Yale News

After over three decades, five academic studies and one thousand pages, a team led by Yale Professor Sam Raskin has solved a part of what some consider math’s “Rosetta Stone.”

Raskin led a nine-person team that proved the geometric portion of the Langlands conjectures, a theoretical framework for fundamental relationships between three branches of math: number theory, harmonic analysis and geometry. This achievement has far-reaching 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 worked closely with Raskin on his team. “I thought it would take decades to prove it, and suddenly they cracked it.”

The Langlands program — originally proposed in 1967 by former Yale doctoral student and professor Robert Langlands — is a series of conjectures that suggest deep connections between seemingly unrelated areas of math. These conjectures have been fundamental to modern mathematics and unlock new ways of thinking about mathematical ideas.

Raskin, a professor of mathematics in Yale’s Faculty of Arts and Sciences, proved the geometric Langlands program. His research specifically focuses on algebraic geometry: using shapes and geometry to imagine algebraic equations. Raskin’s team formulated Langlands’ number theory conjecture in terms of geometry, and then proceeded to prove it. 

The monumental accomplishment is the product of over thirty years of research in the field of the geometric Langlands conjectures. Given its highly detailed and abstract nature, Gaitsgory noted that it would take months, if not years, to simply explain all the definitions needed to understand the work, and that the scope of the achievement is almost impossible for people without an advanced mathematics background to comprehend.

“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 closely with Raskin in the past. 

Raskin was first introduced to this research as an undergraduate student at the University of Chicago, where he worked with Beilinson and Vladimir Drinfeld, mathematicians who first explored the idea of the geometric conjecture. He later completed his doctorate at Harvard, where he further pursued research in the field with Gaitsgory, his Ph.D. advisor. 

Raskin said he has always been interested in the field of Langlands conjectures, and has explored different approaches to research throughout his career. He compares his work to experimental science in the sense that he closely monitors other researchers’ contributions to the fields and chooses less-attempted approaches to delve into.

“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.

One of the main breakthroughs came during a tumultuous time in his life. A few weeks after Raskin and Joakim Faergeman, a Yale graduate student, had published a crucial paper, Raskin drove his wife to the hospital, where she stayed for six weeks before the birth of their second child. 

During this time, Raskin used the long hours driving from home, school and the hospital, to call Gaitsgory and 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.” 

Gaitsgory explained another potential significance for science: physicists Anton Kapustin and Edward Witten independently realized that the geometric Langlands conjecture was a consequence of quantum field theory. Therefore, according to Gaitsgory, this research offers mathematical proof for particular behaviors in quantum field theory. 

Aside from the final proof, the research that Raskin and his collaborators have been producing over the past decades has shaped the field of Langlands conjectures, and has unlocked 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.”

In the future, Raskin and Gaitsgory plan to continue working in the field of Langlands conjectures, and they feel that there is still much information to be discovered. 

Sam Raskin received his Ph.D. from Harvard in 2014.