A mathematician couple solves a major group theory problem after 20 years of work, making incremental progress on one of the biggest open problems in the field.
A Math Couple Solves a Major Group Theory Problem—After 20 Years of Work
The Conjecture That Refused to Die
In 2003, a German graduate student named Britta Späth encountered the ‘McKay conjecture‘ , one of the biggest open problems in group theory. At first, her goals were modest: she hoped to prove a theorem or two that would make incremental progress on the problem. But over the years, she was drawn back to it again and again.
The Power of Primes
The ‘McKay conjecture’ began with the observation of a strange coincidence. John McKay, a brilliant mathematician, hypothesized that if mathematicians wanted to compute a crucial quantity that would help them characterize their group, they’d just have to look at one of a particular set of Sylow normalizers: The number of representations would be the same for both the group and its Sylow normalizer.
A Breakthrough
A year after McKay first observed the coincidence, Marty Isaacs proved that it held for a large class of groups. But then mathematicians got stuck. They were able to show that it held up for one specific group or another, but there were still infinitely many groups left to tackle.
The Reformulation
In 2004, mathematicians finally succeeded in showing that all the building blocks must fall into one of three categories, or else belong to a list of 26 outliers. This classification would help simplify problems such as the ‘McKay conjecture’ . Maybe they didn’t have to prove the conjecture for all finite groups. Maybe they only had to prove an alternative statement covering the 29 types of building blocks—or for some related set of straightforward groups—that would automatically imply the full ‘McKay conjecture’.
The McKay conjecture is a mathematical statement that describes the relationship between finite simple groups and their associated sporadic groups.
Proposed by John McKay in 1980, it suggests that there are only two infinite families of finite simple groups: alternating groups and Chevalley groups.
The conjecture has significant implications for group theory and has been studied extensively since its proposal.
Despite efforts to prove or disprove the McKay conjecture, it remains an open problem in mathematics.
The Obsession
Britta Späth arrived at the University of Kassel in 2003 to start her doctorate with Gunter Malle. She was almost perfectly suited for working on the ‘McKay conjecture’ : she could spend days or weeks on a single problem. After completing her graduate degree, she decided to use that expertise to continue chipping away at the ‘McKay conjecture’.
The Partnership
A few years later, in 2010, Späth started working at Paris Cité University, where she met Cabanes. He was an expert in the narrower set of groups at the center of the reformulated version of the ‘McKay conjecture‘ , and Späth often went to his office to ask him questions.
The Final Case
The fourth kind of Lie group had so many difficulties, so many bad surprises. But gradually, Späth and Cabanes managed to show that the number of representations for these groups matched those of their Sylow normalizers—and that the way the representations matched up satisfied the necessary rules. The last case was done. It followed automatically that the ‘McKay conjecture’ was true.
A Spectacular Achievement
In October 2023, they finally felt confident enough in their proof to announce it to a room of more than 100 mathematicians. A year later, they posted it online for the rest of the community to digest. ‘It’s an absolutely spectacular achievement,’ said Radha Kessar of the University of Manchester.
Looking Ahead
Mathematicians can now confidently study important properties of groups by looking at their Sylow normalizers alone—a much easier approach to making sense of these abstract entities, and one that might have practical applications. And in the process of establishing this connection, Navarro said, the researchers developed ‘beautiful, wonderful, deep mathematics’.
Mathematics is a language and discipline that deals with numbers, quantities, and shapes.
It encompasses various branches, including algebra, geometry, calculus, and statistics.
Mathematicians use logical reasoning and mathematical proofs to develop and apply mathematical theories and models.
The subject has numerous applications in science, technology, engineering, and mathematics (STEM) fields, as well as economics, finance, and social sciences.
The Mystery Remains
Some mathematicians still don’t understand why a comparatively tiny set is enough to tell you so much about its larger parent group. There has to be some structural reason why these numbers are the same. But that remains a mystery for now.
Späth and Cabanes are moving on, each searching for their next obsession. So far, according to Späth, nothing has consumed her like the ‘McKay conjecture’ . ‘If you have done one big thing, then it’s difficult to find the courage, the excitement for the next,’ she said.
