Yale Professor Sam Raskin and Team Solve Geometric Portion of Langlands Conjecture After Decades of Work

After more than three decades, five academic studies, and a thousand pages of research, a team led by Yale Professor Sam Raskin has made a breakthrough in solving a crucial aspect of what some call math’s “Rosetta Stone.” Raskin’s team succeeded in proving the geometric portion of the Langlands conjectures, a theoretical framework that connects three major branches of mathematics: number theory, harmonic analysis, and geometry. This accomplishment carries profound implications for mathematics, physics, and quantum field theory.

“We always knew that there was some very big mystery, and until we solve that we won’t be able to do the full proof,” said Dennis Gaitsgory, director of the Max Planck Institute for mathematics in Bonn, Germany, who closely collaborated with Raskin. “I thought it would take decades to prove it, and suddenly they cracked it.”

The Langlands program, initially proposed by former Yale doctoral student and professor Robert Langlands in 1967, is a set of conjectures that reveal deep connections between seemingly unrelated mathematical fields. These conjectures have had a transformative influence on modern mathematics, providing new perspectives and methods for thinking about mathematical relationships.

Raskin, a professor in Yale’s Faculty of Arts and Sciences, is known for his work in algebraic geometry, a field where geometric methods are applied to study algebraic equations. Raskin and his team formulated Langlands’ conjecture from the field of number theory in geometric terms before proceeding to prove it, making a monumental contribution to the Langlands program.

This milestone is the result of over thirty years of research in the geometric Langlands conjectures. Due to the highly abstract and detailed nature of this research, Gaitsgory emphasized that explaining all the necessary definitions could take months, if not years. The significance of the achievement is difficult to fully grasp for those without a deep background in mathematics.

“It is extremely beautiful, beautiful mathematics, which is connected very much with other mathematics and with mathematical physics,” said Alexander Beilinson, a University of Chicago professor who has worked with Raskin in the past.

Raskin’s journey in the field began during his undergraduate years at the University of Chicago, where he collaborated with Beilinson and Vladimir Drinfeld, mathematicians who explored the idea of the geometric Langlands conjecture. Later, at Harvard, Raskin completed his doctorate under Gaitsgory’s supervision, continuing his work in this field.

Raskin’s long-standing interest in Langlands’ conjectures has driven his career. He describes his approach to research as similar to experimental science, in that he observes developments by other mathematicians and then takes alternative approaches to advance the work.

“Mathematical research isn’t necessarily geared towards big problems, but it’s geared towards incremental progress and understanding things a little bit better,” Raskin said. “And sometimes you have a new idea which is interesting, and you play with it; if you get really lucky, then it connects to some big stuff.”

A key breakthrough occurred during a particularly challenging time in Raskin’s personal life. A few weeks after Raskin and Joakim Faergeman, a Yale graduate student, published an important paper, Raskin faced a difficult situation. He was driving his wife to the hospital, where she stayed for six weeks before the birth of their second child.

During this period, Raskin found time to call Gaitsgory, using the long drives between home, school, and the hospital to discuss ideas for the proof.

“There’s been a lot of progress, but there have been certain hurdles no one’s ever really been able to get past,” Raskin said. “Somehow, somewhere in there, in essentially the worst week of my life, I managed to get past the last hurdle.”

The significance of this breakthrough extends beyond mathematics. Physicists Anton Kapustin and Edward Witten independently realized that the geometric Langlands conjecture is a consequence of quantum field theory. This connection, according to Gaitsgory, provides mathematical proof for particular behaviors in quantum field theory, opening new avenues for exploration.

Aside from the ultimate proof, Raskin and his collaborators have made significant contributions to the field of Langlands conjectures over the years, shedding light on new relationships in modern mathematics.

“Even that process of just contributing knowledge [to] the field without solving the full proof is what 90 percent of my life consisted of,” Gaitsgory said. “But it was satisfying enough.”

Looking ahead, Raskin and Gaitsgory plan to continue their work in the field of Langlands conjectures. They remain confident that there is much more to discover and that this breakthrough is just the beginning of a deeper understanding of the subject.

Sam Raskin received his Ph.D. from Harvard University in 2014, and his career continues to impact the world of mathematics.