However one among Malle’s graduate college students was on the case. Britta Späth.
“Our Obsession”
In 2003, Späth arrived on the College of Kassel to begin her doctorate with Malle. She was nearly completely fitted to engaged on the McKay conjecture: Even in highschool, she might spend days or even weeks on a single downside. She notably reveled in ones that examined her endurance, and he or she fondly recollects lengthy hours spent looking for “methods which can be, in a means, not even so deep.”
Späth spent her time learning group representations as deeply as she might. After she accomplished her graduate diploma, she determined to make use of that experience to proceed chipping away on the McKay conjecture. “She has this loopy, actually good instinct,” mentioned Schaeffer Fry, her pal and collaborator. “She’s capable of see it’s going to be like this.”
Courtesy of Quanta Journal
Just a few years later, in 2010, Späth began working at Paris Cité College, the place she met Cabanes. He was an skilled within the narrower set of teams on the heart of the reformulated model of the McKay conjecture, and Späth typically went to his workplace to ask him questions. Cabanes was “all the time protesting, ‘These teams are difficult, my God,’” he recalled. Regardless of his preliminary hesitancy, he too ultimately grew enamored with the issue. It grew to become “our obsession,” he mentioned.
There are 4 classes of Lie-type teams. Collectively, Späth and Cabanes began proving the conjecture for every of these classes, they usually reported several major results over the subsequent decade.
Their work led them to develop a deep understanding of teams of Lie kind. Though these teams are the most typical constructing blocks of different teams, and due to this fact of nice mathematical curiosity, their representations are extremely tough to check. Cabanes and Späth typically needed to depend on opaque theories from disparate areas of math. However in digging these theories up, they supplied among the finest characterizations but of those necessary teams.
As they did so, they began courting and went on to have two youngsters. (They ultimately settled down collectively in Germany, the place they take pleasure in working collectively at one of many three whiteboards of their house.)
By 2018, that they had only one class of Lie-type teams left. As soon as that was accomplished, they might have proved the McKay conjecture.
That remaining case took them six extra years.
A “Spectacular Achievement”
The fourth sort of Lie group “had so many difficulties, so many unhealthy surprises,” Späth mentioned. (It didn’t assist that in 2020, the pandemic stored their two younger youngsters house from college, making it tough for them to work.) However progressively, she and Cabanes managed to point out that the variety of representations for these teams matched these of their Sylow normalizers—and that the way in which the representations matched up glad the mandatory guidelines. The final case was accomplished. It adopted mechanically that the McKay conjecture was true.
In October 2023, they lastly felt assured sufficient of their proof to announce it to a room of greater than 100 mathematicians. A yr later, they posted it online for the remainder of the neighborhood to digest. “It’s a fully spectacular achievement,” mentioned Radha Kessar of the College of Manchester.
