Portrait of John Terilla, Professor of Mathematics at Queens College and on the Doctoral Faculty at the CUNY Graduate Center, Member of the Friends of IHES Honorary Committee

Relationship with IHES

My relationship with IHES started long before I visited the Institute. My PhD advisor Mike Schlessinger was a graduate student at Harvard during the years when Alexandre Grothendieck was visiting Harvard from IHES. Mike worked closely with Grothendieck on the foundations of deformation theory, solving a problem whose solution was published in a paper called “Functors of Artin Rings” and formed the basis of his PhD Thesis.

While I was a beginning graduate student at Chapel Hill, some of the most interesting articles that I encountered were written by IHES professors like Pierre Deligne or published in the Publications Mathématiques de l’IHÉS, like Raoul Bott’s “Morse Theory Indomitable.” Later, I became interested in certain deformation theory problems that came from mathematical physics. Certain models in physics could be understood best by looking at how they can be deformed. Mike and I discovered that Maxim Kontsevich at IHES was doing the most interesting, most insightful research in this area, and settling great problems, like resolving the Formality Conjecture. I was getting the feeling that IHES was a magical place where the very best foundational research was being conducted.

After graduating from Chapel Hill, I went to Stony Brook where my postdoctoral advisor was Dennis Sullivan, who had been a permanent professor at IHES for 23 years. Dennis also held the Einstein Chair of Science at the City University of New York (CUNY) and I spent a great deal of time at the CUNY Graduate Center attending the Einstein Chair Seminar. There, I attended seminar talks by IHES professors like Alain Connes and Misha Gromov which had a great influence on me. In fact, it was Sullivan’s Einstein Chair Seminar that drew me to New York City and to my permanent position here at the CUNY.

A few years after joining CUNY, I visited IHES for the first time. This was in the mid 2000s when I was a young mathematician just starting my career. It was there at IHES that I proved what was, in my opinion, one of my best theorems. I was still interested in deformation theory problems coming from physics. The problems I was thinking about were governed by something called a BV algebra. This was the mathematical object at work behind the scenes. The theorem I proved while I was visiting IHES was a smoothness theorem for differential BV algebras. It says that the vanishing of a minor obstruction has a kind of domino effect that makes all the obstructions vanish. Nowadays, people understand this result as a piece of a large picture about actions of moduli spaces extending to actions of compactified moduli spaces, but for me it really clarified a situation I was trying to understand for years — why the deformation problems coming from physics were so often unobstructed.

The experience of being at IHES and working on this problem was absolutely fantastic. I would get up early, have a coffee and get to work. Then I’d have lunch. I really appreciate good food, and it was really special to have a beautifully cooked meal served at lunch and to share this moment with other scientists.

I would work again in the afternoon and I’d relax in the evening by reading. The next day, I would repeat this experience. It was perfect. At some point, Maxim Kontsevich suggested that the smoothness theorem I was working on was related to something called the “Non-commutative Hodge-to-de Rham degeneration conjecture.” Of course, that turned out to be completely correct.

It took me a couple of months after I left IHES to really understand this connection. In the end, when I published the result in the Journal of Topology, it was from the perspective suggested by Kontsevich, framing the theorem in terms of the non-commutative Hodge-to-de Rham degeneration conjecture.

So this was my first visit at IHES: I was really well taken care of so I could focus and work hard, had great interactions with other researchers there, got a great idea from an amazing scientist which I took away from IHES and it fueled my work for quite a long time after my visit. This experience perfectly describes why IHES is such a fantastic place.

Friends of IHES adventure

Later, when I was back in New York, and I was talking to Bernard Saint-Donat about IHES, he brought up the idea of helping the institute from the US by getting involved with Friends of IHES. Shortly thereafter I was elected to the Board of Directors.

I have really enjoyed working with the Friends of IHES over the years. The Board changed and evolved, but there were always a great group of personalities dedicated to helping IHES. Coming to mind right now are some individuals who did so much, who have unfortunately passed away. Richard Bernstein, Stanley Deser, Robert Zimmer, Nicola N. Khuri, Vaughan Jones. I really miss them!

I believe I have been the longest serving board member of Friends of IHES so far. During the years I served, I was able to provide some continuity. I bridged the time from before the founding director Bernard Saint-Donat retired, though the time Mike Douglas joined the board, the many years I served as Treasurer, and into the current period in which Marilyn and Jim Simons have taken over from Mike. There have been a lot of dynamic periods of growth and enthusiastic leadership. I’ve stepped away now, but I am thrilled to now be part of the Honorary Committee, to stay involved, and continue helping the institute.

Some of my favorite events

I always enjoy the the scientific events, often celebrating great discoveries like the Higgs particle. I particularly enjoyed an event Friends hosted at CUNY. Thibault Damour talked about the theory of colliding black holes on the occasion that LIGO detected such an event with gravitational waves. The gala events are always great with so many scientists, artists and civic leaders in attendance. There have been many opportunities at Gala events to talk to mathematicians and physicists who are a little bit outside my area. It was fascinating to learn about spin glasses from Jennifer Chayes or about percolation from Hugo Duminil-Copin. During the 2016 gala, organized at the Intrepid Air and Space Museum, I started a discussion with Yiannis Vlassopoulos, who had been working at IHES for about a decade. This conversation continued over the following years and led to a very productive collaboration that is still going on today.

What I’m working on now

Currently, I am trying to understand the mathematics that is at work in natural language. Text data consists of a sequences of symbols. It’s a very mathematical object and it’s clear there is a lot of structure involved. For example, the meaning of what people intend to communicate is different from, but inseparable from, the characters they write. Semantics and structure are intertwined. There are other structures in language too: parts of speech, distributional information, and more. It’s clear that there is a great mathematical structure at work, but it’s not part of captured mathematics. It’s something new and it’s well hidden, so it’s a very captivating problem. And it’s work that first began at a Friends of IHES event a few years ago!