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 nearly completely suited to engaged on the McKay conjecture: Even in highschool, she may spend days or even weeks on a single downside. She notably reveled in ones that examined her endurance, and she or he fondly remembers lengthy hours spent looking for “tricks that are, in a way, 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 crazy, really good intuition,” mentioned Schaeffer Fry, her good friend and collaborator. “She’s able to see it’s going to be like this.”
Courtesy of Quanta Journal
A number of 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 “always protesting, ‘Those groups are complicated, my God,’” he recalled. Regardless of his preliminary hesitancy, he too finally 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 a number of main outcomes over the subsequent 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 due to this fact of nice mathematical curiosity, their representations are extremely troublesome to review. Cabanes and Späth usually needed to depend on opaque theories from disparate areas of math. However in digging these theories up, they offered a few of the finest characterizations but of those necessary teams.
As they did so, they began courting and went on to have two youngsters. (They finally settled down collectively in Germany, the place they get pleasure from working collectively at one of many three whiteboards of their house.)
By 2018, they’d only one class of Lie-type teams left. As soon as that was carried out, they might have proved the McKay conjecture.
That ultimate case took them six extra years.
A “Spectacular Achievement”
The fourth type of Lie group “had so many difficulties, so many bad surprises,” Späth mentioned. (It didn’t assist that in 2020, the pandemic stored their two younger youngsters house from faculty, making it troublesome for them to work.) However regularly, 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 required guidelines. The final case was carried out. It adopted routinely 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 an absolutely spectacular achievement,” mentioned Radha Kessar of the College of Manchester.
