One of Malle’s doctoral students took up the challenge: Britta Späth.
“Our Obsession”
In 2003, Späth began her doctoral studies at the University of Kassel under Malle’s guidance. She was exceptionally well-suited for tackling the McKay conjecture; even during her high school years, she spent weeks immersed in single problems. She particularly enjoyed those that tested her perseverance, recalling long hours spent searching for “tricks that are, in some sense, not even very profound.”
Späth dedicated herself to thoroughly studying group representations. Upon completing her graduate studies, she resolved to further pursue the McKay conjecture using her expertise. “She possesses this remarkable, almost uncanny intuition,” remarked her friend and collaborator, Schaeffer Fry. “She can instinctively sense the direction things will take.”
Courtesy of Quanta Magazine
A few years later, in 2010, Späth accepted a position at Paris Cité University, where she crossed paths with Cabanes. He specialized in a narrower subset of groups central to the reformulated version of the McKay conjecture, and Späth frequently visited his office to pose her questions. Cabanes often exclaimed, “Those groups are so complicated, my goodness!” Despite his initial reluctance, he too became captivated by the challenge. It transformed into “our obsession,” he mentioned.
There are four classifications of Lie-type groups. Together, Späth and Cabanes embarked on proving the conjecture for each of these classifications, yielding several significant results over the course of the next decade.
Their efforts led them to cultivate a profound comprehension of groups of Lie type. While these groups serve as essential building blocks for other theoretical constructs and thus attract significant mathematical interest, their representations are extraordinarily challenging to analyze. Cabanes and Späth often relied on complex theories drawn from various mathematical domains. However, by excavating these theories, they provided some of the most insightful characterizations of these pivotal groups.
As they progressed, their relationship blossomed, leading to two children. (They eventually settled in Germany, where they enjoy collaborating at one of their home’s three whiteboards.)
By 2018, they had only one category of Lie-type groups remaining. Once that was resolved, they would have established the truth of the McKay conjecture.
That final category took them an additional six years.
A “Spectacular Achievement”
The last type of Lie group presented numerous challenges and unexpected setbacks, Späth recalled. (The situation was exacerbated by the pandemic in 2020, which kept their two young children at home from school, hindering their work.) Yet gradually, she and Cabanes demonstrated that the quantity of representations for these groups corresponded with those of their Sylow normalizers—and that the manner in which the representations aligned met the required criteria. With that, the final case was completed. It followed naturally that the McKay conjecture was proved true.
In October 2023, they finally felt sufficiently assured in their proof to present it to an audience of over a hundred mathematicians. A year later, they shared it online for the broader community to examine. “It’s an absolutely outstanding accomplishment,” praised Radha Kessar from the University of Manchester.
