are more likely near a recent earthquake.s epicenter, so too are crimes, as criminals do indeed return to, or very close to, the scene of a crime. Cities employing this approach have seen crime rates drop and studies are underway to measure predictive policing.s part in that drop. One fact that has been determined concerns the nature of hot spots. Researchers using partial differential equations and bifurcation theory have discovered two types of hot spots, which respond quite differently to increased patrols. One type will shift to another area of the city while the other will disappear entirely. Unfortunately the two appear the same on the surface, so mathematicians and others are working to help police find ways to differentiate between the two so as to best allocate their resources.

Cutters of diamonds and other gemstones have a high-pressure job with conflicting demands: Flaws must be removed from rough stones to maximize brilliance but done so in a way that yields the greatest weight possible. Because diamonds are often cut to a standard shape, cutting them is far less complex than cutting other gemstones, such as rubies or sapphires, which can have hundreds of different shapes. By coupling geometry and multivariable calculus with optimization techniques, mathematicians have been able to devise algorithms that automatically generate precise cutting plans that maximize brilliance and yield. The goal is to find the final shape within a rough stone. There are an endless number of candidates, positions, and orientations, so finding the shape amounts to a maximization problem with a large number of variables subject to an infinite number of constraints, a technique called semi-infinite optimization. Experienced human cutters create finished gems that average about 1/3 of the weight of the original rough stone. Cutting with this automated algorithm improved the yield to well above 40%, which, given the value of the stones, is a tremendous improvement. Without a doubt, semi-infinite optimization is a girl.s (or boy.s) best friend.

Once a problem only in the developed world, obesity is now a worldwide epidemic. The overwhelming cause of the epidemic is a dramatic increase in the food supply and in food consumption not a surprise. Yet there are still many mysteries about weight change that can.t be answered either inside the lab, because of the impracticality of keeping people isolated for long periods of time, or outside, because of the unreliability of dietary diaries. Mathematical models based on differential equations can help overcome this roadblock and allow detailed analysis of the relationship between food intake, metabolism, and weight change. The models. predictions fit existing data and explain such things as why it is hard to keep weight off and why obese people are more susceptible to further weight gain. Researchers are also investigating why dieters often plateau after a few months and slowly regain weight. A possible explanation is that metabolism slows to match the drop in food consumed, but models representing food intake and energy expenditure as a dynamical system show that such a weight plateau doesn.t take effect until much later. The likely culprit is a combination of slower metabolism and a lack of adherence to the diet. Most people are in approximate steady state, so that long-term changes are necessary to gain or lose weight. The good news is that each (enduring) drop of 10 calories a day translates into one pound of weight loss over three years, with about half the loss occurring in the first year. For More Information: Quantification of the effect of energy imbalance on bodyweight, Hall et al. Lancet, Vol. 378 (2011), pp. 826-837.

Some of the simplest and most well-known curves parabolas and ellipses, which can be traced back to ancient Greece are also among the most useful. Parabolas have a reflective property that is employed in many of today.s solar power technologies. Mirrors with a parabolic shape reflect all entering light to a single point called the focus, where the solar power is converted into usable energy. Ellipses, which have two foci, have a similar reflecting property that is exploited in a medical procedure called lithotripsy. Patients with kidney stones and gallstones are positioned in a tank shaped like half an ellipse so that the stones are at one focus. Acoustic waves sent from the other focus concentrate all their energy on the stones, pulverizing them without surgery. Math can sometimes throw you a curve, but that.s not necessarily a bad thing. Parabolas and ellipses are curves called conic sections. Another curve in this category is the hyperbola, which may have the most profound application of all the nature of the universe. In plane geometry, points that are a given distance from a fixed point form a circle. In space, points that are a given spacetime distance from a fixed point form one branch of a hyperbola. This is not an arbitrary mandate but instead a natural conclusion from the equations that result when the principle of relativity is reconciled with our notions of distance and causality. And although a great deal of time has elapsed since the discovery of conic sections, they continue to reap benefits today. For More Information: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas, J. W. Downs, 2010.

Some of the simplest and most well-known curves parabolas and ellipses, which can be traced back to ancient Greece are also among the most useful. Parabolas have a reflective property that is employed in many of today.s solar power technologies. Mirrors with a parabolic shape reflect all entering light to a single point called the focus, where the solar power is converted into usable energy. Ellipses, which have two foci, have a similar reflecting property that is exploited in a medical procedure called lithotripsy. Patients with kidney stones and gallstones are positioned in a tank shaped like half an ellipse so that the stones are at one focus. Acoustic waves sent from the other focus concentrate all their energy on the stones, pulverizing them without surgery. Math can sometimes throw you a curve, but that.s not necessarily a bad thing. Parabolas and ellipses are curves called conic sections. Another curve in this category is the hyperbola, which may have the most profound application of all the nature of the universe. In plane geometry, points that are a given distance from a fixed point form a circle. In space, points that are a given spacetime distance from a fixed point form one branch of a hyperbola. This is not an arbitrary mandate but instead a natural conclusion from the equations that result when the principle of relativity is reconciled with our notions of distance and causality. And although a great deal of time has elapsed since the discovery of conic sections, they continue to reap benefits today. For More Information: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas, J. W. Downs, 2010.

Mathematics contributes in many ways to the process of converting wind power into usable energy. Large-scale weather models are used to find suitable locations for wind farms, while more narrowly focused models incorporating interactions arising from factors such as wake effects and turbulence specify how to situate individual turbines within a farm. In addition, computational fluid dynamics describes air flow and drag around turbines. This helps determine the optimal shapes for the blades, both structurally and aerodynamically, to extract as much energy as possible, and keep noise levels and costs down. Mathematics also helps answer two fundamental questions about wind turbines. First, why three blades? Turbines with fewer blades extract less energy and are noisier (because the blades must turn faster). Those with more than three blades would capture more energy but only about three percent more, which doesn.t justify the increased cost. Second, what percentage of wind energy can a turbine extract? Calculus and laws of conservation provide the justification for Betz Law, which states that no wind turbine can capture more than 60% of the energy in the wind. Modern turbines generally gather 40-50%. So the answer to someone who touts a turbine that can capture 65% of wind energy is "All Betz" are off. For More Information: Wind Energy Explained: Theory, Design and Application, Manwell, McGowan, and Rogers, 2010.

Mathematics contributes in many ways to the process of converting wind power into usable energy. Large-scale weather models are used to find suitable locations for wind farms, while more narrowly focused models incorporating interactions arising from factors such as wake effects and turbulence specify how to situate individual turbines within a farm. In addition, computational fluid dynamics describes air flow and drag around turbines. This helps determine the optimal shapes for the blades, both structurally and aerodynamically, to extract as much energy as possible, and keep noise levels and costs down. Mathematics also helps answer two fundamental questions about wind turbines. First, why three blades? Turbines with fewer blades extract less energy and are noisier (because the blades must turn faster). Those with more than three blades would capture more energy but only about three percent more, which doesn.t justify the increased cost. Second, what percentage of wind energy can a turbine extract? Calculus and laws of conservation provide the justification for Betz Law, which states that no wind turbine can capture more than 60% of the energy in the wind. Modern turbines generally gather 40-50%. So the answer to someone who touts a turbine that can capture 65% of wind energy is "All Betz" are off. For More Information: Wind Energy Explained: Theory, Design and Application, Manwell, McGowan, and Rogers, 2010.

The heart.s function of pumping blood may seem fairly simple but the underlying mechanisms and electrical impulses that maintain a healthy rhythm are extremely complex. Many areas of mathematics, including differential equations, dynamical systems, and topology help model the electrical behavior of cardiac cells, the connections between those cells and the heart.s overall geometry. Researchers aim to gain a better understanding of the normal operation of the heart, as well as learn how to diagnose the onset of abnormalities and correct them. Of the many things that can go wrong with a heart.s rhythm, some measure of unpredictability is (surprisingly) not one of them. A healthy heartbeat is actually quite chaotic not regular at all. Furthermore, beat patterns become less chaotic as people age and heart function diminishes. In fact, one researcher recommends that patients presented with a new medication should ask their doctors, "What is this drug going to do to my fractal dimensionality?" For More Information: Taking Mathematics to Heart: Mathematical Challenges in Cardiac Electrophysiology, John W. Cain, Notices of the AMS, April 2011, pp. 542-549.

The heart.s function of pumping blood may seem fairly simple but the underlying mechanisms and electrical impulses that maintain a healthy rhythm are extremely complex. Many areas of mathematics, including differential equations, dynamical systems, and topology help model the electrical behavior of cardiac cells, the connections between those cells and the heart.s overall geometry. Researchers aim to gain a better understanding of the normal operation of the heart, as well as learn how to diagnose the onset of abnormalities and correct them. Of the many things that can go wrong with a heart.s rhythm, some measure of unpredictability is (surprisingly) not one of them. A healthy heartbeat is actually quite chaotic not regular at all. Furthermore, beat patterns become less chaotic as people age and heart function diminishes. In fact, one researcher recommends that patients presented with a new medication should ask their doctors, "What is this drug going to do to my fractal dimensionality?" For More Information: Taking Mathematics to Heart: Mathematical Challenges in Cardiac Electrophysiology, John W. Cain, Notices of the AMS, April 2011, pp. 542-549.

It.s often a challenge to get from Point A to Point B in normal circumstances, but after a disaster it can be almost impossible to transport food, water, and clothing from the usual supply points to the people in desperate need. A new mathematical model employs probability and nonlinear programming to design supply chains that have the best chance of functioning after a disaster. For each region or country, the model generates a robust chain of supply and delivery points that can respond to the combination of disruptions in the network and increased needs of the population. Math also helps medical agencies operate more efficiently during emergencies, such as an infectious outbreak. Fluid dynamics and combinatorial optimization are applied to facility layout and epidemiological models to allocate resources and improve operations while minimizing total infection within dispensing facilities. This helps ensure fast, effective administering of vaccines and other medicines. Furthermore, solution times are fast enough that officials can input up-to-the-minute data specific to their situation and make any necessary redistribution of supplies or staff in real time. For More Information: Supply Chain Network Economics: Dynamics of Prices, Flows, and Profits, Anna Nagurney, 2006.

It.s often a challenge to get from Point A to Point B in normal circumstances, but after a disaster it can be almost impossible to transport food, water, and clothing from the usual supply points to the people in desperate need. A new mathematical model employs probability and nonlinear programming to design supply chains that have the best chance of functioning after a disaster. For each region or country, the model generates a robust chain of supply and delivery points that can respond to the combination of disruptions in the network and increased needs of the population. Math also helps medical agencies operate more efficiently during emergencies, such as an infectious outbreak. Fluid dynamics and combinatorial optimization are applied to facility layout and epidemiological models to allocate resources and improve operations while minimizing total infection within dispensing facilities. This helps ensure fast, effective administering of vaccines and other medicines. Furthermore, solution times are fast enough that officials can input up-to-the-minute data specific to their situation and make any necessary redistribution of supplies or staff in real time. For More Information: Supply Chain Network Economics: Dynamics of Prices, Flows, and Profits, Anna Nagurney, 2006.

Experts are adept at answering questions in their fields, but even the most knowledgeable authority can.t be expected to keep up with all the data generated today. Computers can handle data, but until now, they were inept at understanding questions posed in conversational language. Watson, the IBM computer that won the Jeopardy! Challenge, is an example of a computer that can answer questions using informal, nuanced, even pun-filled, phrases. Graph theory, formal logic, and statistics help create the algorithms used for answering questions in a timely manner.not at all elementary. Watson.s creators are working to create technology that can do much more than win a TV game show. Programmers are aiming for systems that will soon respond quickly with expert answers to real-world problems.from the fairly straightforward, such as providing technical support, to the more complex, such as responding to queries from doctors in search of the correct medical diagnosis. Most of the research involves computer science, but mathematics will help to expand applications to other industries and to scale down the size and cost of the hardware that makes up these modern question-answering systems. For More Information: Final Jeopardy: Man vs. Machine and the Quest to Know Everything, Stephen Baker, 2011.

Nothing can prevent a tsunami from happening they are enormously powerful events of nature. But in many cases networks of seismic detectors, sea-level monitors and deep ocean buoys can allow authorities to provide adequate warning to those at risk. Mathematical models constructed from partial differential equations use the generated data to determine estimates of the speed and magnitude of a tsunami and its arrival time on coastlines. These models may predict whether a trough or a crest will be the first to arrive on shore. In only about half the cases (not all) does the trough arrive first, making the water level recede dramatically before the onslaught of the crest.

Nothing can prevent a tsunami from happening they are enormously powerful events of nature. But in many cases networks of seismic detectors, sea-level monitors and deep ocean buoys can allow authorities to provide adequate warning to those at risk. Mathematical models constructed from partial differential equations use the generated data to determine estimates of the speed and magnitude of a tsunami and its arrival time on coastlines. These models may predict whether a trough or a crest will be the first to arrive on shore. In only about half the cases (not all) does the trough arrive first, making the water level recede dramatically before the onslaught of the crest.

Chartier and Martin talk about they used math to show that a triple cork snowboarding maneuver was possible.