Oscar Lanford was not only a remarkable and deep mathematician; he was also a wonderful friend. We met in the fall of 1961 during our graduate years at Princeton; Oscar had arrived one year before me. Our friendship became marked by our office desks in Palmer 420 being within talking distance, and we did a great deal of that. I came to understand that Oscar had read and understood every mathematical text or paper that I was interested in, and Oscar routinely cleared away every cobweb in my puzzlements. Over time Oscar became my closest friend in Princeton.

He was already together with Regina, and eventually Lizabeth was also on the scene. Here are Oscar and Regina with Lizabeth in the 1965-6 academic year, visiting my home in Kingston, New Jersey, near Princeton. Years afterward, Lizabeth became my advisee when she concentrated in physics as a Harvard undergraduate, before becoming a pediatric cardiologist.

We both worked on our theses with Arthur Wightman on the question of the existence of non-linear quantum field theories. A successful resolution of this problem would show the mathematical (logical) compatibility of special relativity with quantum theory and interaction—a major open problem in theoretical science as well as in mathematics. Success would also give the first non-trivial example of a relativistic quantum field with non-trivial scattering, which also satisfied the axioms of Wightman.

For the 1963-4 academic year, Wightman helped IHES start in Bures, by advising Léon Motchane on the organization of a year devoted to mathematical quantum field theory. It was good fortune that Wightman invited both Oscar and me to join him in that journey. IHES had moved to Bois Marie only one year earlier. Louis Michel has just joined the institute, and David Ruelle would arrive the following year.

While today all the professors at IHES have offices in the scientific buildings, in that early year most professors had offices in what is now the administrative building; a few had offices in the separate building that housed the kitchen and lunchroom. Construction had only just begun on what is now the scientific building, with the upper floor being completed half-way through our year there. The music pavilion served multiple functions as library, seminar room, and workspace for visitors. In January 1964, we moved into offices on the upper floor of the first scientific building, while the lower level remained an empty shell.

While some permanent members lived in Bures, most of the visitors for that year were housed in apartments in the Résidence Chevreuse in Orsay. Those long-term visitors that year included several students: Oscar, Sergio Doplicher, Edwardo de Raphael, Klaus Pohlmeyer, and Klaus Hepp on occasion, as well as Jurko Glaser, Res Jost, Harry Lehmann, Jean Lascoux, François Lurçat, Henri Epstein, and Michael Artin. Annie Rolland put everything together, accomplishing what we thought comprised the work of four persons.

This was a wonderful era for mathematical physics. New connections arose between physics and mathematics, and Oscar was at the center of much of this work. Before Regina arrived, we often had dinner together at the Croix de Bures on the Route de Chartres, which served a traditional French menu that we enjoyed while discussing mathematical physics. I always took the crème de marron as dessert.

In our thesis work, Oscar studied on the case of quantum fields with Yukawa interaction between bosons and fermions, while I concentrated on non-linear boson self-interactions. Eventually Oscar and I published one joint paper in 1967 giving our early results with Wightman on cut-off quantum field theories [2]. The full solution required the removal of these cutoffs, which I did for two-dimensional space-time in a long series of papers with James Glimm, and later also with Thomas Spencer. Members of our research groups later obtained the Yukawa model and boson self-interaction in three dimensions. Whether one can find any complete example in four-dimensional space-time remains a fundamental open problem.

When Oscar and I left Princeton in 1966, we wanted to continue our scientific discussions. We both had offers of faculty positions at Berkeley, and we thought that would be the way to go on. Somehow, I got side-tracked and ended up at Stanford. It turned out that the distance between Palo Alto and Berkeley was practically further than what it appeared to be on a map. So our interactions were less frequent, and the next year I moved back to the east coast.

Some sixteen years after our student encounters, Oscar left Berkeley to return to IHES as a professor, where he spent seven happy years. He had major collaborations with David Ruelle and others during that time, leading to the Dobrushin-Lanford-Ruelle equations [3]. Oscar also became interested in understanding the mathematical structure of the Feigenbaum fixed point, also studied by Collet and Eckmann [1,4,5]. In Oscar’s work on using a computer computation as part of a mathematical proof, he insisted on understanding exactly what the computer did in performing a calculation. This meant that Oscar would only program his computations in machine language, in order to be able to rely on error bounds with complete certainty. I believe this may have been the first “computer-assisted proof” about which there could be no doubt of its veracity.

Oscar also moved from statistical physics to dynamical systems, where he made fundamental progress. Eventually he moved to ETH Zurich. My close friend Konrad Osterwalder, who had become Rektor there, confided to me that one of his proudest moments was to recruit Oscar. I often visited ETH where it seemed inevitable that I would run into computer problems. The person who always spent his generous time to provide the way to solve them was Oscar.

I had lunch together with Oscar and Regina on August 21, 2013, at their home in Thalwil, along the right side of the lake of Zurich; at the time Oscar was quite ill. At lunch I took this snapshot (the cover picture) and gave them a copy of the photo in Kingston from our Princeton student days almost 50 years earlier. That lunch was on a beautiful afternoon, and I remember, when walking to the train, reliving in my mind many of our previous interactions. While I miss him sorely, his wonderful contributions to mathematics and physics will live forever.

References

1. Pierre Collet, Jean-Pierre Eckmann, and Oscar E. Lanford III; Universal properties of maps on an interval. Comm. Math. Phys. 76 (1980), 211–254.

2. Arthur M. Jaffe, Oscar E. Lanford III, and Arthur S. Wightman; A general class of cut-off model field theories. Comm. Math. Phys. 15 (1969) 47–68.

3. Oscar Lanford and David Ruelle; Integral representations of invariant states on B* algebras. J. Mathematical Phys. 8 (1967) 1460-1463.

4. Oscar E. Lanford III; Remarks on the accumulation of period-doubling bifurcations. Mathematical problems in theoretical physics (Proc. International Conf. Math. Phys., Lausanne, 1979), pp. 340-342, Lecture Notes in Phys., 116, Springer, Berlin-New York, 1980.

5. Oscar E. Lanford III; A computer-assisted proof of the Feigenbaum conjectures, Bull. A.M.S. (New Series), 6 (1982) 427–434.

Arthur Jaffe
Cambridge, Massachusetts
September 25, 2024

It is with great sadness that IHES shares the news of the passing of Pierre Cartier, aged 92, this Saturday, August 17, 2024.

Pierre Cartier, graduate of École Normale Supérieure and CNRS research director in mathematics, was a researcher at IHES from 1971 to 1982. He remained close to the Institute ever since.

An active member of the Bourbaki group from 1955 to 1983, he is known in particular for his theory of commutative formal groups, as well as for his work on typical curves, quantum groups, symmetric functions in combinatorics, and Hopf algebras.

Tribute to Pierre Cartier by Michel Broué (in French)

Tribute to Pierre Cartier by Francis Brown

Tribute to Pierre Cartier by Alain Connes (in French)

Tribute to Pierre Cartier by Luc Illusie

Tribute to Pierre Cartier by Cédric Villani (in French)

Tribute to Pierre Cartier by Michel Waldschmidt (in French)

Pierre Cartier, A Visionary Mathematician : Tribute by Alain Connes and Joseph Kouneiher

Read Pierre Cartier’s text written for the book The Unravelers by Jean-François Dars, Annick Lesne and Anne Papillault here.

It is with great sadness that IHES shares the news of the passing of Henri Epstein, aged 92, on Thursday, August 15, 2024.

Henri Epstein, a graduate of École Polytechnique (X53), discovered theoretical physics thanks to Louis Michel, a permanent professor at IHES from 1962 to 1992, who was then at Polytechnique.

After several visits to the Institute, Henri Epstein, CNRS research director in physics, officially joined IHES in 1971. A theoretical physicist basing his research on sophisticated mathematical models, he worked in particular on the quantum theory of relativistic fields and on dynamical systems, areas in which he made numerous contributions.

Fascinated by computers and machines, he also helped the Institute in its IT equipment.

CNRS emeritus research director since 1999, he continued to play a major role at IHES in facilitating and developing contacts between physicists and mathematicians.

—

Tribute to Henri Epstein by Jürg Fröhlich

Memories of Henri Epstein by Giovanni Gallavotti

Tribute to Henri Epstein by Arthur M. Jaffe

Henri Epstein and Mathematical Physics by Ugo Moschella and Slava Rychkov

Reminiscences of Henri Epstein by Dennis Sullivan

Laureate of the 2022 Fields Medal, Hugo Duminil-Copin, Permanent Professor in Mathematics at IHES and the University of Geneva, participated in the closing ceremony of the International Final of the Mathematical and Logical Games Championship, held at École Polytechnique’s Poincaré Lecture Hall in Palaiseau on August 26th.

Passionate about mathematical games, particularly the game of HEX, which is the focus of one of his public lectures, Hugo Duminil-Copin sponsored the 38th edition of the Mathematical and Logical Games Championship, whose final took place on August 25th and 26th.

With more than 27,000 participants, including 500 finalists aged 7 to 77, this year’s edition of the Games was particularly successful.

During the award and closing ceremony, organized by the French Federation of Mathematical Games (FFJM), Antoine Vanney, President of the FFJM, had the honor of welcoming several distinguished guests alongside Hugo Duminil-Copin. These included Laura Chaubard, President and General Director of École Polytechnique, Claire Waysand, Executive Vice President and General Secretary of ENGIE, and Philippe Marie-Jeanne, AXA’s representative on the IHES Board of Directors.

ENGIE, a major partner of the Mathematical and Logical Games Championship in 2024, has also been supporting IHES through its Foundation by funding visits from women scientists to the Institute and contributing to the budget of the annual IHES summer school for young researchers since 2023.

To promote gender equality in mathematics, two new initiatives supported by ENGIE were introduced this year by the FFJM: a mixed 2+2 team championship and the ENGIE Cup for Girls in High School. During the final, the Cup finalists had the opportunity to meet Hugo Duminil-Copin, as well as several representatives from ENGIE, who shared their career experiences.

These initiatives are already yielding significant results for the FFJM: with more than 40% female participation in the French Championship, the event counts a much higher proportion of girls compared to the percentage of female STEM students in France.

IHES congratulates all the finalists, and especially the girls who participated, for their involvement in this inspiring event!

The Institute also extends an invitation to the inauguration of the “Maths and Games” exhibition, organized by [S]cube, and also sponsored by Hugo Duminil-Copin, on December 5th 2024.

Photo credit : © École polytechnique – J. Barande

I worked at IHES and lived in Bures-sur-Yvette for four and a half years, from 1978 to 1982. The Institute was truly a paradise back then, and I consider the time I spent there to be one of the best and most delightful periods of my life. One of the inhabitants of that paradise was Henri Epstein.

But let’s go back a little farther into the past. I first met Henri at the famous 1970 Les Houches summer school on Statistical Mechanics and Quantum Field Theory. My PhD advisor, Klaus Hepp, along with other luminaries in mathematical physics, lectured on renormalized perturbation theory in quantum field theory (BPHZ). Henri presented his ongoing work with Jurko Glaser on a new approach to renormalized perturbation theory, that made very clever use of locality and causality and avoided encountering infinities. Their work was not published, yet, but would later become famous. The Les Houches school was also the occasion when I first met further top mathematical physicists, including some of those who were or would become inhabitants of the paradise at Bures-sur-Yvette, in particular Alain Connes, the late Oscar E. Lanford III, and David Ruelle.

In 1970, I was a beginning PhD student, and I must say, I found Henri quite intimidating—a feeling that never entirely left me. He seemed to insist on ultimate mathematical precision and conceptual clarity, both in his thinking and in discussions with colleagues, which made it difficult for me to interact with him.

A year or two later, I met Henri at the University of Zurich, where he gave a great lecture on an application of his work with Glaser to quantum electrodynamics. Most of the theorists in Zurich were hopeful that Henri would be offered a professorship at the university. He shared scientific interests—particularly in axiomatic quantum field theory in the spirit of Arthur S. Wightman—with two of my most admired teachers, Res Jost and Klaus Hepp. Res held Henri in the highest esteem, describing him as “pure gold.”

Unfortunately, for reasons I’d prefer not to go into, the professorship offer never materialized—a significant loss for theoretical physics in Zurich.

In the fall of 1972, I became a postdoctoral researcher at the University of Geneva. Almost every week, Jean-Pierre Eckmann, Jurko Glaser, and Raymond Stora would meet for a working seminar in mathematical physics, and I was one of the regular listeners. This provided yet another opportunity to meet Henri and learn about his work with Glaser on renormalization theory.

In 1974, I encountered Henri again at Princeton, shortly after he had published his new proof of the strong subadditivity property of quantum-mechanical entropy—a result first established by Lieb and Ruskai. Henri’s analysis made use of holomorphic functions, an area of mathematics in which he was an unparalleled master. I found his proof both very illuminating and, relatively easy to follow.

In the summer of 1975, I returned to Europe for a few months. Jean-Pierre Eckmann and Henri were working on a problem that I had also become interested in: proving that perturbation theory is asymptotic to the S-matrix in a family of relativistic scalar quantum field models in two space-time dimensions, known as P(ϕ)₂-models. They showed me a preliminary draft of a paper that was intended to contain a proof of this result. However, there was still a small gap in their argument, which I was able to fill. This is how I became a co-author of a beautiful paper for which Henri deserves ninety-nine percent of the credit.

The paper showcases Henri’s remarkable knowledge of the properties of chronological, retarded, and advanced Green functions in quantum field theory. He had previously applied this expertise in significant work with the late Jacques Bros and others on the properties of scattering amplitudes in relativistic quantum field theory.

During my time at IHES in the late 1970s and early 1980s, I witnessed the beginnings of Henri’s fascination with computers and his growing interest in the theory of dynamical systems. I was deeply impressed by his proof, with Massimo Campanino, and building on partial results by David Ruelle, of the existence of a fixed point in Feigenbaum’s equation—a proof that didn’t rely on any computer assistance. In hindsight, I regret not following Henri more closely in his explorations of dynamical systems and sharing his fascination with computers.

In 2015, I organized a special trimester at IHES on new developments and open problems in quantum physics, entitled “Trimestre sur le Monde Quantique.” I believe this was a successful program, addressing some of the truly deep questions in contemporary physics. My impression was further validated by Henri’s comments—he mentioned that he found the lectures on quantum physics, both theoretical and experimental, to be highly educational and entertaining.

In recent years, Henri and various colleagues, mostly worked on properties of quantum field theory on de Sitter and anti-de Sitter space. The little I learned on these matters from Henri and his friends inspired some results in a paper entitled “KMS, Etc.” I wrote with a student, twenty-three years ago.

Let me conclude with a few remarks on some well-known traits of Henri’s personality, cherished by his friends. Above all, Henri exemplified outstanding intellectual honesty. He possessed a highly critical mind and insisted on accuracy and precision in his professional interactions. He was deeply cultured, and despite being more knowledgeable and intelligent than most of us, he remained modest and kind. Henri appreciated style in all areas of life—whether in conversations, discussions, cultural pursuits, literature, or culinary adventures. He was remarkably loyal and respectful in his relationships with colleagues and friends, and he had a fine sense of humor.

To close this testimonial, I would like to remind readers that Henri’s youth was profoundly shaped by the precarious circumstances in Europe between 1933 and 1945. We all hope such times will never return.

I will miss Henri.

(I would also like to add that I will miss the pleasant encounters and conversations I had with Pierre Cartier, who passed away just a few days after Henri.)

Jürg Fröhlich (ETH Zurich)
September 2024

I first met Henri in the early ’70s at IHES. His approachability quickly eased my hesitation about engaging with a scientist of his stature, and he almost immediately offered strong encouragement for my research on the renormalization group in quantum field theory. Since then, we met regularly, at least until the COVID-19 pandemic.

I often visited IHES and was always eager to spend time with Henri to seek his advice and ideas on a wide range of topics. I was particularly impressed by his formidable expertise in holomorphic functions. In the early ’80s, Henri, Pierre Collet, and I began working on SL(2,R), which eventually led us to develop a KAM theory of perturbations of geodesic flows on surfaces of constant negative curvature. This was a long project to which Henri contributed essential ideas and techniques. I remember those two years as an exciting time, filled with frequent new developments, as everyone working in non-Euclidean geometry would expect.

After that, we continued to meet in various places such as Les Houches, Zurich, and, of course, Paris. He helped me understand Mitchell Feigenbaum’s work on universality by patiently explaining it through the lens of his own contributions, often referring back to applications of holomorphic functions. He also explained his work with Jean-Pierre Eckmann, which allowed them to recover classical results in stability theory and normal form theorems for PDEs.

In the ’90s, Henri introduced me to the Linux operating system, demonstrating his remarkable open-mindedness. I owe him particular gratitude for the time he dedicated to this—though brief, it was very intense and effective. During this period, he also returned to quantum field theory, revisiting his early career interests with Jacques Bros and Vladimir Glaser in perturbation theory and scattering amplitudes—their early papers had impressed me even before we first met. He continued to work steadily on quantum fields in anti-de Sitter geometry, introducing new and original ideas and techniques.

I still have vivid memories of my last brief technical discussion with him during the daily tea break at IHES. He was determined to understand the details of a proof on the location of the roots of a class of self-inversive polynomials on the unit circle. I also remember our frequent gourmet meetings at Paris restaurants or at his home with mutual colleagues, often accompanied by his sister Evelyne. Together, we frequently discussed world affairs (particularly French politics), often in Italian—a language he mastered perfectly.

Kindness and availability characterized his personality and served as an example for many. Several young postdocs benefited from his teachings and were inspired by his conferences. The theoretical physics community will miss him. I miss you already, caro amico Henri.

Giovanni Gallavotti, Roma, August 2024

Henri Epstein was a deep thinker who made important progress both in quantum field theory and in dynamical systems. He was also a dear friend.

We first met over sixty years ago, during the 1963—64 academic year at IHES, marking its beginning in Bures-sur-Yvette. During that time the first part of the scientific building was only under construction, and researchers had to find other places to work. But the tearoom in the main building (then with most of the professors’ offices) provided a wonderful place to interact. My teacher, Arthur Wightman, helped organize a year-long workshop on mathematical foundations of quantum field theory, so it was wonderful to be included while still a student. Jurko Glaser was there as well, and Henri was engrossed in their program with Jacques Bros to understand the analyticity properties in quantum field theory that were a consequence of the Wightman axioms.

During that period that the three of us wrote a small paper showing that the energy density in any Wightman quantum field theory could not be positive. It is remarkable that one still finds many citations of this result. Later Henri became interested in the Feigenbaum fixed point, and he gave an ingenious convergence proof.

Henri and I got to know each other much better during the IHES year by attending weekly (along with Jean Lascoux) the course of Laurent Schwartz, held at the Institut Henri Poincaré. We often did something together after the class, and Henri became my expert advisor on where to dine in Paris. Some years later I discovered that Henri (as well as Nicolas Sourlas and Jean Iliopolous) were wonderful cooks. During subsequent visits to Paris we enjoyed several meals that they prepared, rivaling the best addresses they could recommend.

In 1964—65 Henri came to Princeton as a post-doctoral fellow, where his insights were highly appreciated, despite his enormous modesty. We interacted a great deal not only with Wightman, but also with Klaus Hepp, Gian-Fausto Dell Antonio, and Hans Borchers. I took a photo (above) of Henri with this group while boarding a plane on the way to a conference in Cambridge, Massachusetts organized by I. Segal. The second photo (below) is in the yard of the home of Edward Nelson one Princeton afternoon, also with Arthur Wightman.

A few years later I gave a colloquium at CERN; at the time Henri was a member of the theory division. He suggested that we meet the next day for lunch. Henri picked Shelly Glashow and me up in his car that morning, and without explanation, off we went to Paul Bocuse in Lyon. Henri had reserved a table; it was a beautiful afternoon on the terrace! Through this extraordinary outing, Henri introduced me for the first time to “3-star” French cuisine.

I saw Henri and his sister last during the 2022 Ising meeting at IHES, spending an afternoon at their home on the rue de Vaugirard. Along with his other acquaintances, we will miss Henri’s humor, his generosity, his friendship, as well as his great insights into mathematical physics.

Arthur Jaffe
Cambridge, Massachusetts
September 10, 2024

Thomas Edison wrote that genius is one percent inspiration, ninety-nine percent perspiration. Had he met Pierre Cartier, he would not have omitted the third, and probably the most important ingredient: conversation. Cartier was interested in absolutely everything and loved nothing more than to chat with people, no matter what their background was. Through such discussions and his teaching, he influenced and inspired an entire generation of mathematicians.

Cartier had an encyclopaedic knowledge of mathematics and physics, not to mention history, politics and a whole range of other subjects. He always made time to talk to students, and his conversations were always interspersed, to their delight, with anecdotes about mathematical figures going back half a century or more. He was gifted at rescuing doctoral students who were struggling, or had lost their way, by taking them under his tutelage (I was one of them, and will forever be grateful).

One of my most enduring impressions of Cartier was his open-mindedness. He had the kindness to see mathematicians first and foremost as people, treating them all equally, whether they were students starting out on their career or colleagues at the very top of their field.

Cartier’s reputation led to many invitations from colleagues all over the world well into his eighties (he was already past the age of retirement when I first met him, but he remained active and very energetic for many years afterwards). During one such visit, his host sent a student to collect him at the train station who wasn’t entirely sure who he was supposed to be meeting. After asking his name, the student responded in disbelief “But are you related to THE Cartier?” – it was inconceivable to him that the Pierre Cartier from his textbooks was in front of him. After giving some thought as to how best he should reply, Pierre modestly answered, “Mmmm…. Yes! He was my uncle”.

Cartier believed strongly in the unity of mathematics and physics. He had an unusually broad vision of both subjects, and could see each new development in its proper historical and intellectual context. He taught us that the most important questions are those which have applications in many different fields, and that one should embrace all available techniques to tackle them. He would often say that one should use both hands when doing mathematics, and not keep one hand tied behind one’s back.

Cartier was an ever-constant presence in Bures and for many of us, it is hard to imagine the IHES without him. One always knew he was around from the sound of his booming voice and immediately recognizable laughter. He will be greatly missed.

Francis Brown

Photo credit: © Jean-François Dars / IHES

It was with great sadness that I learned of the passing of Pierre Cartier. An immensely talented mathematician has left us, someone I knew well starting in the 1960’s. We first met at a seminar by Grothendieck. I was deeply impressed by the depth and encyclopedic breadth of his knowledge, but I had no idea at that time that his work would play a pivotal role in my research for decades to come.

In the early 1970s, crystalline cohomology—a new cohomology theory envisioned by Alexander Grothendieck in the late 1960s, and developed by Pierre Berthelot in his doctoral thesis—was attracting a lot of attention from many geometers. A conjecture of Katz, predicting certain inequalities on p-adic valuations of the Frobenius, was particularly studied. I was astonished when I realized that the solution, under certain assumptions by Mazur, and later in full generality by Ogus, ultimately relied on what is now called the Cartier isomorphism—a reformulation by Katz of a construction originally published by Cartier in a 1957 note in the Comptes Rendus de l’Académie des Sciences. This isomorphism, for the de Rham complex of a smooth variety in positive characteristic, which links its components to its cohomology sheaves, would play a key role in differential calculus in characteristic p or mixed characteristic for years to come.

However, yet another construction of Cartier led to a decisive turning point in my research. This was
Cartier’s theory of typical curves, a more flexible and powerful approach to the classical theory of Dieudonné modules. A bold application of Cartier’s theory by Spencer Bloch to Quillen’s K-theory groups led to the construction of an explicit complex with remarkable structure and properties, particularly concerning the p-adic valuations of the Frobenius mentioned earlier. This complex computed the crystalline cohomology of smooth projective varieties of dimension less than p, for any p > 2. Following a suggestion of Deligne, I proposed a K-theory-free, more general construction of this complex, with no restrictive hypotheses, using purely differential geometric means. Here again, the Cartier isomorphism proved to be an essential tool. This complex, which I named the de Rham-Witt complex, is still generating a lot of research today.

Let me share one more memory, where the Cartier isomorphism was to strike yet again. I recall Cartier’s surprise and joy (perhaps tinged with slight frustration?) when, in 1986, I told him that Deligne and I had just discovered that a reinterpretation of his isomorphism in terms of deformation theory led to a purely algebraic proof of one of the fundamental theorems of Hodge theory—namely, the degeneration of the Hodge to de Rham spectral sequence for a complex smooth projective variety. Various questions and conjectures related to this result have been very much studied in recent years.

I much admired Cartier’s mathematical rigor, and the deeply moral rigor of which it was a manifestation, combined with his keen sense of humor and his genuine simplicity. Cartier was also a wonderful storyteller. I could listen to him for hours. He had juicy anecdotes about many subjects, particularly about the Bourbaki group. It was a joy to see him mimic Dieudonné’s Homeric fits of anger. I still hear his deep, warm voice.

Luc Illusie

Acknowledgement: I heartily thank Nick Katz and Richard Kruel for the English translation of my text “L’isomorphisme de Cartier.”

Photo credit: Jean-François Dars / IHES

Henri Epstein met Louis Michel in 1955, during his final year as a student at the École polytechnique. This pivotal encounter steered him towards a lifelong career as a theoretical physicist. After completing his thesis, Henri was staff member of CERN’s theoretical physics department from 1967 to 1970. During this time, he continued to make regular visits to IHES, eventually joining the Institute as a CNRS researcher in 1971.

In 1957-58, he attended Arthur Wightman’s course on axiomatic quantum field theory at the Collège de France, a topic that became the focus of his first research endeavors. In the 1960s, Henri achieved remarkable results, such as the proof of the non-positivity of energy density (with Vladimir Jurko Glaser and Arthur Jaffe) and the establishment of analyticity properties, including the crossing symmetry relation, for 2→2 scattering amplitudes (with Jacques Bros and Glaser). The latter, now a classic result, forms one of the foundations of the S-matrix bootstrap program, which has seen a significant revival in recent years.

In the 1970s, Henri’s work on quantum field theory led to the famous Epstein-Glaser construction of causal perturbation theory, which offers a novel approach to handling UV divergences in Feynman diagrams. The full potential of this method, particularly for perturbations of conformal field theories, still remains an active area of research.

In the 1980s, Henri’s focus shifted to discrete dynamical systems, and particularly Feigenbaum’s universality properties. He notably came up with an ingenious new proof for the existence of fixed points for the Feigenbaum-Cvitanović equation. Unlike the original argument by Oscar Lanford III, his proof, based on the Schauder-Tikhonov fixed point theorem, does not require computer assistance.

In the latter part of his life and scientific career, Henri devoted himself primarily to the study of quantum fields in de Sitter and anti-de Sitter universes, collaborating with Jacques Bros, Ugo Moschella, and occasionally with Michel Gaudin and Vincent Pasquier. De Sitter universes, which are solutions to Einstein’s cosmological equations without matter, play a central role in contemporary theoretical physics. Mathematically, they are analytic Lorentzian manifolds, structures particularly well suited to the application of methods from the theory of analytic functions of several complex variables—a field in which Henri was the last surviving master of a glorious era.

The results Henri and his colleagues achieved over 25 years range from general structural properties of de Sitterian quantum field theories to the derivation of concrete formulas such as the Källén–Lehmann spectral representations. The latter are non-trivial, exact formulas with surprising implications for particle stability and the existence of bound states.

Henri continued his research as long as he could, up until the last days of his life. His final paper was published on the very day of his passing. He loved beauty and elegance in his research and in all aspects of life. He had a deep appreciation for literature and music. His sense of humor and clarity of mind made working with him a true pleasure and joy. Until the end, he remained youthful in spirit, enthusiastic, sincere, and generous. Mozart and Schubert, whom he loved so much, accompanied him on his last journey.

Ugo Moschella and Slava Rychkov

 

This year, IHES hosted yet another summer school in physics, organized by Bruno Le Floch, Elli Pomoni, and Masahito Yamazaki. Following the 2018 edition on supersymmetric localization and exact results, they joined forces with Zohar Komargodski to organize two weeks focused on symmetries and anomalies.

While the 2018 school aimed to overview calculation methods in supersymmetric gauge theories—a field kick-started by the computation of the instanton partition function for four-dimensional theories with N=2 supersymmetry by Nikita Nekrasov and of their sphere partition function by Vasily Pestun, and which has links to matrix models, integrable systems, conformal field theory and holography—the situation for this year’s school was quite different. “We were thinking about organizing a school on modern notions of symmetry for quite some time now. If I remember correctly, our first email exchanges on this idea go back to 2021,” says Bruno Le Floch. “Back then, the subject was just starting, and we didn’t imagine that it would generate the amount of activity it did in the last two to three years. This is, of course, very fortunate, and the Symmetries and Anomalies summer school couldn’t have come at a better time,” explains Masahito Yamazaki.

A symmetry is a certain invariance of a physical system that can be helpful in its analysis. For instance, a symmetry may reduce the number of parameters one has to consider. It has recently become apparent that the usual notion of symmetry can be widely extended with the introduction of more general notions of symmetries such as higher-form, higher-group or non-invertible symmetries. Just as ordinary symmetries are understood in terms of groups, the language of these generalized symmetries is that of fusion n-categories.

An anomaly appears when a symmetry does not carry over from the classical to the quantum world. In quantum systems ranging from condensed matter to high-energy particle physics, symmetries can feature different types of anomalies, which may constrain the dynamics or ruin a model’s consistency. This can give important clues on possible extensions to the Standard Model or new topological phenomena in quantum materials. Anomalies have also played an essential role in the modern developments of supersymmetric quantum field theories and string theory.

The 2024 summer school aimed to introduce students to the physical and mathematical underpinnings of anomalies, including its more mathematical aspects on topological quantum field theory and characteristic classes, with a view toward recent applications to topological phases of matter and strongly coupled gauge theories. The overarching idea was to have courses from three points of view that build upon each other: that of a mathematician, a high-energy physicist, and a condensed matter physicist.

A notable feature of the symmetries and anomalies community is that its current leaders are often still at an early stage of their careers. “Knowing that many top contributors are not much older than us is really motivating. It shows that we could come up with new results as well,” remarked a participant.

During the two weeks at IHES, participants not only had the opportunity to enjoy lectures from leading researchers in the field, but also to meet one of the field’s originators, Xiao-Gang Wen, who introduced topological order in the 1980s and was present at the school as a member of the scientific committee. “I really enjoyed the mix of intense lectures and exercise classes with more informal sessions, such as the panel discussion with some of the organizers and lecturers. The summer school was a very welcome opportunity to catch up with my research community in the wonderful setting of IHES,” said another student.

The organizers wish to thank the sponsors, ENGIE, FMJH, QRT, and Société Générale, as well as the entire IHES staff for their work and dedication in making this summer school a success.

Photo credit: © Chris Peus / IHES

I was only 20 when I first encountered Jim Simons’ work. It was in February 2001, in the freezing cold of Chernogolovka, a small Soviet scientific town. I was in the middle of my senior year at the Moscow Institute of Physics and Technology, and I had taken a long, two-hour bus ride to Chernogolovka to attempt to be admitted to the quantum field theory section of the famous Landau Minimum Series of Theoretical Physics exams, established by Lev Landau himself in 1933. The tradition of these notoriously difficult exams, which test the mathematical skills of graduate students in physics at the cutting-edge of theoretical physics research, was carried on first by Landau, then by his students and their students ever since.

The examiner, Igor Polybin, wrote “$S = \frac{k}{4\pi} \int$ tr $(AdA + \frac{2}{3} A^3)$ in $3d$” on a small sheet of paper and asked me if I recognized the Lagrangian. While my coursework had covered quantum electrodynamics and quantum Yang Mills theory in 3+1-dimensional space-time, I couldn’t identify this one. “No“, I said. “Well, it’s called Chern-Simons theory,” he explained, “and this Lagrangian makes sense in 2+1-dimensional space-time—at least classically. Could you check if this Lagrangian also makes sense in quantum theory, and in particular, what happens to the coupling constant $k$ if you compute one-loop quantum effects?” I understood the question and began my calculations. I quickly encountered subtle “infinity minus infinity” quantum field theory Feynman diagram-style cancellations that seemed to produce a reliable finite result in this theory no matter how I approached them. While the Lagrangian described a continuous, soft, and wavy theory, the computations in Chern-Simons theory had a discrete, deterministic, rigorous, and well-defined feel.

I solved the problem in Chern-Simons theory and passed my student exam, unaware at the time of the depth of this theory and its connections to everything from pure mathematics (algebraic and differential topology, knot theory) to theoretical physics (one could view string theory as a generalization of Chern-Simons theory), and quantum computing. I didn’t realize then how much my life would be influenced by Jim.

I also didn’t know that two years later, in 2003, I would move to Princeton to pursue a PhD in theoretical physics with Edward Witten, the inventor of Chern-Simons topological quantum field theory. Witten based his work on the Chern-Simons three-form and named it after Chern and Simons. After two years in Princeton, I wrote The Hitchin Functionals and the Topological B-Model at One Loop, which is set in 5+1-dimensional space-time but is heavily influenced by ideas on the quantization of 2+1-dimensional space-time Chern-Simons theory.

In August 2005, I attended the Simons Summer Workshop in Stony Brook, Long Island, NY. This month-long gathering brought together a crowd of junior and senior string theorists working on various geometrical problems arising from string theory. I presented my paper in a seminar, and a few days later, I met Jim at the large social event that he hosted every summer for the Simons Summer Workshop participants at his property in the woods near Stony Brook. I was well aware of the depth and importance of Jim’s prolific contributions to mathematics, but I knew very little about his other work. In 2008, I learned that a new institute, the Simons Center for Geometry and Physics, had been founded in Stony Brook. In 2011, I even received an offer for a three-year position there but decided to stay at the Institute for Advanced Study in Princeton. I now realized that Jim contributed significantly to many areas beyond pure mathematics. At the time, I had heard about Renaissance Technologies but didn’t know much about it, except that its profits contributed to the Simons Foundation, which, with its multi-million-dollar funding, generously supports science and mathematics.

In 2014, I accepted a permanent professorship at IHES and moved to Bures-sur-Yvette, in France, where I continued my work in mathematical physics and string theory. My research focused on understanding the emergence of precise and rigid, discrete-like structures supported by the infinite-dimensional continuous realm of quantum field theories, much like the way Chern-Simons theory connects continuous path integration and discrete topological knot invariants.

While my work was primarily funded by the European Research Council, IHES also received significant donations from the Simons Foundation, supporting visitor programs, conferences, and large grant collaborations. This funding was essential to the rapid pace of scientific discovery at IHES, and I was able to enjoy a perfect research environment thanks to the support of public funds, taxpayers, and private donors such as Jim and Marilyn.

In 2022, I finally joined Renaissance Technologies and moved to Stony Brook with the goal of helping to channel market inefficiencies into funding for math and science. In doing so, I aimed to form a new topological link with Jim Simons’ path and align with his legacy and farsighted vision.

Jim will not only be remembered for his extraordinary beautiful and highly impactful scientific work. Even more impressive was his ability to foster thriving work environments for others—whether it be at the Stony Brook Department of Mathematics, the Simons Center for Geometry and Physics, the Simons Collaborations at IHES, or at Renaissance Technologies. Jim had a unique gift for transforming his additive personal talent into a multiplicative one, i.e. a force that multiplies the strengths of everyone and everything around him. I feel incredibly grateful and fortunate that my own path was touched by his magic as well.

Vasily Pestun
New York, August 2024