Reminiscences by a student of Langlands

TL;DR

This paper shares personal memories of Thomas Hales about his graduate years at Princeton under Robert Langlands, providing historical insights into the development of Langlands' influential mathematical program.

Contribution

It offers a personal historical account that enriches the understanding of the origins and development of the Langlands program from a firsthand perspective.

Findings

Personal recollections of Thomas Hales about Langlands' mentorship

Historical context of Langlands' work at Princeton

Insights into the early development of the Langlands program

Abstract

This article gives some memories of Thomas Hales of his years at Princeton as a graduate student under Robert Langlands. It has been prepared for the book "The Genesis of Langlands' Program," edited by Dr. Julia Mueller and Dr. Freydoon Shahidi.

Full text

Reminiscences by a student of Langlands

Thomas Hales

We are in a forest whose trees will not fall with a few timid hatchet blows. We have to take up the double-bitted axe and the cross-cut saw, and hope that our muscles are equal to them. – R. P. Langlands

Bob Langlands was my thesis advisor at Princeton, 1984-1986. No mathematician has shaped my research career so profoundly as he has. To use Weil’s memorable phrase, I offer a few souvenirs d’apprentissage.

1. Leap to Generality

A stack exchange discussion asks for the largest “leap-to-generality” in mathematical history. Suggestions include the notion of category theory (Eilenberg Mac Lane), the rise of abstract algebraic structures, Cantor’s set theory, mathematics of the infinite (starting with Archimedes’ use of the method of exhaustion), Aristotelian logic, the Turing machine, the foundations of probability (Kolmogorov), and the Langlands program.

Harish-Chandra, Grothendieck, and Kolmogorov were Langlands’s early “models for emulation.” In Langlands’s own words, “not satisfied with partial insights and partial solutions, they [Harish-Chandra and Grothendieck] insisted – not so much in the form of intentions or exhortations as in what they brought to pass – on methods that were adequate to establishing the theories envisaged in their full natural generality.”

2. Princeton, 1983

By the time I arrived in Princeton as a first-year graduate student in the fall of 1983, I had already acquired interests including Lie theory, representation theory, and the trace formula (thanks to Paul Cohen), $p$-adic analysis (thanks to J.W.S. Cassels), and modular forms (thanks to John Thompson in the heyday of moonshine).

From my first days at Princeton, I visited the Insititute for Advanced Study (IAS) once or twice a week, attending the Borel-Miličić $D$-modules seminar, Borel’s 60th birthday conference, and the Harish-Chandra memorial conference.

Eclipsing everything else that year were the momentous Morning and Afternoon Seminars on the trace formula. It is hard for me to convey how deeply formative those seminars were for me, even if I was not then at a stage to appreciate their full significance. Experts – especially Arthur, Clozel, and Rogawski – encouraged me and taught me the basics.

I first met Bob Langlands in person in January 1984 at a dinner arranged by Helaman Ferguson – somebody that I spent considerable time with my first year at Princeton. (Helaman’s son, Samuel, later became my coauthor on the proof of the Kepler conjecture.) By that spring, Langlands had become my advisor, and I had burrowed my way into the Corvallis conference proceedings. By arriving on scene after the 1977 Corvallis conference, which was still the subject of spirited conversations, I was made to feel I had missed a major event in the history of the Langlands program.

My other main reference that first year was Les Débuts (or the purple turtle as we called it), where the the fundamental lemma was first stated. My research problem, broadly stated, was to use Igusa theory to understand the transfer of $p$-adic orbital integrals between a reductive group and its endoscopic groups.

This remained my primary research interest for ten years. It has been a great adventure to witness the trajectory of endoscopy over the decades, culminating in Ngô Bao Châu’s proof of the fundamental lemma. In research posted to the arXiv last year, the fundamental lemma has finally emerged in its natural geometric context, as expressing that some dual abelian varieties have the same $p$-adic volume.

Already by the time I met him, Langlands had a towering reputation for his mathematical achievements. He once published an unforgettable critical book review with lede, “This is a shallow book on deep matters” that compounded his formidable reputation. I found that he was more mellow in person than his reputation might suggest, and he embodied the Institute’s ideal of curiosity driven research. It was heartwarming for me to see Bob last year at the Abel Prize conference in Minneapolis after many years.

3. apprenticeship

As a graduate student, the mathematical facts I learned mattered far less than my apprenticeship as a researcher under Langlands. I arrived with good work habits, a disposition for long calculations, and ambition. Here are a few things my apprenticeship gave me.

3.1. taste

American popular culture failed miserably in conveying great mathematical ideas to me. In my teenage years, my (undeveloped) idea of research mathematics was a confusing amalgamation of general relativity, Thom’s catastrophe theory, the Penrose staircase, and stunning continued fraction expansions. I remember wandering through the library stacks and wondering which of these books matter most?

There is no question that my mathematical taste improved enormously under Langlands. More broadly, lectures by Langlands, Serre, Weil, Borel, Kottwitz, Iwasawa, Thurston, and Witten developed my tastes.

3.2. seclusion

Today, Google Scholar, the ArXiv, Wikipedia, and MathOverflow give us nearly instant answers; polymath offers instant collaboration.

Then, there was a widespread belief that serious mathematical research required long periods of intense work in relative seclusion. In Flexner’s vision, the Institute “exists as a paradise for scholars who …have won the right to do as they please and who accomplish most when enabled to do so.” I did not see Langlands summers, when he went into work-related retreat in Montreal. I developed my own routines of seclusion: research retreats to a family cabin at the Sundance ski resort in Utah, secret study areas, and long runs along the Raritan Canal.

I have never seen Langlands at any mega-conference, and he did not push the chores of professional service. He discouraged rapid publication. He valued mathematical substance and frowned on veneer. Details of proofs mattered. He advised me to pick jobs based on educational merit rather than salary or prestige.

3.3. complexity

There was a communal belief that we were building a monumental edifice that would take many decades to complete. One hundred page research papers were the norm (and still are).

As documented in Wikipedia’s list of long mathematical proofs, it is no coincidence that some of the longest papers in mathematics are in this field: Langlands (Eisenstein series), Arthur (trace formula), Waldspurger (stable trace formula), Lafforgue (Langlands conjecture for the general linear group over function fields); or in neighboring fields in papers by Grothendieck, Hironaka, Harish-Chandra, Cartan, and Deligne.

As his student, I learned how to hold onto a problem that might take years to solve. I learned how to build evidence to support a hunch, how to follow a lead, and how to bury a fruitless idea and move on. These skills have transferred to my other large-scale research projects.

Researchers in the Langlands program were expected to rapidly assimilate many research fields. See Knapp’s nine-page reading list “Prerequisites for the Langlands Program,” which expands to thousands of pages of readings in algebraic geometry, Lie theory and algebraic groups, representation theory, algebraic number theory, and modular forms.

Yet book lists misguide us. Library scholarship is not mathematical research, and Langlands himself gave me just a few required readings. When we met, it was entirely focused on what I was able to calculate or figure out and on his suggestions for figuring things out better.

3.4. exoticism

Langlands has described his recurrent dream of escaping – in T.E. Lawrence style – “into the life and language of some exotic land and beginning anew,” leading to his expeditions to Ankara. When I was his student, he had just completed his French lectures on stabilization of the trace formula and was collaborating with Rapoport on their German masterpiece “Shimuravarietäten und Gerben.”

Langlands’s exoticism is inseparable from his mathematical oeuvre. To me, it explains how he went straight from a Yale thesis on PDEs and analytic semigroups, to teaching a graduate course at Princeton on class-field theory (“I still knew almost nothing about the subject, had only two weeks to prepare, was very young, and scared stiff”). With minimal ado, he would jump into an exotic field, then reconnect it with the ever expanding Langlands program. His new beginnings are memorable, such as when he wrote that if his proofs seem clumsy, it was because he “has not cocycled before and has only minimum control of his vehicle.”

3.5. examples and generalization

Langlands made many detailed studies of special cases, to shed light on the general theory. There was base change for $GL(2)$, Jacquet-Langlands for $GL(2)$, Labesse-Langlands for $SL(2)$, Igusa theory and endoscopy for $SL(3)$, representations of abelian algebraic groups, and a partial stable trace formula for $SL(n)$.

There were several axes of generalization. What works for $SL(2)$ (or even $GL(1)$) should work for all reductive groups. What works for the field of rational numbers should work for all global fields. What works for one local field should work for all. There should be a parallel between local and global theories, related by local-global principles.

Methods were always under assessment: could they encompass the general case?

3.6. freedom

Langlands’s interests were already shifting to percolation theory when I was his student. I was to be the last of his PhD students in the Langlands program for decades.

The winter of the Langlands program lasted several years, starting with Langlands’s switch to percolation theory, and continuing until Wiles’s announcement of Fermat’s Last Theorem. Although significant activity continued during the winter years, insiders and outsiders alike had become increasingly disenchanted by the glacial pace towards solutions to the central problems of the program.

At the time, Langlands would sometimes baffle his audience and say speak on percolation theory to an audience clearly expecting automorphic representations. As his student, I have claimed the same freedom to pursue my mathematical interests wherever they lead, however baffling. On one level, I have left the Langlands program behind. But on another level, I have remained a true student of Langlands by claiming this freedom.