But one of Malle’s postgraduate students was in the case. Britta Späth.
“Our obsession”
In 2003, Späth arrived at the University of Kassel to start her doctorate with Malle. She was almost perfectly suited to work on the McKay assumption: Even at high school she could spend days or weeks to a single problem. She remained especially in those who tested her endurance, and she remembers very long hours spent looking for ‘tricks that in a way are not even so deep.’
Späth spent her time studying group representations as deeply as possible. After completing her graduate degree, she decided to use the expertise to continue to get away from the McKay. “She has this crazy, very good intuition,” says Schaeffer Fry, her friend and co -worker. “She could see that it was going to be that way.”
Courtesy of Quanta Magazine
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 in the middle of the reformulated version of the McKay assumption, and Späth often went to his office to ask him questions. Cabanes has “always protested”, these groups are complicated, my God, “he remembers. Despite his initial hesitation, he also finally touched the problem. It became “our obsession”, he said.
There are four categories of lie groups. Collectively, Späth and Cabanes began to prove the assumption for each of these categories, and they reported several important results in the next decade.
Their work has led them to develop a profound understanding of groups of lying type. Although these groups are the most common building blocks of other groups, and therefore of great mathematical interest, their representations are incredibly difficult to study. Cabane and Späth often had to rely on opaque theories from different math areas. But when they unearthed the theories, they delivered some of the best characterization of these important groups.
As they did, they started dating and had two children. (They finally sat down in Germany, where they enjoyed working together at one of the three whiteboards in their home.)
In 2018, they have only one category of lie-type groups left. After this was done, they would have proven the McKay assumption.
The final case took them another six years.
A “spectacular performance”
The fourth kind of lie group “had so many problems, so many bad surprises,” Späth said. (It did not help that the pandemic kept their two young children home from school in 2020, which made it difficult for them to work.) But she and Cabanes gradually succeeded in showing that the number of representations for these groups corresponded to those of their Sylow normalize – and that the way the representations matched the necessary rules. The last case was done. This automatically followed that the McKay assumption was true.
In October 2023, they finally felt confident enough in their proof to announce it in a room of more than 100 mathematicians. A year later, they placed it online for the rest of the community to digest. “It’s an absolutely spectacular achievement,” said Radha Kessar of the University of Manchester.