Mathematical Moments from the American Mathematical Society

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. One of the primary goals of researchers is to create an efficient, streamlined process that integrates design and construction parameters so that early on architects can assess the feasibility of a given idea. Currently, implementing a plan involves extensive (and often expensive) interplay on computers between subdivision.breaking up the entire structure into manageable manufacturable pieces.and optimization.solving nonlinear equations in high-dimensional spaces to get as close as possible to the desired shape. Designers and engineers are seeking new mathematics to improve that process. Thus, in what might be characterized as a spiral with each field enriching the other, their needs will lead to new mathematics, which makes the shapes possible in the first place. For More Information: .Geometric computing for freeform architecture,. J. Wallner and H. Pottmann. Journal of Mathematics in Industry, Vol. 1, No. 4, 2011.

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. One of the primary goals of researchers is to create an efficient, streamlined process that integrates design and construction parameters so that early on architects can assess the feasibility of a given idea. Currently, implementing a plan involves extensive (and often expensive) interplay on computers between subdivision.breaking up the entire structure into manageable manufacturable pieces.and optimization.solving nonlinear equations in high-dimensional spaces to get as close as possible to the desired shape. Designers and engineers are seeking new mathematics to improve that process. Thus, in what might be characterized as a spiral with each field enriching the other, their needs will lead to new mathematics, which makes the shapes possible in the first place. For More Information: .Geometric computing for freeform architecture,. J. Wallner and H. Pottmann. Journal of Mathematics in Industry, Vol. 1, No. 4, 2011.

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.

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.

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, 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.