School of Mathematics

Lens of Computation on the Sciences

What do quantum interference, flocking of birds, Facebook communities, and stock prices have in common?

Many natural and social phenomena may be viewed as inherently computational; they evolve patterns of information that can be described algorithmically and studied through computational models and techniques. A workshop on the computational lens, organized by Avi Wigderson, Herbert H. Maass Professor in the School of Mathematics, highlighted the state-of-art and future challenges of this interaction of computational theory with physics, social sciences, economics, and biology.

Birth of a Theorem

By Cédric Villani 

Cédric Villani in the Institute woods in 2009 (Photo: Claire Calmet)

Five months at IAS, two-hundred-fifty pages, and a Fields Medal

Cédric Villani, Member in the School of Mathematics in the spring of 2009 and currently Professor at Université Lyon I and Director of the Institut Henri Poincaré, has called his stay at the Institute one of his most productive periods, during which more than 250 pages were written. In his Member report to then-Director Peter Goddard at the end of his stay, Villani wrote of his collaboration with Clément Mouhot from Paris, “Writing up the paper on Landau damping was one of the most intense experiences of my professional life: for three months in a row, we kept unlocking seemingly untractable obstacles on a weekly basis. Our 180-page-long paper solves a fifty-year-old open problem.” A year after his IAS visit, Villani was awarded the 2010 Fields Medal, in part for the work that he did at the Institute on his proof of nonlinear Landau damping. Following are excerpts from Birth of a Theorem, translated by Malcom DeBevoise (Farrar, Straus and Giroux, 2015), originally published in 2012 as Théorème Vivant (Éditions Grasset & Fasquelle), which describe his fervent, halting, and very human experience in trying to obtain the proof.

 

Princeton, January 1, 2009

Finally, the Institute for Advanced Study—the IAS, as everyone calls it—comes into view. A little like a castle rising up in the middle of a forest. We had to go around a large golf course in order to find it. . . .

It is here that Einstein spent the last twenty years of his life. True, by the time he came to America he was no longer the dashing young man who had revolutionized physics in 1905. Nevertheless, his influence on this place was deep and long-lasting, more so even than that of John von Neumann, Kurt Gödel, Hermann Weyl, Robert Oppenheimer, Ernst Kantorowicz, or John Nash—great thinkers all, whose very names send a shiver down the spine.

 

Yitang Zhang’s Spectacular Mathematical Journey

Curiosity and Persistence by Unknown Mathematician Leads to Fundamental Breakthrough 

Member Yitang Zhang looking out at the Institute pond    (Photo: Amy Ramsey)
Member Yitang Zhang looking out at the Institute pond (Photo: Amy Ramsey)

A year ago April, the editors of the Annals of Mathematics, a journal published by the Institute and Princeton University, received an email with a submission by an unknown mathematician. “Bounded Gaps Between Primes” by Yitang Zhang, an adjunct professor at the University of New Hampshire, immediately caught the attention of the editors as well as Professors in the School of Mathematics. It was refereed by mathematicians who were visiting the Institute at the time and was accepted three weeks later, an unusually expedited pace.

“He is not a fellow who had done much before,” says Peter Sarnak, Professor in the School of Mathematics. “No-body knew him. Thanks to the refereeing process, there were a lot of vibes here at the Institute long before the newspapers heard of it. His result was spectacular.”

The Origins and Motivations of Univalent Foundations

by Vladimir Voevodsky 

Diagram by Voevodsky
This three-dimensional diagram is an example of the kind of ­”formulas” that Voevodsky would have to use to support his arguments about 2-theories.

Professor Voevodsky’s Personal Mission to Develop Computer Proof Verification to Avoid Mathematical Mistakes

In January 1984, Alexander Grothendieck submitted to the French National Centre for Scientific Research his proposal “Esquisse d’un Programme.” Soon copies of this text started circulating among mathematicians. A few months later, as a first-year undergraduate at Moscow University, I was given a copy of it by George Shabat, my first scientific adviser. After learning some French with the sole purpose of being able to read this text, I started to work on some of the ideas outlined there.

In 1988 or 1989, I met Michael Kapranov who was equally fascinated by the perspectives of developing mathematics of new “higher-dimensional” objects inspired by the theory of categories and 2-categories.
The first paper that we published together was called “∞-Groupoids as a Model for a Homotopy Category.” In it, we claimed to provide a rigorous mathematical formulation and a proof of Grothendieck’s idea connecting two classes of mathematical objects: ∞-groupoids and homotopy types.

Later we decided that we could apply similar ideas to another top mathematical problem of that time: to construct motivic cohomology, conjectured to exist in a 1987 paper by Alexander Beilinson, Robert MacPherson (now Professor in the School of Mathematics), and Vadim Schechtman.

In the summer of 1990, Kapranov arranged for me to be accepted to graduate school at Harvard without applying. After a few months, while he was at Cornell and I was at Harvard, our mathematical paths diverged. I concentrated my efforts on motivic cohomology and later on motivic homotopy theory. My notes dated March 29, 1991, start with the question “What is a homotopy theory for algebraic varieties or schemes?”

The field of motivic cohomology was considered at that time to be highly speculative and lacking firm foundation. The groundbreaking 1986 paper “Algebraic Cycles and Higher K-theory” by Spencer Bloch was soon after publication found by Andrei Suslin to contain a mistake in the proof of Lemma 1.1. The proof could not be fixed, and almost all of the claims of the paper were left unsubstantiated.

My Random Walks with Pólya and Szegő

By Olga Holtz 

Olga Holtz lecturing at the Women and Mathematics Program at the Institute for Advanced Study (2014)
Olga Holtz (Photo: Dan Komoda)

The Making of a Mathematician

My love affair with George Pólya began when I was seventeen. It was in Chelyabinsk, Russia, and my first year at the university was coming to an end. I had come across a tiny local library with an even tinier math section, which nobody ever seemed to visit, and had taken out most of those math books one by one before I came across The Book. It was George Pólya’s Mathematics and Plausible Reasoning.

By that time I was a total bookworm, having devoured almost a thousand volumes of my parents’ home library, mostly fiction. My familiarity with math books was much poorer although, growing up, I had enjoyed Yakov Perelman’s popular books for children on math and physics. I was a proud graduate of a specialized math and physics school, the only one in town, and had had a few wins at local olympiads in math and science. A top kid in class as far back as I could remember, I was arrogant as hell.

I read the introduction to Mathematics and Plausible Reasoning and its Chapter I standing up next to the bookshelf. It read like a novel. A cerebral one alright, which made you pay quick attention. Chapter I started out in the least orthodox way, comparing mathematical induction to a domino chain. The book endeavoured to explain not only what was mathematically true but how and why. I was hooked. Chapter I ended with a list of problems. I solved a couple of them still standing up but quickly came to a halt on Problem 3.

The arrogance kicked in––I had to solve those problems. I still remember carrying that book home after I checked it out. It was late spring, gorgeous weather, bird songs in the air, romantic couples––you get the picture. I was besotted with The Book.

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