In a remarkable tale of love and intellectual perseverance, a German graduate student, Britta Späth, and her partner, mathematician Marc Cabanes, have recently made headline news by solving the McKay conjecture—one of the most significant unsolved problems in group theory—after two decades of dedicated work.

The journey began back in 2003 when Späth first encountered the McKay conjecture during her graduate studies. Little did she know, her initial goal of proving a few theorems to contribute to the problem would evolve into an all-consuming quest. “It was like a magnetic pull; I simply couldn’t divert my attention away from it,” she reflected.

While focusing on such a challenging problem posed risks to her academic career, Späth poured her heart and soul into it, which eventually led her to meet Cabanes at the Institute of Mathematics of Jussieu in Paris. Their collaboration blossomed not just mathematically, but personally as they fell in love and started a family.

The McKay conjecture itself revolves around the idea that to comprehensively understand complex mathematical groups—structures that encapsulate symmetries and are essential in fields ranging from physics to chemistry—one need only examine a small, specific portion of them. This principle echoes historical mathematical practices, like Eratosthenes' method of measuring the Earth's circumference through shadow observations in distant cities.

First proposed in the 1970s by mathematician John McKay, the conjecture suggests that the representations of a finite group and its Sylow normalizer—a specific subset of the group—share crucial numerical characteristics. McKay’s insights opened up new avenues of mathematical research, inspiring numerous mathematicians, yet a complete proof of the conjecture seemed an elusive goal for decades.

In 2004, a major advancement emerged through the monumental effort of classifying all finite groups' building blocks, which hinted that proving the McKay conjecture could be simplified. Subsequent works, particularly by Marty Isaacs, Gabriel Navarro, and Gunter Malle, offered a structured approach that allowed researchers to focus on specific groups, opening the door to further breakthroughs.

Späth, who began her academic journey under Malle’s guidance, found herself uniquely equipped to tackle the McKay conjecture. Her affinity for intense problem-solving was complemented by her ability to intuitively navigate the complexities of group representations—a skill that would soon prove invaluable.

As Späth transitioned to Paris Citée University, she teamed up with Cabanes, diving deep into the challenging yet fascinating world of Lie groups. Their efforts led them down an intricate path filled with challenges that often required crossing disciplinary boundaries, drawing techniques from various areas of mathematics—much to their credit.

The couple balanced rigorous research with family life, welcoming two children along the way while continuing to nurture their mathematical companionship.

Fast forward to October 2023, after six long years spent on the final category of Lie-type groups, Späth and Cabanes celebrated a monumental achievement: they had proven the McKay conjecture. Breaking the news to their peers, the mathematical community erupted in applause. “It’s an absolutely spectacular achievement,” lauded Radha Kessar from the University of Manchester.

This accomplishment not only resolves a major puzzle in mathematics but also stands as a testament to the power of dedication, collaboration, and love. As mathematicians worldwide delve into the implications of their work, the story of Späth and Cabanes serves as an inspiration: sometimes, the most complex problems require just the right mix of passion and perseverance to solve, proving that even in the abstract world of mathematics, human connections can yield extraordinary results.

Stay tuned as this enchanting story continues to unfold, and the impact of their findings ripples across the mathematical landscape!