After 20 Years, Math Couple Solves Major Group Theory Problem
Arthur T Knackerbracket has processed the following story:
In 2003, a German graduate student named Britta Spath encountered the McKay conjecture, one of the biggest open problems in the mathematical realm known as group theory. At first her goals were relatively modest: She hoped to prove a theorem or two that would make incremental progress on the problem, as many other mathematicians had done before her. But over the years, she was drawn back to it, again and again. Whenever she tried to focus on something else, she said, it didn't connect."
There was a risk that such a single-minded pursuit of so difficult a problem could hurt her academic career, but Spath dedicated all her time to it anyway. It brought her to the office of Marc Cabanes, a mathematician now at the Institute of Mathematics of Jussieu in Paris who, inspired by her efforts, became consumed by the conjecture, too. While working together, the pair fell in love and eventually started a family.
The problem that absorbed them takes a key theme in mathematics and turns it into a concrete tool for group theorists. Math is full of enormously complicated abstract objects that are impossible to study in their entirety. But often, mathematicians have discovered, it's enough to look at a small fragment of such an object to understand its broader properties. In the third century BCE, for instance, the ancient Greek mathematician Eratosthenes estimated the circumference of the Earth - roughly 25,000 miles - by measuring shadows cast by the sun in just two cities about 500 miles apart. Similarly, when mathematicians want to understand an impossibly convoluted function, they might only need to look at how it behaves for a small subset of possible inputs. That can be enough to tell them what the function does for all possible inputs.
The McKay conjecture is another example of this principle. It says that if you want to formulate a thorough description of a group - an important mathematical entity that can get prohibitively difficult to study - you only need to look at a tiny piece of it.
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