However certainly one of 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 virtually completely suited to engaged on the McKay conjecture: Even in highschool, she may spend days or perhaps weeks on a single drawback. She notably reveled in ones that examined her endurance, and she or he fondly recollects lengthy hours spent looking for “methods which are, in a method, not even so deep.”
Späth spent her time finding out group representations as deeply as she may. 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 good friend and collaborator. “She’s in a position to 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 knowledgeable within the narrower set of teams on the middle of the reformulated model of the McKay conjecture, and Späth usually went to his workplace to ask him questions. Cabanes was “at all times protesting, ‘These teams are sophisticated, my God,’” he recalled. Regardless of his preliminary hesitancy, he too finally grew enamored with the issue. It turned “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 a number of main outcomes over the following decade.
Their work led them to develop a deep understanding of teams of Lie sort. Though these teams are the most typical constructing blocks of different teams, and subsequently of nice mathematical curiosity, their representations are extremely tough to review. Cabanes and Späth usually 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 essential teams.
As they did so, they began courting and went on to have two kids. (They finally settled down collectively in Germany, the place they get pleasure from working collectively at one of many three whiteboards of their residence.)
By 2018, they’d only one class of Lie-type teams left. As soon as that was finished, they’d have proved the McKay conjecture.
That last case took them six extra years.
A “Spectacular Achievement”
The fourth sort of Lie group “had so many difficulties, so many dangerous surprises,” Späth mentioned. (It didn’t assist that in 2020, the pandemic stored their two younger kids residence from college, making it tough for them to work.) However progressively, she and Cabanes managed to indicate 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 required guidelines. The final case was finished. It adopted robotically 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 12 months later, they posted it on-line for the remainder of the group to digest. “It’s a fully spectacular achievement,” mentioned Radha Kessar of the College of Manchester.
