Geometric group theory – the study of the symmetries of objects – is a relative newcomer to the math world, having truly become its own area of study in the late 1980s. Georgia Tech School of Mathematics Professor Dan Margalit figured that a different approach is needed to teach undergraduates about the theory. That’s why he has co-edited a new book, “Office Hours With A Geometric Group Theorist” (Princeton University Press).
The print version of the book publishes on July 11; a digital version is available now.
“What is most novel about the book is that, unlike most math textbooks, the tone is very informal,” Margalit says. “The different chapters – or ‘office hours’ – are all written by different authors.” Margalit wanted his authors to explain these concepts to his students in an accessible, relatable way.
Margalit, who edited the book with University of Arkansas Associate Professor Matt Clay, says geometric group theory has many important applications within mathematics, such as the award-winning University of California, Berkeley mathematician Ian Agol’s groundbreaking work into the possible shapes of three-dimensional spaces. Real-world applications include the coordination of robots as they move on factory floors, the creation of secure cryptosystems, and even the design of more efficient blenders.
But that’s not why Margalit studies geometric group theory. “There is beauty and depth in the symmetries,” he says, “that one can only appreciate by taking the time to learn the subject.”
As an example, he points to an illustration from the book – a Farey graph. The swirling and folding loops look like something created with a Spirograph toy from the 1970s, but Margalit sees art in its symmetries. “There are many facts about matrix multiplication that you can understand just by considering this beautiful object,” he says.
Dan Margalit received his Ph.D. from the University of Chicago and did postdoctoral work at the University of Utah before arriving at Georgia Tech in 2010. He researches the intersection of low-dimensional topology and geometric group theory, focusing on the symmetries of surfaces.
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