August 8, 2014
Math professor solves a fundamental open problem in algebra
Kansas State University mathematics professor Andrew Chermak has solved an important open problem at the interface between algebra and topology. His breakthrough has given rise to new tools and techniques that go under the name of "partial groups." Chermak's article, "Fusion Systems and Localities," has appeared in ACTA Mathematica, a prestigious and highly selective journal founded in 1882 by the Swedish mathematician Mittag-Leffler.
A group is an abstract object that mimics the properties of numbers that we all learn in school. Elements of a group can be multiplied to yield a new element in the group. In fact, strings of elements can be multiplied together and usually it doesn't matter in which order this is done. Groups that have only a finite number of elements are especially important because they often relate to hidden symmetries.
About 15 years ago abstract algebra, and more specifically the theory of groups, was revolutionized by the influx of new ideas from a completely different field of mathematics called homotopy theory, which is a branch of topology and abstract geometry.
"This was very surprising," Chermak said. "So I set out to study and learn this new point of view."
After becoming interested in this new connection, Chermak realized that there were a couple of important outstanding unsolved problems. As is often the case in research, working on one of these problems lead Chermak to the solution of the other one.
"I got to a point where I felt that I was pushing beyond my capabilities," Chermak said. "I had never had that feeling before." He quoted a famous Zen saying, "If you want to be enlightened, you must feel like drinking a hot iron ball that you can neither swallow nor spit out."
Chermak said, "Partial groups retain some of the properties of groups, but they are stripped of anything that is not necessary towards the geometric structure of interest. You can still multiply certain strings of elements together, but not always. And the result is that for any given prime number p we get a local understanding of the group."
These new mathematical objects could be useful to blaze an alternative approach to the problem of classifying all the finite simple groups. This is a task that was only recently completed, but whose solution has left many experts unsatisfied and begging for a deeper understanding. Chermak, with help from K-State postdoctoral fellow Alex Gonzalez, plans to apply his new techniques to the classification theory of finite simple groups and many other open problems in algebra.