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.

Exploring Thermodynamics with Billiards

The American Mathematical Society's Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture. Listen to researchers talk about how they use math, from creating realistic animation to beating cancer.

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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.

Smashing Particles up Against Mathematics

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.

Supporting Wildlife with Statistics

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.

Explaining Wildfires Through Curvature

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.

Bridges and Wheels, Tricycles and Squares

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.

Exploring Thermodynamics with Billiards

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