In Memory of Igor Krichever

In Memory of Igor Krichever Alexander Braverman, Pavel Etingof, Andrei Okounkov, Duong Phong, and Paul Wiegmann The Work of Igor Krichever Professor Igor M. Krichever of the Mathematics Department of Columbia University passed away in New York City on December 1, 2022. He was born on October 8, 1950 in Kuybyshev in the former Soviet Union. He graduated in 1972 from the Department of Mechanics and Mathematics (MechMat) of the Moscow State University, under the direction of Professor Sergei P. Novikov. Throughout his career, he held research positions at the Krzhizhanovsky Energy Institute, the Institute for Problems in Mechanics, and the Laudau Institute for Theoretical Physics. In 1992, he became a professor at the Independent University of Moscow and later visited Columbia University as the Eilenberg Chair of Mathematics in 1996, becoming a permanent faculty member in 1997. He significantly contributed to the development of the Columbia Mathematics Department and served as its chair from 2008 to 2011. He also taught at the Higher School of Economics in Moscow and served as deputy director of the Institute for Problems of Information Transmission of the Russian Academy of Sciences. In 2016, he founded the Alexander Braverman is a professor of mathematics at the University of Toronto and the Perimeter Institute. His email address is sashabraverman@gmail .com. Pavel Etingof is a professor of mathematics at the Massachusetts Institute of Technology. His email address is [email protected]. Andrei Okounkov is a professor of mathematics at Columbia University. His email address is [email protected]. Duong Phong is a professor of mathematics at Columbia University. His email address is [email protected]. Paul Wiegmann is a professor of physics at the University of Chicago. His email address is [email protected]. Communicated by Notices Associate Editor Daniela De Silva. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2909 APRIL 2024 Center for Advanced Studies at Skoltech, which now bears his name. The central theme in Igor Krichever’s research was the theory of solitons, where he made groundbreaking contributions that shed new light on a wide range of topics in mathematics and physics, notably algebraic geometry, quantum integrable models, statistical physics, condensed matter theory, and string theories. He received numerous awards throughout his career, including the Prize of the Moscow Mathematical Society, and was invited to speak at several International Congresses of Mathematicians, including as a plenary one-hour speaker at the 2022 ICM. Columbia University held week-long conferences in his honor in 2011 and 2022. Krichever’s scientific legacy is profound, extensive, and diverse. It is nearly impossible to encapsulate the full breadth of his contributions within the format of a memorial article. Nonetheless, we can highlight some of his most influential achievements: - Systematic construction of algebro-geometric solutions of integrable models, notably the KadomtsevPetviashvili (KP) equation, based on the concept of BakerAkhiezer function. - Development of what is now known as the KricheverNovikov algebra, which extends the Virasoro algebra to a Riemann surface with two marked points (punctures). - Introduction of a Lax pair for the elliptic CalogeroMoser system, linking the dynamics of its poles with the KdV equation, which later played a crucial role in SeibergWitten theory. - Development of the Whitham semiclassical approach to nonlinear waves in soliton theory. - Integrable structure of Laplacian growth. - Construction of a universal symplectic form for soliton equations based on the Lax pair, leading to a new Hamiltonian theory of solitons applicable to 2D equations. - Proof of Welters’s conjecture of the early 1980s that an indecomposable principally polarized abelian variety is NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 483 the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety. - Characterization of Prym varieties as polarized abelian varieties with Kummer varieties admitting a pair of symmetric quadrisecants. - Complete solution of Peierls instability leading to the formation of charge density waves, and analysis of finiteband structure in electronic crystals. The above is only a small sampling of Igor’s scientific legacy. More is detailed in the individual recollections below, offering different perspectives on the aforementioned works. Igor Krichever was truly a great mathematician. Clearly, mathematics in general, and the theory of integrable models with its applications to algebraic geometry and theoretical physics in particular, owe him a lot. However, those of us who had the privilege of knowing him personally or working with him owe even more. He provided us with a model of honesty, kindness, and generosity, and demonstrated equanimity and fortitude in the most trying circumstances. For this, we shall always be most grateful. Memories Enrico Arbarello There was something of a Chagall’s quality in the watercolor that Tanya painted during her first visit to Rome with her parents, Igor and Natasha, over thirty-five years ago. She was very young, and I remember admiring not just her talent, but the subtle layering of culture, over generations, that had brought her to that point. My mother was still alive then, and this intangible cultural kinship flowed in all of our conversations, across generations and across wildly different experiences. It was as if my mother had always known Igor and Natasha: they shared a deep humanity and sense of humor that knows no limits of age or language. I had a similar experience some twenty years earlier, when, as a young man just landed in New York, I found a safe haven in the home of Mary and Lipa Bers, whose friendship and warmth melted away any feeling of unfamiliarity as soon as I stepped through their door. In the early 1970s, years before Igor’s first visit to Rome, I attended a very crowded lecture by Sergei Novikov at the CNR (Consiglio Nazionale delle Ricerche). It was something completely new to me. He talked about KdV equations and hyperelliptic curves. I did not get much Enrico Arbarello is a professor of mathematics at the Sapienza University of Rome. His email address is [email protected]. 484 out of that lecture, other than the determination to study those beautiful relations from an algebro-geometric point of view. When Igor arrived, he talked indiscriminately with geometers and physicists, like my friend Francesco Calogero, and this presented the opportunity to deepen my understanding of those relations. I recognized his versatility and openness as one of the wonderful traits of the Russian school, which I have observed throughout the years in meeting all the Russian mathematicians passing through Rome. In retrospect, one of my deepest regrets is that we, in Italy, could not somehow find a way to hold on to all those great mathematicians leaving the Soviet Union. Igor, who was deeply rooted in Moscow, was not among them at that time, but many others were. We could not find a way to attract them to stay in Italy. And it should have been easy! In fact it was clear that the general disorganization of a country like Italy made them feel quite at ease. The cumbersome bureaucracy, the endless forms to fill out, the myriad of obstacles thrown in the way whenever you try to get anything done, as well as the consequent cleverness in finding ways around all these difficulties, beating the system at its own game: this aspect of Italian life was all too familiar to anyone coming from Russia. So I wish that, when Igor decided to get a position abroad, that the very same bureaucracy had not stood in the way when we should have jumped at offering a desirable job to Igor, whose presence would have influenced Italian mathematics in wonderful ways. But Columbia University won, without even a fight. In the subsequent years, when talking with Igor, there was this constant willingness to understand each other’s points of view: the dynamical system point of view, and the algebro-geometrical point of view. From Igor, and in fact from the Russian school, I learned how to keep both points of view in my mind, and it was instructive to see how the miraculous mathematical procedures that Igor often produced had a neat, but also concealed, algebrogeometrical counterpart. A typical one was his use of the Baker-Akhiezer function. The last time we were in Moscow together was 2018, and it was a truly memorable stay. It was a privilege to enjoy art, theater, music, ballet, and the city in general, with Igor and his friend Irina (who went by Ira). I had the feeling of a very sophisticated city, whose deep culture enriched all aspects of life, and became a wonderful background to our interesting mathematical conversations in the Skoltech Institute, as well as strolling through Moscow’s alleys and avenues, to end up at the lively seminars at HSE. We enjoyed museums with Tanya, and amazing architectural gems with Ira and Igor. Mika, Igor’s grandson, was my guide to ungentrified areas of Moscow, to listen to great jazz and enjoy that vibrant unofficial cultural life that is the NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 hallmark of any great city. That feeling of familiarity and kinship, started so long ago in Rome, has continued, and has been passed on to the next generations. The friendship that tied me to Igor and his family will continue to thrive. Serguei Brazovski I had the good fortune to meet and collaborate with Igor in the 1980s due to the intersection of our interests in the theory of solitonic lattices. For me, this was the problem of spontaneous translational symmetry breaking in a multi-fermion system upon a deformable background (the Peierls-Frohlich model or Gross-Neveu one in field theory), with a special interest in the formation of solitons and their superstructures. Simple cases were found to be solvable by a naive ansatz, and it surprised me that the multi-fermionic self-consistent problem allows for exact solutions. The mathematical theory of S.P. Novikov and coauthors on quasi-periodic solutions of KdV and nonlinear Schrödinger PDEs gave us the opportunity to study physically necessary doubly periodic structures such as overlapping soliton lattices, and envelope and embedded solitons—all this in continuum models. At some point, I realized that the well-known exact solution of the Toda lattice equations gives us for the first time the opportunity to solve multi-fermion problems in a physical discrete system. This perspective resonated with old ideas of I. E. Dzyaloshinskii (my former teacher) about the effect of locking commensurability. I. E., who worked parttime at the MechMat of the Moscow University, shared the idea with Novikov, who suggested Igor Krichever to help us. “Help” turned out to be a vast but lightning work by Igor, which completely subjugated “simple physicists” by introducing incredible elegance in operating with Riemann surfaces, recovering from them all the distributions we needed in real space. For me, a representative of the generic solid state theory, this transcendental view of hyperelliptic functions seemed like some kind of miracle (as well as the whole theory laid down by Novikov). After the first publications treating the most pressing physical questions, I left the team—not wanting to be a coauthor in works whose technique I could not even reproduce. More versatile and mathematically oriented, I. E. continued to work with Igor in the more difficult extensions beyond exact solutions. This scientific path went through circumstances which were not entirely favorable for Igor Moiseevich Krichever. Such a Jewish patronymic, and especially the infamous Serguei Brazovski is a research director emeritus at the Laboratoire de Physique Théorique et des Modèles Statistiques - LPTMS, CNRS, Orsay, France. His email address is [email protected]. “line five” in the passport1 could not but spoil life in the Soviet Union. Igor was lucky to be young enough to enter the university in the somewhat tolerant 1960s. But the events of 1967 (the Six-Day War) and 1968 (Prague) dramatically changed the climate, putting an end to the remnants of the Khrushchev Thaw. In the early 1970s, despite his already demonstrated talents, Krichever had no chance of staying at Moscow University, or even at a less prestigious university, or at any of the numerous academic institutions of Moscow. Eventually a position for him was found at the industrial Energy Institute, where there was no demand for fundamental science and even less for higher mathematics. According to my recollections, Igor worked there (for 13 years!) practically as a system administrator with a big computer (he wrote computational algorithms and even drivers for printers). It was necessary to have great mental stability, versatility, and speed in order to keep in shape under these conditions, and even more so to keep working at the cutting edge of modern mathematics. This resilience, inner strength, and complete self-control were clearly visible in everything that Igor did. These characteristics were wonderfully combined with seasoned humor, certain skepticism, and sometimes evasiveness, and with the high culture of the Moscow intelligentsia. Returning to history, Novikov’s recommendation of Krichever had a subtext: the prospect of Igor’s employment at the Landau Institute (the famous Novikov and Sinai already worked there part-time). Unfortunately, nothing came out of it at that time—we faced a double opposition. Firstly, the director of the institute was under enormous pressure for exceeding the (unofficial, but enforced by authorities) ethnic quota—the number of Jews was already several times higher than any other academic institution could afford. Secondly, among the “old guard” there was an opposition to the deviation from the original purpose of the condensed matter theory, and even more so to the shift towards mathematics. All the obstacles fell by 1990 thanks to perestroika, the doors to academic institutions were opened for Igor, and he even finally became a member of the Landau Institute for several years. Igor did not hold a grudge for the injustice (e.g., even when he was in the US, he was one of the few who invariably came to the annual meetings of the “Landau days”). Then followed an enchanting series of duties in three different new prestigious institutes in Moscow, when not only his scientific, but also his organizational talents were discovered and in demand. And, even more importantly, the human qualities of Igor Krichever were appreciated. 1 In the Soviet Union, line five in the internal passport was ethnicity. APRIL 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 485 Alexander Braverman I knew the name Igor Krichever since my early years in graduate school. But then in 1999, I also became his sonin-law. Both fortunately and unfortunately my interaction with him throughout the years was concentrated around family matters much more than around mathematical discussions. So, I will try to gather here some (rather erratic) thoughts about Igor’s mathematical life, but I will be writing it more in the capacity of a family member than that of a mathematician. A long time ago, V. I. Arnold said that mathematicians could be divided into two categories: those who on their arrival to a new untouched land try to climb the highest mountain and those who start by building roads. According to Arnold, the most obvious examples of mathematicians of the two kinds were Kolmogorov and Gelfand (he later wrote that neither of them were happy with this metaphor). In my opinion, Igor Krichever did not fall into either category: he definitely was not just a road constructor (he always cared about the result, not just about the process), but he was not a climber either. In his nonmathematical life he was a semi-professional ping-pong player, and it seems to me that his approach to mathematics was partly governed by that. His philosophy of pingpong was, roughly speaking, as follows: you must train a lot to bring your technical abilities to a very high level, and after that is done you need to invent your own unique shot—this is what will eventually allow you to win important games. I guess, in mathematics his approach was similar: after getting a very broad education during the “golden years” of the Moscow MechMat, and after acquiring a really remarkable level of technical ability2 he did design his own unique shot: it was the bridge between integrable PDE’s and geometry of algebraic curves, which started with the famous “Krichever construction” of finite-zone solutions of the KdV hierarchy and then got extended to a huge variety of other situations. As with ping-pong, although the shot itself was public knowledge, nobody could master it at the level of its original designer—mostly because of a lot of small things that had to accompany “the shot” (for example, I still remember the days when, going to an important game, he spent a lot of time gluing the rubber to the racket blade; according to him, the result of the game very much depended for example on the type of glue used). 2 For example, his computational skills never ceased to amaze me—essentially until his last day; a couple of weeks before his death he complained that a certain mathematical problem would probably remain unsolved forever: “Nobody except me can get through this calculation, and I can’t do it because I can’t use a pen anymore.” 486 It is interesting to note that although for many years this “shot” was applied in one particular direction (using algebraic geometry to construct solutions to integrable PDE’s and their deformations) one of his more recent big cycles of papers did exactly the opposite: Igor was able Figure 1. Igor with his to prove the so-called Welgrandson Mika. ters’s conjecture (which provided a very geometric characterization of Jacobians of curves among all principally polarized abelian varieties) using the theory of integrable equations.3 Igor was very proud of this work—he used to say that finally the theory of integrable systems paid back to algebraic geometry for all the good it received from it during the previous 40 years. Igor was definitely a mathematician and not a physicist, but physics always attracted him; I think he regretted that he did not spend more time working with physicists. For example, he always talked about the time of his collaboration with Dzyaloshinskii and others on the so-called Peierls models (in the beginning of 80’s) as one of the happiest periods of his scientific life. Unlike so many physicists and mathematicians, Igor had absolutely no element of paranoia about having his results stolen from him. He always told everyone everything—including everything unpublished. It seems that he understood well that no one could steal really big results from him anyway—since no one except him would be able to figure out all the details (continuing with the ping-pong analogy—what kind of glue you need to apply on the rubber to fight today’s opponent), and “small” things did not bother him so much. In fact, it is completely unclear when Igor was doing mathematics (e.g. in the last 23 years he was babysitting one of his grandchildren for about 2/3 of the time). He was very fond of his office at Columbia, but he was not an “armchair scientist.” He spent time in the office mainly in order to write something down, but he knew how to think “on the go.” Even on the day of his death, he recalled how many years ago he was walking along Leninsky Prospekt in Moscow and suddenly he came up with a very interesting idea (“the Lax matrix for the elliptic 3 It is worth emphasizing that unlike, say, the famous Novikov conjecture proved by Shiota, where the sought-for characterization of Jacobians is built into the formulation, Welters’s conjecture is formulated purely in the language of elementary algebraic geometry—thus it is highly nonobvious that integrable equations have anything to do with it. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 Calogero-Moser system”—nowadays it is a very common thing, but 40 years ago it was new), after which he decided to immediately tell it to his former adviser S. P. Novikov who lived nearby. But it turned out that Novikov had a birthday that day, so instead of talking about the Lax matrix, they got terribly drunk. His status in the mathematical community was of course very high and it was important to him in a good sense. He was very proud to get an invitation to deliver a plenary talk at ICM-2022. He viewed this invitation as an opportunity to summarize his mathematical legacy and to try to explain it to a wide mathematical audience. At the same time he did remember some of his “failures:” for example it always amused me that even many decades later he was still a little disturbed by the fact that during the International Mathematical Olympiad in 1967 he won only a silver medal, rather than a gold medal. Igor always had a hyperdeveloped sense of duty, so when it was his turn to be the chair of the Mathematics Department in Columbia, he did not refuse, but initially perceived this position as being sentenced to something between hard labor and scaffold. But in the end he unexpectedly got a taste for administrative work—apparently because he realized that he actually had the ability to do good things in such a post. So in a few more years, when he was invited to create a mathematical center at Skoltech in Moscow, he plunged into this job very deeply (and into Skoltech itself too—for example, for several years he headed the promotion and tenure committee of the entire Skoltech). The center, it seems to me, was a phenomenally successful project, although realizing its original idea (to be a lively and very international mathematical center) is certainly impossible in today’s Russia, definitely not after Russia launched the aggression against Ukraine; and it was very sad to see Igor, who was already quite ill, witness the destruction of many of his efforts. After Igor passed away, the center was given his name—it is now called The Igor Krichever Center for Advanced Studies, and I truly hope that some day it will regain the international character it had before the war. One last thing: Igor, of course, was not in any sense an applied mathematician, but at least in recent years some applied questions attracted him very much. For example, he was fascinated by the area of computational geometry, in particular, questions like how to approximate complicated surfaces by small flat pieces (it turns out that integrable systems also arise there; it is also not a coincidence that he was very fond of the works of Zaha Hadid—she was his favorite architect during the last years of his life). I am sure that he would have managed to do something interesting in this area if his illness hadn’t interfered. APRIL 2024 Vladimir Drinfeld Igor Krichever’s work on commutative rings of differential operators was motivated by the theory of integrable systems. Unexpectedly, it greatly influenced the development of the Langlands program for function fields of characteristic 𝑝. To explain how this happened, let me first recall two classical algebraic constructions. If 𝐵 is a commutative ring equipped with a derivation 𝑓 ↦ 𝑓′ then one can form the ring of differential operators 𝐵[𝐷]; this is the associative algebra generated by 𝐵 and an element 𝐷 with the defining relations 𝐷𝑓 = 𝑓𝐷 + 𝑓′ for 𝑓 ∈ 𝐵. On the other hand, if 𝑝 is a prime and 𝐶 is a commutative 𝔽𝑝 -algebra then one can form the associative algebra 𝐶{𝜏} generated by 𝐶 and an element 𝜏 with the defining relations 𝜏𝑐 = 𝑐𝑝 𝜏 for 𝑐 ∈ 𝐶. The two constructions are somewhat parallel. In a 1976 article, Krichever described commutative subrings of 𝐵[𝐷] in terms of vector bundles on smooth projective curves. Around the same time I was studying so-called elliptic modules, which are commutative subrings of 𝐶{𝜏}; I used elliptic modules to prove a particular case of the Langlands conjecture for global fields of characteristic 𝑝. It turned out that a variant of Krichever’s theory about the relation between vector bundles and commutative subrings of 𝐵[𝐷] has an analog for commutative subrings of 𝐶{𝜏} (this was realized independently by D. Mumford and me). This led to the notion of a shtuka, which is a generalization of the notion of an elliptic module (more details can be found in an expository article by D. Goss in the Notices of the AMS 2003, vol. 50, no. 1, pp. 36–37). Shtukas were then used by L. Lafforgue and V. Lafforgue to prove the Langlands conjecture for global fields of characteristic 𝑝. Anton Dzhamay Igor Krichever came to Columbia University in Fall 1997. At the time, I was a graduate student there interested in applications of gauge theory to geometry through the theory of Donaldson and Gromov–Witten invariants. I had already been very intrigued by the appearance of the 𝜏function of the KdV hierarchy in the statement of Witten’s conjecture, so I was very eager to take Igor’s course on soliton equations. In that course, Igor explained his unique and beautiful approach to integrable systems through algebraic geometry and the notion of the Baker-Akhiezer function. It took me many years to truly appreciate the depth Vladimir Drinfeld is a professor of mathematics at the University of Chicago. His email address is [email protected]. Anton Dzhamay is a professor of mathematics at the University of Northern Colorado. His email address is [email protected]. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 487 and significance of Igor’s ideas. Over the years Igor became my PhD advisor, colleague, and friend, but above all I consider Igor to be my mentor and a true role model. have been very fortunate for the opportunity to know him for so many years. His passing leaves a huge void that cannot be filled, and I miss him deeply. Pavel Etingof Figure 2. Igor Krichever with some of his students (from the left: Dmitry Zakharov, Yury Volvovskiy, Fedor Soloviev, Igor Krichever, and Anton Dzhamay). When I think of Igor, one of the first things that comes to mind is his ability to stay calmly positive even in the most difficult circumstances. I cannot recall Igor ever raising his voice or losing his temper. But with this quiet determination Igor was able to achieve a lot. For three years, Igor was a very effective and respected Chair of the Department of Mathematics at Columbia University. He was also instrumental in the creation of the Center of Advanced Studies at Skoltech and it is very appropriate that this center is now named after Igor. I hope that the Center will survive the present difficult times and Igor’s legacy will continue. In addition to his enormous mathematical legacy, Igor left us many deep and unfinished ideas that, unfortunately, may be difficult to develop completely without his deep and very original insight and technical skills. Still, I am sure that efforts spent in understanding Igor’s mathematics will be very fruitful and I hope that the new generation of mathematicians will continue developing these ideas. When I think about Igor, I always think about Natasha, Tanya, and the rest of his beautiful family. For Igor, the family was very important, he was a very dedicated father and grandfather. It is unbelievable how Igor managed to take care of his grandkids, perform a manifold of administrative duties, and at the same time continue doing new and original mathematics. Igor Krichever was a true pillar of strength to his family, friends, colleagues, and many students. Igor’s support made a huge impact on my life during some of its most challenging moments, and for that I am eternally grateful. Igor Krichever was an outstanding mathematician and a great human being. People like him are very rare and I 488 I met Igor in 1988 when he started working at the Institute for Problems in Mechanics in Moscow. At that time, I was a fifth-year undergraduate working there on my master thesis advised by V. M. Entov. My topic was Laplacian growth with zero surface tension. This problem exhibits an infinite family of integrals of motion which allows one to construct many explicit solutions. Thus my adviser and I wondered how it is related to soliton theory, in which similar phenomena arise. So I was very excited and slightly intimidated to talk to Igor, who was already a world-famous mathematician and a leading expert in the subject. But although I was young, Igor listened very seriously to my story, read my texts and gave me great advice which helped me a lot in future research, in particular when I wrote my first book on this subject with A. Varchenko. He also thought deeply about this topic and later wrote (with P. Wiegmann, A. Zabrodin, and others) a series of papers connecting Laplacian growth to mainstream soliton theory. These works, among others, have made this once fairly specialized subject into a vibrant area deeply connected to integrable systems, random matrices, stochastic processes, quantum field theory, etc. During this time I recall coming to Igor’s apartment, where I met his 13-year-old daughter Tanya. She is now a professor of literature and a dear friend of mine, along with her wonderful family—her husband and my coauthor Sasha Braverman and their children Mika, Asya, and Natasha. In 1999, Igor called me and asked if I’d like to join the Columbia Mathematics Department. This was my first tenured appointment, and I spent a great year there with Igor and his colleagues, learning a lot from him about mathematics and beyond. One of my last major interactions with Igor, which I enjoyed a lot, was in the summer of 2019 when he invited me to teach a minicourse at the first International Summer School at the Center for Advanced Studies in Skoltech, which he had founded three years before. Igor had nothing but profound contempt for Putin’s regime, as he did for the Soviet regime decades earlier, but he believed that fundamental science and international collaboration should be fostered even in adverse political climates, and he hoped to hold such a school every summer. Sadly, this was not to be—the following year covid arrived, and in two more years Russia unleashed a devastating full-scale war against Ukraine, which in particular wiped out much NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 of what Igor had built. Yet the seeds he sowed are bearing fruit—the students raised by him at the Center are now doing mathematics in many different parts of the world. The day before his death, I wrote Igor a short message of gratitude, and he responded “Thank you. You have always been dear to me.” These were his last words for me; in a few hours he was no more. As I remember these words, I think how fortunate I am to have met Igor. Throughout my whole mathematical life, I have been blessed to learn from his wisdom and enjoy a warm professional and personal relationship with him. I will always admire him as one of my teachers and a wonderful human being. I miss him greatly. Samuel Grushevsky I first met Igor Krichever in 2003, more recently than many other contributors to this article. I was then a fresh PhD— at the beginning of my life as an independent researcher. Some time later, we were discussing mathematics regularly, and soon thereafter we collaborated on our first joint paper. Our collaboration then continued until Igor’s untimely death, with further projects unfinished. My path in mathematics was completely altered by working with Igor. Beyond sharing his mathematical expertise and intuition freely, Igor was a very close friend, and knowing him made me a better human being. His departure from this world is an unhealing wound for me, while his memory will continue to provide mathematical and personal inspiration. For my PhD dissertation I studied the Schottky problem—the question of characterizing Jacobians of curves among all abelian varieties. This is a field that was transformed by Igor’s breakthrough 1970s construction of solutions of integrable hierarchies (Functional Analysis and Its Applications, 1977), using the theta functions of Jacobians.4 This further led to Novikov’s conjecture proven by Shiota in his 1986 paper, showing that the Schottky problem is solved by the KP equation: if the theta function of a principally polarized abelian variety satisfies the KP equation, then it is a Jacobian of a curve. In 2003, I finished my first post-PhD project arXiv:0310085, and being aware of Igor’s stature in the field, I shared the preprint with him, hoping he might find something appealing in our computations with derivatives of theta functions. In response Igor pointed me to the conjecture on addition theorems for theta functions from his paper with Buchstaber. I was eventually able to resolve this conjecture, in arXiv:0503026. By chance I finished the argument during a visit with Riccardo Salvati Manni in Rome, at the Samuel Grushevsky is a professor of mathematics at the University of New York at Stony Brook. His email address is [email protected]. 4 see also his 1977 paper in the Russian Mathematical Surveys. APRIL 2024 same time that Igor was visiting Enrico Arbarello there. Thus I was able to explain the argument to Igor in person, and had my first ever dinner with him (and our Italian colleagues). While I expected that a visiting mathematician would spend all his days at the math department, Igor was instead very diligent in visiting the outstanding art museums (then not as crowded as now), while he was still able to share much mathematical insight over coffees and dinners. From then on Igor and I became friends and collaborators. One thing that always amazed me in working with Igor was how playful his mathematics was. While sometimes I would come to our next meeting (which occurred regularly for years) to discover pages upon pages covered by computations, most frequently I would observe how Igor would come up with an idea just in passing, while walking, doing the dishes, smoking, or even between the acts of an opera. A lot of hard work and difficult computations went into our eventual first joint paper arXiv:0705.2829 (and Igor’s computational prowess was clearly manifest), but many of the basic ideas and concepts arose just casually, and occurred to Igor naturally as he was exploring the circle of ideas around his celebrated proof of Welters’s trisecant conjecture, arXiv:0605625. That proof of the statement that if the Kummer image of a principally polarized abelian variety has one trisecant line, the abelian variety is Jacobian, is a tour de force that still has not been understood via methods of algebraic geometry. While Igor’s methods come from integrable systems, and his related statements for degenerate trisecants (arXiv:0504192) have recently found an algebro-geometric explanation (arXiv:2009.14324), the full statement, and Igor’s construction of the family of trisecants and integrable hierarchy, starting with just one trisecant line, remain mysterious. Igor was interested in curves with a differential for decades, partly motivated by numerous problems from physics, e.g., see his work with Phong arXiv:9604199. While much of the work in this area had been on applying algebro-geometric methods to solve questions in mathematical physics, in arXiv:0810.2139 we applied realnormalized differentials (meromorphic differentials on Riemann surfaces with all periods real) to reprove a statement in algebraic geometry: Diaz’s theorem (1984) that a compact complex subvariety of the moduli space of genus 𝑔 curves has dimension at most 𝑔 − 2. Our proof does not require advanced technology, and the result and the proof can essentially be explained to a good undergraduate student. This is another amazing property of Igor’s oeuvre—the sheer number of different novel ideas that he has come up with, and applied broadly in mathematics. While some of NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 489 Igor’s papers utilize advanced machinery and show heavy computations, whenever Igor was asked to explain something, the answer was never “you compute for ten pages and then you see.” All of Igor’s work, even our very technical study with Norton of degenerations of real-normalized differentials arXiv:1703.07806, had a clear underlying idea and philosophy in Igor’s mind, which—when followed through to the end—yielded a result, however difficult or technical at a first glance. Igor’s ability to intuitively see through the computation and divine the final result, and then to relentlessly follow such a path to success, was unsurpassed. In everything Igor did, in mathematics and in life broadly, he exuded the impression of unhurried and benevolent strength. All his numerous accomplishments—personal, mathematical, and administrative, did not appear to come to him through blood, sweat, and tears. Igor loved life, loved mathematics, loved people around him, and enjoyed creating new knowledge and sharing his vision. This vision will continue to shape geometry for years to come, and the memory of Igor will continue to push all those who knew him to become better people. Nikita Nekrasov The hot air outside the Columbia housing block on 113th street was thick. The parking next to the building was being renovated, the asphalt drill making a terrible noise. The building elevator broke down. It was August 9, 2022. I climbed the narrow staircase to the fourth floor, and entered the apartment. The door was unlocked. Igor was waiting for me, dressed in a short-sleeved shirt and a pair of slacks. His always trim figure was now too small even for small-sized clothes. A few days prior, thanks to the eloquent pitch of my girlfriend Nina, Igor agreed to have a recorded conversation. We assembled a film crew, carried a ton of equipment up those stairs, and set up the cameras and sound equipment. Igor’s daughter Tanya and her son Mika were helping to get the crew some air, my son Boris helped the sound operator. Family, friends, and mathematics were inextricably mixed up, it was always like that with Igor, both last year, and thirty three years ago. . . 𝐍. I think I saw you for the first time at your lecture at the Moscow Mathematical Society. Topological gravity and KdV equations were gaining popularity, and you were presenting something on the topic. I remember Sergei Novikov was sitting in the front row and saying something Nikita Nekrasov is a professor of physics at the Simons Center of Geometry and Physics, Stony Brook, New York. His email address is nikitastring@gmail .com. 490 like “there is something funny with your limit of small 𝜀. . . ” 𝐈. I think it was my review of the 1990 ICM. Those who attended the Congress were to review the papers at the MMS. 𝐍. But I think you were lecturing about your own work... 𝐈. Perhaps it was when I got interested in Whitham dispersionless hierarchies, partly because of the genus zero topological gravity of Witten, I don’t know. . . −−−−−−−−−−−−−−− 𝐍. Could you tell us how you came to be a mathematician? When did your interest in mathematics start? Do you recall? 𝐈. I don’t remember when I wasn’t interested. 𝐍. You don’t remember when you weren’t interested? 𝐈. I’ve always been interested in solving problems. 𝐍. And where did you get the problems from? 𝐈. There were problem books that I read. But it probably wouldn’t have worked out if I hadn’t come across a good math teacher in the city of Taganrog. This wasn’t your 57th school, yet she was a great teacher. After only a couple of classes she told me: “Sit quietly in class, do what you want, don’t participate, solve whatever problem you want—it’s your business,” and then set me up with one of her former students, who had gone away to the Kolmogorov Internat (a special science oriented boarding school in Moscow) the prior year. 𝐍. Did your parents just let you go, or what was it like? 𝐈. I still don’t understand why my mother let me go. Because I was very. . . 𝐍. Timid child? 𝐈. Totally timid. . . 𝐍. Did your parents have anything to do with science? 𝐈. No, they didn’t. 𝐍. They didn’t. How did you find out about the Boarding School? 𝐈. It all happened in the city of Taganrog. There were no special schools, nothing there. The first one, the only, sort of, special, English school was opened, when I was in the fourth grade. My parents, of course, as expected, sent me there. At one point I got an 𝐹 in math for a two-digit number problem. I wrote “10𝑎 + 𝑏” and got an 𝐹 for it. My math teacher had a long argument with me in front of the whole class. She said: “Look, for the number 75 we don’t write 10 ⋅ 7 + 5, so you must write 𝑎𝑏. You see? And from that we got something completely wrong.” When I told my mother about the argument I had with the teacher she pulled me out of that school the same day and transferred lucky me to a nice simple school where Anton Chekhov studied. It was an old grammar school. I can’t say that my NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 teacher was from those Chekhov days, but she was very aware of her limitations. I felt very comfortable with her. She did not make me do “ab” under the guise of double digits. And because one of her students had gone to the Internat the previous year (it was the first or second round), I found out about it. 𝐍. Can you not believe that there is such a thing, a plan, a meaning to the meeting? How can one not be surprised by the weaves of fate? By living through such encounters which define everything. . . . How did you meet S. Novikov and what was your style of communication with him? 𝐈. At the end of my second year at MechMat, I had to choose a division and an advisor. I decided to go to Novikov, despite the endless warnings of many of my friends. They said, “Novikov doesn’t remember his students, he doesn’t know what they look like.” It turned out not to be true. In fact, he remembered everything about everyone, and what’s more, he had a ledger in his head. When I first came to him, he told me: “if you are hoping to get a problem from me, don’t get your hopes up. If I know a good problem. . . ” 𝐍. I will solve it myself! 𝐈. “I will solve it myself.” 𝐍. Yes. 𝐈. again, that’s. . . 𝐍. This a perfectly logical thought! That’s what I myself say to everyone. Where would I get a good problem? I kind of have my own. . . 𝐈. You know, Nikita, it works for some people. It suited me. In all my life I did not get a single problem from Novikov. All our work together occurred when something came up, he never told me “here’s the problem, it needs to be solved.” I myself tried the Figure 3. Igor in August 2022. same practice on several of my students. It turns out it works for some people and not for others. I can’t say anything specific, so to speak, about what Novikov taught me. But he instilled in me a taste for what is good and what is bad. And that is the most valuable thing. 𝐍. Is it ever the case that you understand something, but not because you can build up a logical chain, but because it’s somehow there, yet maybe you can’t keep track of the precise logic. 𝐈. Nikita, that’s quite a complicated question, because I’m deeply a nonreligious person. Yet there is a deeprooted belief inside of me that there is some kind of harmony in the world. APRIL 2024 𝐍. Harmony? 𝐈. Yes, at times I have stubbornly tried to prove some nonsense because I felt that there was one brick missing for beauty and that something must be right because it is beautiful. It’s another belief, I don’t know what you call it, but in general it’s not a chain, it’s a feeling that. . . 𝐍. that there is a pattern, an ornament! 𝐈. . . . the world is somehow harmonious. . . you see it, and it must be that way to be good. . . − − − − − − − − − − − − − − − − −− A few months later, at our last meeting, I wanted to ask him about his feelings. He knew what was ahead of him. He said: “Nikita, this is not a place for words.” And in the same breath: “The Lax flows of the integrable system we are discussing, on the other hand, deserve further discussion.” Today, seven months later, that Columbia apartment is empty. Igor is no longer with us. Yet he is with us, the Lax flows continue, and so does the flow of ideas he weaved so masterfully. Sergei Novikov Igor Krichever started to interact with me in the late 1960s. He was an undergraduate student at that time. In the early 1970s, he became a graduate student and did very good work in topology. He studied actions of compact groups on manifolds, using the whole machinery of algebraic topology—including cobordisms, formal groups, and so on. In 1974, I invited him to work on the theory of solitons and nonlinear waves. In 1975/76 he did very good work constructing algebro-geometric solutions of the KP equation which is a natural 2D extension of the famous KdV equation. Its integrability was established by Driuma-Zakharov-Shabat in 1974 who started to study it. They found the “Lax pair” for it. The method of Krichever was based on the pair of commuting ordinary differential (OD) operators of relatively prime orders. The role of commuting OD operators in the theory of the KP equation became well-known. In 1977–1978 some British mathematicians found forgotten works done in the 1920s (Burchnall-Chaundy) where commuting operators of relatively prime orders were investigated algebraically. Similarity with some modern studies was impressive. One should say that classical people never considered systems of nonlinear PDE, nor periodic or rapidly decreasing potentials in quantum mechanics. Indeed, while in 1940s reflectionless potentials were classified (Bargmann), the algebraic background remained hidden until the 1970s. Sergei Novikov is a Distinguished Professor Emeritus at the University of Maryland College Park. His email address is [email protected]. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 491 Famous periodic and quasiperiodic finite-gap potentials were discovered and classified in the 1970s—Dubrovin, Matveev, Its, and myself. The algebraic background became completely clear after Krichever’s work. Then began the study of 2D problems. I. 2D Schrödinger operators generate 2 + 1-dimensional nonlinear systems which we call “Manakov’s L-A-B triples” instead of 1+1-dimensional Lax pairs. They use eigenfunctions of the Schrödinger operator restricted to one energy level only. This was developed by several authors including Dubrovin, Krichever, Veselov, Grinevich, myself, and others. II. 𝜃-functional solutions of the KP system can be used, according to my conjecture, to recognize 𝜃-functions associated with Riemann surfaces. This is a classical problem known since the nineteenth century. This problem was solved by Dubrovin, Arbarello-De Concini, and completed by Shiota. Krichever improved this approach, replacing it by the operators of Lax pairs. III. The development associated with commuting operators of nonrelatively prime orders is extremely interesting. The classics (Burchnall and Chaundy) worked with pairs of relatively prime order OD operators. Concerning the nonrelatively prime case they wrote: “This problem (commuting operators of nonrelatively prime orders) is trancendental.” Indeed, it is obvious that unlike the relaFigure 4. Igor Krichever. tively prime case, the common eigenfunction cannot be found explicitly in quadratures. But Krichever and I started to work on this in 1978–1979. Drinfeld and Mumford also began working on this problem. It became completely clear that holomorphic vector bundles over Riemann surfaces play a fundamental role. There exist two different methods to study holomorphic vector bundles over algebraic curves—Tyurin’s and Mumford’s. Drinfeld and Mumford used Mumford’s approach and made useful observations. However, their results were noneffective. Krichever and I, on the other hand, used Tyurin’s approach based on “framed” bundles. Choose the Chern class 𝑐1 = ℓ𝑔 (where ℓ is the rank of the vector bundle, 𝑔 the genus of the base). Framing means selection of ℓ holomorphic sections. Avoiding details, let me say that we developed an effective method how to calculate the coefficients of the OD operators. Even for the pairs of orders 492 (4,6) and (6,9) the situation is nontrivial. They depend on arbitrary (free) functional parameters of one variable 𝑥. The whole pair depends on two variables (𝑥, 𝑦). Each commuting pair defines a solution of the KP equation. Time dependence of the KP-equation leads to time dependence only of free functions of one variable entering the operators. It leads sometimes to remarkable (𝑥, 𝑡) systems. The case (4,6) made it possible to correct a mistake in the classification made by the school of Shabat: their list of “integrable” KdV-type systems (i.e. 𝑐𝑡 = 𝑐𝑥𝑥𝑥 + 𝑓(𝑐, 𝑐𝑥 , 𝑐𝑥𝑥 )) was not complete—the most complicated system was missing. In the late 1980s, Krichever did very good work in statistical physics working jointly with some of the best theoretical physicists in the Landau Institute—Dzyaloshinskii and Brazovski. His last work with me was done around the year 2000. It was dedicated to commuting difference operators. Igor was one of my best students and a remarkable mathematician. He left a lasting legacy in mathematics. Andrei Okounkov Igor Krichever was a bright pulse of light. Just the kind of nonlinear natural phenomenon that can be seen on a poster of a conference on integrable systems, his main field of interest. Except, this pulse was the opposite of solitary. Most of the time, he was this warm smiling globe of light that instantly made everybody feel understood, supported, and loved. And in those, sometimes rare, moments when he was free to do his mathematics, all that light could focus like a laser to dissect any mathematical difficulty and illuminate the core issue from within. About the balance of the two, I don’t know if it was a conscious duty or just nature for him, but he always had an unlimited budget of time and energy for helping others. For his family, his many friends, his colleagues, neighbors, et cetera, et cetera, he was just like a bright sun, the source of everyday energy, vitality, and good humor. A source that never ran dry and never failed to be there for them every single day. Until he was suddenly gone. While always pointing towards big goals and ideas, his compass never pointed away from people. He was very passionate about the future of mathematics and, later in his life, he devoted a very large part of himself to administrative work. Everybody knows it is not easy to provide the institutional foundations on which mathematics can flourish: there will be some difficulties every single day, and it may be hard to resist approaching these difficulties with a formal, corporate logic. Igor was that rare kind of leader who always put people first. Somehow, there never was an obstacle or a disagreement that couldn’t be put behind by the combination of his smile and his wisdom. For that, he NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 was always dearly loved by his colleagues, staff, superiors, and maybe most of all, by the mathematical youth. For the young people, he was both an inspiration as a brilliant researcher and a caring senior colleague, whose rock-solid support they could count on in all possible real-life situations. I hope they will carry that feeling and the memory of Igor with them into the future. And “sorry” is not really the word to describe the fact that most of Igor’s dreams were crushed just before the sudden illness overcame him. But nothing can take away the legacy of Igor the original thinker. It is overwhelming to contemplate the brilliance and the fundamental nature of his contributions to mathematics. Many of them are the central pillars of building bridges between different fields, with a lot of ideas traveling in both directions. For me, he personified the fact that mathematical physics can interact and should interact with every branch of pure mathematics. I always loved discussing mathematics with him. In addition to Igor’s universally celebrated landmark papers, I had my own personal favorites among the lesser known ones. While thinking about the monodromy of quantum connections, I spent many months with his “Analytic theory of difference equations … ” by my side. Igor’s work on the spectral theory of 2-dimensional periodic difference equations had a big influence on my work with Richard Kenyon on planar dimers and Harnack curves. Igor’s elliptic genera and their rigidity were a huge inspiration for elliptic stable envelopes and many rigidity-based computations in enumerative 𝐾-theory. My understanding of many other areas of mathematics was really deepened and sharpened in discussions with Igor, which continued until the tragic day came. The closest we came to writing a paper together was when we discussed the limit shapes for planar dimers and their quantum analogs, which I had defined in 2009. ˆ are defined for domains of finite Quantum limit shapes 𝑄 size and the fact that they converge to the classical limit shape as the size of the domain grows to infinity remained a conjecture at the time. Eric Rains and I constructed ˆ changes with Painlevé-like equations that describe how 𝑄 the domain. With Igor, we proved an averaging result for these dynamics. We both liked it. It involved realnormalized differentials, 2-dimensional quasiperiodic difference operators, and Igor’s other favorite objects. It implied the convergence to the classical limit shape. Still, other projects kept us from completing this one. As I look at the slides of my 2010 lectures about this work with Igor, I feel devastated by the size of the dark void that is left by his departure in mathematics, my life, and many, many other people’s lives. APRIL 2024 Duong Phong Igor Krichever was for me a very dear friend and colleague. I often marvel about how mathematics can bring together people whose paths were most unlikely to meet. The first time I heard about Igor was in 1976, from a beautiful description of his work by David Mumford at a colloquium at the University of Chicago. My admiration for Igor’s work only increased in subsequent years, when I participated in the year-long seminar held by Lipman Bers at Columbia on integrable systems and Riemann surfaces. Even so, Igor was a world away, and I did not even dream that we could be colleagues and friends some day. Things changed drastically in the early 1990s. Many outstanding scientists from the former Soviet Union had begun emigrating to the West, and the Columbia mathematics department had lost, through some unfortunate circumstances, many faculty who had to be replaced. But with the economic crisis of 1992, many other Columbia departments were facing difficulties of their own, and the new Columbia administration at that time installed a new policy, where the renewal of faculty slots would not be automatic, but would have to be won by each department on its own merit. This meant that the mathematics department had to propose a truly world-class candidate. So it was with great anticipation that it learnt that Igor could perhaps be interested, and the main task became that of building a strong enough case for the Columbia administration to give priority to his appointment. There I am very happy to report that his letters far exceeded all expectations, and I still recall vividly many enthusiastic comments, including that a particular contribution of his was “an epoch-making work.” Thus Igor did join Columbia, where he became a key figure in its renewal, serving as chair in the early 2010s, until his untimely passing away in 2022. It was a great stroke of luck for me that, at the time of Igor’s arrival at Columbia, our scientific interests happened to overlap. Nathan Seiberg and Edward Witten had just made their great breakthrough on supersymmetric gauge theories with spectacular applications to topology, and the central role of symplectic forms and integrable models had begun to emerge with works of Ron Donagi and Witten, and Emil Martinec. At the same time, Igor had just completed his work on the Whitham hierarchy, and had also recognized, in joint work with Alexei Morozov and others, the Seiberg-Witten solution of the SU(2) theory as the spectral curve for the Toda model. Eric D’Hoker and I had been investigating instanton corrections. Motivated by the theory of integrable models, Igor and I decided to focus on the construction of moduli spaces of pairs of differentials and their symplectic forms. We succeeded in this goal, and obtained a unified approach to NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 493 all the Seiberg-Witten solutions known at that time in the literature. But rather unexpectedly, this work ultimately led to something that we had not even hoped for, namely a universal symplectic form 𝜔 expressible in terms of Lax pairs (𝐿, 𝐴), 𝜔 = Res∞ ⟨Ψ∗ 𝛿𝐿 ∧ 𝛿Ψ⟩ 𝑑𝑘, and a new approach to hierarchies of 2D integrable models. Here Ψ and Ψ∗ denote the Bloch and dual Bloch functions for 𝐿, and Res∞ denotes taking the residue at ∞. In particular, it is very different from the Hamiltonian approach pioneered by Ludwig Faddeev and Leon Takhtajan, and has perhaps the advantage over the approach of Mikio Sato of not involving the infinite number of coefficients of a pseudo-differential operator. But Igor was not just a colleague with whom I arrived at some of my most cherished works. We became the most mutually trusting of friends, and our families grew to be very close as well. Igor was very generous and thoughtful, and I cannot count all the times when an unexpected gift or kind gesture of his would show how much attention he paid to the smallest wishes of his friends: the book with the painting of “La Princesse Lointaine” on the Hotel Metropole in Moscow, which he brought back for me when he heard how much I liked that piece of theatre; the CD’s of Boulat Okoudjava; and the Georgian movies and DVD’s of dances from the Caucasus. Equally vivid are the souvenirs of times when I could just drop by his apartment, and be welcomed by his wife Natasha with a warm and aromatic mushroom soup. It is terribly sad to think that there will be no more such occasions, but I can take solace in at least having experienced them with a precious friend such as Igor. work, which I always admired (in my dinner speech at his 60th birthday conference at Columbia in 2011 I said that we shared a “secret love” for the Riemann-Roch theorem). The first Soviet-American Symposium on Solitons in Kiev in the summer of 1979, which Igor and I attended, was a pivotal event for the theory of integrable systems in the Soviet Union and in the USA. I gave a talk on our joint paper with Ludwig Faddeev on the eight-vertex model, solved by Baxter; we successfully applied to this model our recently invented (together with Evgeny Sklyanin) method of the algebraic Bethe Anzatz, which takes full advantage of the Yang-Baxter equation (the term was introduced in our paper). Subsequently, in his 1981 paper, Igor beautifully applied the theory of algebraic correspondences and the Riemann-Roch theorem to the problem of classifying solutions of the Yang-Baxter equation and explained the algebro-geometric meaning of the vacuum vectors in our paper. Leon Takhtajan In 1996, when Igor joined the Mathematics Department at Columbia University and moved with his wife Natasha to New York City, my wife Tanya and I became very close with them. They often visited us Figure 6. Igor giving a talk. on Long Island and we ventured on a few trips to upstate New York (see Figure 5). Our scientific interests became closer; Igor and I had many discussions about different definitions of deformation spaces. Igor preferred the algebro-geometric approach based on his favorite differentials of the second kind with real periods, while I was using the Ahlfors-Bers approach, and we were trying to merge them. Igor and I attended numerous conferences and workshops, including the Summer School on New Principles Igor Krichever was a great friend and his passing came as a terrible loss. My first encounter with Igor was around 1975; he was giving a talk at the Steklov Institute in Leningrad on his (now classical) results on the periodic problem for the Korteweg-de Vries equation. It was the beginning of the Golden Era of numerous interactions between algebraic geometry and mathematical physics, with many discoveries and triumphs. Igor’s paper introducing the Baker-Akhiezer function was one of them. His great gift was a special feeling of perspective in mathematics, distinguishing the foreground and the background, like in art. Using this comparison, Igor belongs to the school of old masters, possibly with some twist of impressionism. Igor realized the great unused potential of the RiemannRoch theorem for curves and masterfully used it in all his Leon Takhtajan is a professor of mathematics at the University of New York at Stony Brook. His email address is [email protected]. 494 Figure 5. Natasha and Igor (and Tanya). Lake Mohonk, New York, 1998. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 in Quantum Field Theory in Cargese in 1991, in Saint Petersburg in 2013, 2014, and 2016, in Ascona in 2015, in Gallipoli in 2017, in Moscow in 2018, and his last birthday conference in New York City in 2022. Igor was a remarkable husband, father, and grandfather. Caring for his family was always his priority even outpacing mathematics. The passing of his wife Natasha in 2013 was a terrible loss after which Igor was even more involved with his daughter Tanya and her family. Igor was an outstanding person with deep moral principles he always followed, courageous and brave, and he always kept his word. He had a great organizational talent, whether as chair of the Mathematics Department at Columbia or as director of the Center of Advanced Figure 7. Igor and L. D. Studies at Skoltech in MosFaddeev. Saint Petersburg, cow. Starting in 2016, 2016. he was able to successfully build a new research center on par with the best mathematics and theoretical physics centers worldwide, now called “The Igor Krichever Center for Advanced Studies.” To summarize, I greatly treasure our friendship, conversations about mathematics and life, and the moments we spend together with Igor and Natasha, and later with his friend Irina. He will continue to live through his mathematics, his family and his friends. The last photograph, taken in 2016, shows Igor with my teacher and friend Ludwig Faddeev. They are seen through a mirror, which forever preserves the moment. Alexander Veselov Alexander Varchenko Igor Krichever was a student at the Moscow boarding high school No. 18 for gifted children organized by Andrei Kolmogorov at Moscow University. The school opened in December 1963. Igor enrolled in the school in 1965, and I enrolled a year earlier. At school, Igor was notable as a winner of mathematical Olympiads. Twenty years later, summarizing the school’s work, a meeting was organized by Kolmogorov with the graduates from the school who became doctors of physical and mathematical sciences. In Russia, there are two scientific degrees: a candidate of sciences degree and a much more exceptional doctor of sciAlexander Varchenko is a professor of mathematics at the University of North Carolina at Chapel Hill. His email address is [email protected]. APRIL 2024 ences degree. By that time, there were only eight doctors of sciences, see Figure 8, which was published in one of the central newspapers of the Soviet Union. During that meeting, we had a chance to chat with Kolmogorov about the recently introduced mathematical Olympiads for undergraduates. I recall that Igor expressed his belief that undergraduates would be better served by engaging in real research rather than simply playing Olympiad games. Igor published his first paper when he was 21. I played table tennis with Igor on different occasions in different countries. In his youth, Igor was on the table tennis team of Moscow State University. Once I asked Igor if he played against J.-P. Serre, who was known as a very good table tennis player among mathematicians. Igor replied that he had Figure 8. First row from left to played against Serre and right: A. Varchenko, A. had actually won. Kolmogorov, V. Temlyakov. I had only one joint paSecond row: Y. Matiyasevich, per with Igor, which was E. Shchepin, I. Krichever, V. Yanchevsky, S. Voronin, S. written in 2019. We conPinchuk. structed a family of commuting flows on the space of solutions of the Bethe ansatz equations in a simplest ˆ 𝑁 XXX model and identified these flows with the flows 𝔰𝔩 of coherent rational Ruijesenaars-Schneider systems. The last time I saw Igor was in October 2022. He said that we need to make the next step in our project, but we did not have time. Who was Igor Krichever to me? Firstly, the elder scientific brother, the referee of my first paper written during my PhD study under the supervision of Sergei P. Novikov. I remember well my visit to Igor’s place at Zhitnaya Ulitsa in Moscow, where he taught me how to write a good paper. I remember also his own PhD defense at Moscow State University, which actually was in the area of algebraic topology. Only recently I had a chance to work with Victor M. Buchstaber on cobordism theory and was able to fully appreciate Igor’s contribution to this area, which was overshadowed by his outstanding achievements in Alexander Veselov is a professor of mathematics at the Loughborough University. His email address is [email protected]. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 495 integrable systems and algebraic geometry. For many years, I was very fortunate to have numerous scientific discussions with Igor, which were always very illuminating and stimulating. Even during our last meeting in New York in October 2022 when I came to say goodbye, Igor used this chance to explain his very revealing understanding of Leon Takhtajan’s talk, which concluded the conference celebrating his remarkable scientific career. Igor was also a close friend with whom I could discuss the most delicate problems of my life. I enjoyed every minute spent with Igor’s wonderful family, especially with his wife Natasha and grandson Mika. I will forever remember Igor Krichever as a very strong and positive person, bringing the sense of optimism to others. He will be sorely missed. Paul Wiegmann With Igor’s untimely passing, I lost a dear and trusted friend with whom I shared my values in science, humanity, and morality. He was three years older, and we belonged to the same generation, experiencing life events from a similar perspective. When I was about twenty years old and starting my diploma work at the Landau Institute, I had a memorable conversation with a friend who chose a career in molecular biology. During that discussion, it occurred to me that besides being drawn to theoretical physics by its general prestige in the Soviet Union, what truly attracted me to this field were the wonderful people who were part of it. Since then, I consciously admit that, for me personally, the social comfort and intellectual closeness that creative work brings among the people around me may be just as valuable, if not more, than the new knowledge that this work creates. This adds to the sense of loss I feel, knowing Igor as a friend and having the privilege of working with him. I knew Igor’s name since my early years at Landau as a prominent student of S. P. Novikov. Back then, Igor was working at some obscure Energy Institute, which I felt was unfair, as I believed that some of us, and primarily, myself, with lesser achievements and promise managed to get into premier institutes like Landau. Later on, I heard from Igor that he was quite content there, enjoying a good degree of academic freedom and facing less peer pressure, albeit at the cost of less prestige. Throughout our roughly four decades of friendship, I never heard Igor complain. He was never driven by ambitions for superficial things like titles or public standing. At the same time he found value in seeing his ideas and results being recognized for their merit. Many people who knew him noticed his consistently positive attitude, and as Andrei Okounkov aptly put it, Igor was a bright pulse of light to many of us. 496 At Landau, S. P. Novikov commanded great respect and admiration. S. P. was known for passionately fostering interaction between physics and pure mathematics (and also for his memorable statements that often carried aphoristic value). When Igor finally joined Landau, the interaction between the two distinct cultures of thinking was a theme of the day, leaving a profound impact on many of us. Igor was a bright pulse not just of light but also of clarity, having the exceptional ability to extract the core essence from abstract complex concepts and deliver it in a straightforward manner. People like myself, who lacked formal mathematical education but regularly bumped into algebraic geometry, owed Igor a great deal for his readiness to remove a mathematical concept from its abstract shell. Igor seemed to derive a sense of pleasure from simplifying an intricate matter to its core. A notable example of such nexus is Igor’s early work of 1982 with Serguei Brazovski and Igor Dzyaloshinskii and his 1983 paper with Dzyaloshinskii on the Peierls model. This model is a case of Peierls instability, an important phenomenon of structural distortions in crystalline materials caused by electronic interaction. After that work, it became a major condensed matter application of “Krichever’s construction” of finite-zone solutions of integrable hierarchies, such as the Toda chain in this case. We grew closer in Spring 1989 when we randomly ran into each other on Blvd. Saint-Michel in Paris. We ended up in the first café that came along, in the midst of all the tourists, discussing the unfolding events in Moscow and the upcoming election of YeltFigure 9. Igor explaining sin the next day. There was mathematics. a sense of euphoria in the air, and a general feeling that our lives were on the brink of a significant change. I remember that Igor’s thoughts were balanced, somewhat subdued. He told me that he had an invitation to stay “longer” in Paris but may travel to Columbia for “reconnaissance.” I mentioned that I was considering going to San Diego and then possibly to Princeton. The next day we went to the embassy somewhere on the edge of the Bois de Boulogne to vote for Yeltsin. It was a momentous occasion for both of us as we had never voted before (and I haven’t voted since). In the mid-1990s, after settling in the US, we became even closer. I met Natasha and Tanya, and we started exchanging visits and having discussions about physics (or mathematics). NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 Our first paper arXiv:9604080, written with Ovidiu Lipan (then a student) and Anton Zabrodin, explored a mysterious connection between the Bethe Ansatz equations for eigenvalues of quantum integrable systems and a special class of solutions (elliptic solutions) of a classical difference Hirota equation. This problem stemmed from discussions with Igor about the Bethe Ansatz solution of the Hofstadter problem, which Anton and I were working on at that time. Hirota’s equation encompasses known integrable hierarchies of classical nonlinear equations when time is treated as a discrete variable. The increment of the discrete time corresponds to the Planck constant of the continuous time quantum equations. In a subsequent paper with Igor and Anton Zabrodin arXiv:9704090, we extended Igor’s algebraic-geometric solution to discrete (or difference) integrable equations. In the late 1990s, Mark Mineev-Weinstein introduced me to the problem of Laplacian growth. This phenomenon, also known as the Hele-Shaw problem in fluid mechanics, has been known to engineers since the mid-19th century. It involves the unstable growth of an interface between viscous and inviscid fluids, resulting in fingering instabilities and the formation of cusplike singularities in finite time. By the mid-twentieth century, the problem had been linked to the deformation of a conformal map of a domain with a variable area and fixed harmonic moments. In its modern context, the focus has shifted to the evolution beyond singularities, leading to the emergence of stable, fractal patterns known as diffusion-limited aggregation. When I discussed this topic with Anton Zabrodin, he noticed a potential connection to Igor’s work on the 𝜏-function of the universal Whitham hierarchy (source: arXiv:9205110). We shared this observation with Igor, and he recognized that the dynamics of the interface is merely a specialization of the deformation of Riemann surfaces with real periods, which he referred to as Boutroux curves. His paper explains that the evolution of such curves is described by the dispersionless limit of the 2D Toda hierarchy (see arXiv:0005259 and arXiv:0311005). Building on this connection, jointly with Razvan Teodorescu and Seung-Yeop Lee, who were students at that time, I conjectured that patterns of diffusion-limited aggregation are real sections of what we now call Krichever-Boutroux curves, experiencing a sequence of changes in genus. An example of this is the pattern of anti-Stokes lines of the asymptote of a genus 1 solution of the Krichever-Novikov string equations. It is quite remarkable that a classical problem in fluid mechanics, dating back 150 years, finds its place in Igor’s realm of algebraic geometry, integrable hierarchies, random matrix models, topological field theories, and more. Igor was fascinated by the geometric appearance of Laplacian growth and the countless connections APRIL 2024 it had to subjects he had previously worked on. He also enjoyed visualizing the various algebro-geometric objects that arose from the study of fluid dynamics. It was wonderful to observe how seemingly different things fit together so harmoniously. Igor effortlessly divided his time between New York and Moscow, showing devoted loyalty and an unwaveringly positive attitude towards both places. Unlike many of his peers with similar backgrounds for whom relocation to a new environment was an abrupt change, Igor’s life gradually evolved between the two countries. I found myself deeply sympathetic to this approach. Under his influence, I began spending more time in Moscow, and he (and Tanya, Igor’s daughter) opened my eyes to the continuous and rapid developments of this metropolis and to the vibrant intellectual and artistic environment a sophisticated city provides to its dwellers. As for other perplexing aspects of life, his attitude toward a lifestyle revolving around the New York-Moscow axis was balanced and generally positive. In the last decade, Igor actively participated in building and revitalizing the environment for fundamental research in mathematics and theoretical physics in Moscow. It was unexpected to see him in this capacity, but he proved to be a creative administrator in science. First at Columbia, and then in Moscow, his actions as a deputy director of the Institute for Problems of Information Transmission, and later as the founder of the Center for Advanced Studies at Skoltech, had a lasting positive effect on the lives of many people dedicated to science. He will be sincerely missed by many. Anton Zabrodin My older friend and coauthor, Igor Krichever, contributed a lot to different areas in mathematics and mathematical physics. Here I would like to concentrate on the topic related to both soliton equations and integrable many-body systems of classical mechanics. I mean the remarkable connection between singular solutions of soliton equations and many-body systems of Calogero-Moser type. The study of singular solutions to integrable nonlinear partial differential equations and their pole dynamics was started in 1977 in the seminal work by H. Airault, H. P. McKean and J. Moser. They considered elliptic and rational solutions to the Korteweg-de Vries equation and discovered that poles of such solutions move like particles of the Calogero-Moser many-body system. However, their Calogero-Moser-like dynamics was subject to some essential restrictions in the phase space. Igor showed in his Anton Zabrodin is a professor of mathematical physics at the Higher School of Economics (Moscow). His email address is [email protected]. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 497 1978 paper that in the case of the more general KadomtsevPetviashvili (KP) equation 3𝑢𝑦𝑦 = (4𝑢𝑡 − 6𝑢𝑢𝑥 − 𝑢𝑥𝑥𝑥 )𝑥 the correspondence between the pole dynamics and dynamics of Calogero-Moser particles becomes a complete isomorphism. So, the connection between soliton equations and integrable many-body systems becomes most natural: the dynamics of the poles 𝑥𝑖 of rational (with respect to the 𝑁 variable 𝑥) solutions 𝑢(𝑥, 𝑦, 𝑡) = −2 ∑𝑖=1 (𝑥 − 𝑥𝑖 (𝑦, 𝑡))−2 to the KP equation as functions of the variable 𝑦 is isomorphic to the Calogero-Moser system of particles with rational interaction potential (𝑥𝑖 −𝑥𝑗 )−2 without any restrictions in the phase space. Elliptic (i.e., doubly-periodic in the complex plane of the variable 𝑥) solutions to the KP equation were studied by Igor in his 1980 paper, where it was shown that the (in general, complex) poles 𝑥𝑖 of elliptic solutions 𝑢(𝑥, 𝑦, 𝑡) = 𝑁 −2 ∑𝑖=1 ℘(𝑥 − 𝑥𝑖 (𝑦, 𝑡)) + 𝐶 are subject to the equations of motion 𝑥𝑖̈ = 4 ∑𝑘≠𝑖 ℘′ (𝑥𝑖 − 𝑥𝑘 ) of the Calogero-Moser system with elliptic potential of pairwise interaction ℘(𝑥𝑖 −𝑥𝑗 ) (here ℘ is the Weierstrass elliptic function and dot means the derivative in 𝑦). The method first suggested by Igor consists in the substitution of the elliptic solution not in the KP equation itself but in the auxiliary linear equation 𝜕𝑦 𝜓 = (𝜕𝑥2 + 𝑢)𝜓. This allows for separation of variables 𝑦 and 𝑡 from the very beginning. In this approach, one should use a special ansatz for the wave function 𝜓 (a linear combination of Lamé functions with some coefficients 𝑐𝑖 ), depending on a spectral parameter. It is then proved that such a wave function is the Baker-Akhiezer function on the spectral curve. The auxiliary linear problem is then equivalent to an overdetermined system of linear equations for the coefficients 𝑐𝑖 which follows from cancellation of poles. This method allows one to obtain the equations of motion for 𝑁 poles in the fundamental domain together with the Lax representation of them: 𝐿 ̇ = [𝑀, 𝐿], where 𝑁 × 𝑁 matrices 𝐿, 𝑀 depend on 𝑥𝑖 , 𝑥𝑖̇ , and on the spectral parameter. The Lax equation, on the one hand, is equivalent to compatibility of the linear system mentioned above while, on the other hand, it means that the 𝑦-evolution of the Lax matrix is an isospectral transformation. The characteristic equation for the spectral parameter dependent Lax matrix 𝐿 defines the spectral curve which is an integral of motion. Igor’s method turned out to be rather general and productive and later was applied to a number of other problems. For example, it was used in Igor’s work arXiv:9411160 with Babelon, Billey, and Talon for the analysis of singular solutions to the matrix KP equation; they turned out to be connected with a spin generalization of the Calogero-Moser system (in the rational case known earlier as the Gibbons-Hermsen system). The spin de- 498 grees of freedom are matrix residues at the poles (in the scalar case they are fixed). In 1995, a similar method was used in our first joint work arXiv:9505039 for investigation of pole dynamics of elliptic solutions to the 2D nonabelian Toda lattice. In this case the auxiliary linear problem is a differential-difference first order equation. The Lax equation is equivalent to equations of motion for poles and spin degrees of freedom which define a new integrable many-body system with internal degrees of freedom—the spin generalization of the RuijsenaarsSchneider model with elliptic interaction. Its Hamiltonian formulation is still not known. In the scalar case, one obtains the Ruijsenaars-Schneider system which is a relativistic deformation of the Calogero-Moser system. The equations of motion have the form 𝑥𝑖̈ = ∑𝑗≠𝑖 𝑥𝑖̇ 𝑥𝑗̇ ℘′ (𝑥𝑖 −𝑥𝑗 ) ℘(𝜂)−℘(𝑥𝑖 −𝑥𝑗 ) , where 𝜂 is a parameter playing the role of the inverse velocity of light and having the meaning of the latttice spacing in the differential-difference Toda lattice equations. Some time ago Igor proposed another method to obtain equations of motion of integrable many-body systems based on the observation that the equations of motion are equivalent, rather mysteriously, to the existence of a meromorphic solution to the linear problem (a linear differential or difference equation). In our last joint paper arXiv:2211.17216, this method was applied to the 𝐵-version of the Toda lattice introduced previously in our work arXiv:2210.12534. In this way, we introduced a new integrable many-body system which is a deformation of the Ruijsenaars-Schneider system. Our last joint work with Igor appeared on the e-print archive on December 1, 2022, the day of his death. He worked till his last day. Since 1995, we completed with Igor more than 10 papers on different topics. Besides the pole dynamics, among them are representations of the Sklyanin algebra, functional relations in quantum integrable systems, Whitham equations in free boundary problems and others. It was a blessing to communicate and work with Igor. Tatiana Smoliarova One never knows what random moment of everyday life will stick in one’s memory forever. I can clearly see: a Moscow morning in early winter. It’s still dark in the street; only the snow on the roofs is glittering. I am eight, my dad is thirty-two. We are doing our regular morning exercises (which, of course, he does for my sake: a serious amateur ping-pong player, he is in excellent shape; I am a plump child). Amid the sit-ups, he says: “By the way, you know, Tatiana Smoliarova is an associate professor of Russian literature at the University of Toronto. Her email address is [email protected]. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 4 yesterday I defended my doctoral thesis.”5 Understanding that this is probably something nice, but also nothing special, I say, “Ok, that’s good?”—and do another sit-up. An eight-year-old, I have no idea what an achievement it is to defend a doctoral thesis, in the grim early 1980s, in the Soviet Union with its state anti-Semitism, for someone who is Jewish, and young to boot. That’s how my father always talked about the various pivotal points of his scientific career—casually, timidly, with an embarrassed smile. That’s how he worked and lived. He loved his comfortable and cozy office at Columbia, overlooking Broadway, but his most cherished study was a tiny room in the attic of our dacha (country house outside Moscow), overlooking pine trees, an old well, and a bed of ferns. It was there that he would start each day at 5 a.m., at his old desk, with his fountain pen and a notepad. Otherwise, as was observed by several conFigure 10. Igor with his tributors to this article, nogranddaughters Asya and Natasha. body knew when he was doing his mathematics. I was always told that my father achieved his most important mathematical results of 1975 while endlessly washing and rewashing my swaddling clothes. When my three kids were born, swaddling clothes were no longer in use, but there was still plenty to do. He was a devoted grandfather, and playing games, reading books, or teaching his grandchildren to ride a bicycle were no less serious creative tasks than teaching students, chairing the department, or writing articles. Yet he was never as super-excited, happy, and proud as when he communicated to me (a nonmathematician) that he had just “come up with a very beautiful thing.” As Nikita Nekrasov has said, my father, a profoundly nonreligious person, believed nevertheless in the ultimate harmony of the world. And I have always lived with a deep belief that even if the entire world were falling apart, my father would come— with a beautiful formula or tool, mathematical or not, or simply with the unique calm and reassuring tone of his voice—and fix everything. Sadly, when the world really began to fall apart, he was already too ill to fix it. 5 As Alexander Varchenko mentioned in his memoir in the present article, in Russia, there are two advanced degrees: Candidate of Sciences (equivalent to a PhD) and the much more exceptional Doctor of Sciences (equivalent to the French or German habilitation). APRIL 2024 ACKNOWLEDGMENT. We thank Slava Gerovitch for reading the article and making useful comments. Alexander Braverman Pavel Etingof Andrei Okounkov Duong Phong Paul Wiegmann Credits Figures 1, 6, 9, and 10 are courtesy of Tatiana Smoliarova. Figure 2 is courtesy of Anton Dzhamay. Figure 3 is courtesy of Igor Krichever. Figure 4 is courtesy of Nikita Nekrasov. Figures 5 and 7 are courtesy of Leon Takhtajan. Figure 8 is courtesy of Alexander Varchenko. Photo of Alexander Braverman is courtesy of Alexander Braverman. Photo of Pavel Etingof is courtesy of Pavel Etingof. Photo of Andrei Okounkov is courtesy of Andrei Okounkov. Photo of Duong Phong is courtesy of Duong Phong. Photo of Paul Wiegmann is courtesy of Paul Wiegmann. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 499