Remembrances of Ciprian Ilie Foias

Ciprian Ilie Foias (1933–2020)

Hari Bercovici, Peter Constantin, Allen Tannenbaum, Roger Temam, Edriss S. Titi

Figure 1.

Ciprian lecturing at the Vrije Universiteit in Amsterdam, The Netherlands, October 2000, on the occasion of receiving an honorary degree.

Ciprian Ilie Foias was born on July 20, 1933, in Resita, Romania. He studied mathematics at the University of Bucharest and finished his dissertation in 1957. Ciprian Foias began teaching at the University of Bucharest in 1954, where he stayed until 1958. In 1958, he was named to the Mathematics Institute of the Romanian Academy, where he taught and conducted research. In 1966, he became professor of mathematics at the University of Bucharest, and in 1968, he was named “doctor docent” in recognition of his already growing international reputation.

Ciprian Foias was an invited speaker at the International Congress of Mathematicians twice: in 1970 in Nice, France, and in 1978 in Helsinki, Finland. During the ICM in 1978, after delivering his invited lecture, Ciprian defected, and he flew to Paris. At that time Laurent Schwartz was very active helping mathematicians in difficulty, and he facilitated the entrance of Ciprian to France where he asked for political asylum. During the next year, while he was in France, his wife and two daughters remained in Romania. It was not clear how and when they would be allowed to leave Romania and join him abroad. Many people helped them, including the French government and the Indiana senators. It took one year for the family to be granted permission to leave Romania. After spending six months in Paris, Ciprian accepted the position of professor of mathematics at Indiana University in Bloomington, Indiana, where he spent the next 20 years of his career. He retired from Indiana University in 2000 at the age of 66. However, retirement was not for him, and he accepted a teaching position at Texas A & M University in College Station, where he became a University Distinguished Professor in 2007, and he stayed there until his retirement in 2016.

Ciprian authored or coauthored 11 books and over 500 refereed professional journal articles. He had 19 graduate students and 202 descendants according to the Mathematics Genealogy Project. He influenced a large number of young mathematicians, and he had over 100 coauthors.

As a prolific researcher, as well as a brilliant mind, he received many accolades and awards. He was named Distinguished Professor of Mathematics at Indiana University in 1983. He was inducted in the Hungarian and Romanian Academies of Science, as well as the American Association for the Advancement of Science. He received honorary doctorate degrees from the University of Timisoara in Romania, and from the Vrije Universiteit in Amsterdam. In 1995, Ciprian was awarded the very prestigious Norbert Wiener Prize from the American Mathematical Society and the Society for Industrial and Applied Mathematics for his extensive research in applied mathematics. After Ciprian passed away, the AMS created the Ciprian Foias Prize in Operator Theory. This Prize “will be awarded every three years beginning in January 2022. The prize will recognize notable work in operator theory published during the preceding six years. Ciprian Foias (1933–2020) was an influential scholar in operator theory and fluid mechanics, a generous mentor, and an enthusiastic advocate of the mathematical community. His colleagues and friends established the prize in his memory.” (AMS :: News from the AMS).

He mainly worked in three directions: operator theory, Navier-Stokes equations and turbulence, and control theory. Nevertheless, he contributed to many subfields in pure and applied mathematics, and he would never miss a challenge. For example, back in Bucharest, due to an error of a typist, a problem on an examination was changed from an easy basic exercise to a very challenging one. Ciprian took the challenge and eventually solved the accidentally invented difficult problem. This led him to introduce a new constant eventually called the Foias constant https://mathworld.wolfram.com/FoiasConstant.html.

Figure 2.

Nicoletta and Ciprian Foias, summer 2000, during a ceremony for the retirement of Ciprian from Indiana University. Behind them, John B. Conway and his wife Ann, and Wing Suet Li.

One of the three areas of mathematics in which Ciprian Foias made fundamental and lasting contributions was operator theory. Roughly half of his published output, including several books, are about operator theory or its applications. Even his work in other fields of mathematics often involved ideas derived from operator theory. Several subjects recognized by Mathematical Reviews originated in this work (47A45, 47B40, 47L45). A complete account of his work in the area would fill many pages, so we restrict ourselves to a few significant items. In his early work, Ciprian focused on the theory of spectral operators where he introduced the generalized spectral operators (that have a functional calculus with sufficiently differentiable functions, in other words, an operator-valued distribution) and decomposable operators. (This last class reappeared, in a slightly modified form, in work of Haagerup and coauthors.) A long and productive collaboration with B. Sz.-Nagy led to the construction of canonical models for contraction operators as well as for operator semigroups. In particular, the celebrated Commutant Lifting Theorem helped unify results of operator theory and interpolation, and it still serves as a model for contemporary developments. These results had many applications, within and without mathematics. In particular, this work was instrumental in the construction of a theory of dual algebras that in turn led to striking results (by Ciprian and collaborators) about invariant subspaces. Another breakthrough occurred in joint work with Apostol and Voiculescu that characterized in purely spectral terms the class of quasitriangular operators. This fundamental result was the starting point of a vast classification theory of arbitrary Hilbert space operators. The quasitriangularity work answered one of Halmos’s most difficult questions, and subsequent work by Ciprian and collaborators answered several other long-standing questions about the approximation of Hilbert space operators. (For instance, quasinilpotent operators are norm-limits of nilpotent operators.)

For the theory of the Navier-Stokes equations, he had two early papers which were the basis for many subsequent developments. Firstly, motivated by the results of Prodi, his first contribution was the monumental paper in two parts, related to statistical solutions of the Navier-Stokes equations Foi72. This paper was the mathematical basis of much subsequent work related to turbulence, including results with Oscar Manley, who always brought an additional physical insight, and work with Ricardo Rosa, including the book FMRT01. Another paper with Prodi FP67 was eventually the starting point of many results showing the finite dimensional behavior of turbulent flows; this includes the finite dimensionality of the attractors and relations between the dimension and the physical constants of the problem, FT79, CF85, CFT88, CFT85. Subsequently this work led to the introduction of the concept of inertial manifolds FST88 and of exponential attractors CFNT89aCFNT89b, EFNT94. Also, a large literature relates to the development of determining modes and determining nodes, which led to applications to data assimilation for weather and climate predictions, and feedback control of dissipative systems. Other research directions initiated by Ciprian and his co-authors include the Gevrey analyticity of the solutions to the Navier-Stokes equations FT89, and the analysis of the asymptotic behavior of regular solutions to the Navier-Stokes equations with potential forces, leading to a Poincaré-Dulac normal form for these equations FS84.

Ciprian made pioneering contributions to control theory leveraging his work in operator theory, especially the Commutant Lifting Theorem. Robust control is the study of feedback systems when one has uncertainty in the plant (the fixed part of the system) DFT92. The problem is to design a controller (the part free to design) to reach some overall performance goal in the closed loop feedback system, e.g., stabilization, tracking, disturbance rejection. One of the key methods for accomplishing this is so-called $H^\infty$ control, proposed independently by J. William Helton, A. Tannenbaum, and George Zames in the late 1970s, early 1980s. In the finite dimensional case for plants modeled by linear time-invariant differential equations, it was recognized very early on that the problem could be solved by Nevanlinna-Pick interpolation. The problem arose then to give a solution in the infinite dimensional case, e.g., for systems with delays or described by partial differential equations (PDEs). It is here that one needed the much more powerful operator theoretic methodologies of Ciprian. The Commutant Lifting Theorem led to a complete solution in the delay case FTZ87 and a new class of operators called skew Toeplitz (done in joint work with Hari Bercovici) to handle the PDE case even for systems involving multiple inputs and outputs BFT88. The methods, together with many real-world control examples, are summarized in the text coauthored by Ciprian FOT95. Without the insights of Foias, robust control would have been restricted only to the finite dimensional case without treating the crucial problem of systems with delays and applied to the very rich world of dynamical models described by partial differential equations.

Ciprian was interested in just about every field of human knowledge and he read widely from molecular biology to horse racing to mountain climbing (which was one of his passions) to history. In his twenties and thirties, he loved mountain climbing. Later in life, he became passionate about horse racing, and, besides his family, colleagues and friends would accompany him driving hours to horse races in Ohio and Kentucky, for the pleasure of the company if not for the horse races themselves. Ciprian saw it all through the eyes of a mathematician, keeping close accounting of horses, horse trainers, and jockeys. He loved to travel and when the family first moved to the US, he took many cross-country trips that he planned very carefully to ensure he saw and showed to the family as many national parks as possible. Ciprian loved his mathematics, his students and colleagues, and he loved his family, and they all loved him.

He died quietly at home, surrounded by his family, at 7:30 pm on Sunday, March 22, 2020. This article is based in part on the obituary written by the family: https://richardsonfuneralhome.org/obituary/Ciprian-Foias. The following articles are individual memories of students, friends, and colleagues, listed in alphabetic order. Most of the authors are younger, sometimes former students. Ciprian survived many of his senior friends and colleagues who are not here to witness. Among those who admired Ciprian and who would visit him regularly, one can mention Ronald Douglas, Israel Gohberg, Oscar Manley, Basil Nicolaenko, and George Sell.

Hari Bercovici is a professor of mathematics at Indiana University Bloomington. His email address is bercovic@indiana.edu.

Peter Constantin is the John von Neumann Professor of Mathematics and Applied and Computational Mathematics, Department of Mathematics, and Director of PACM, Princeton University. His email address is const@math.princeton.edu.

Allen Tannenbaum is a professor in the departments of computer science and applied mathematics at Stony Brook University. His email address is allen.tannenbaum@stonybrook.edu.

Roger Temam is director of the Institute for Scientific Computing & Applied Mathematics, a distinguished professor in the department of mathematics at Indiana University Bloomington, and a professor emeritus at the Université Paris-Sud, France. His email address is temam@indiana.edu.

Edriss S. Titi is a distinguished professor at Texas A & M University; Chair professor of the department of applied mathematics & theoretical physics at the University of Cambridge; a professor in the department of computer science & applied mathematics at the Weizmann Institut for Science, Israel; and a professor emeritus in the department of mathematics, and of mechanical & aerospace engineering at the University of California, Irvine. His email address is est42@cam.ac.uk.

Ciprian Foias: A Memorial Tribute

Robert A. Becker

Ciprian and I published eight papers on mathematical economics. Our most important ones concerned the dynamics of a many-agent model economy following a competitive equilibrium path. The underlying economic problem was motivated in my first published paper based on Frank Ramsey’s 1928 conjecture. I reformulated his problem, introduced a formal mathematical model consistent with his conjecture, and proved the existence of a unique steady-state equilibrium. My late colleague and mentor, Nicholas Spulber introduced me to Ciprian. We knew each other for about a year before we started our joint project. I was an “honorary Romanian” at a regular Wednesday lunch gathering devoted to politics and other newsworthy events. We met by chance for lunch one day at the student union cafeteria. Ciprian asked about my research. I sent him a reprint of my article. He called the next day and with great enthusiasm said he thought we should work on my dynamic conjecture. The ensuing project took many years to come to fruition and was ultimately published in a 1987 issue of the Journal of Economic Theory. It is our most highly cited joint work. We ultimately published five more papers on what Ciprian called the Ramsey equilibrium problem. This included placements in Econometrica, Economic Theory, and the Journal of Mathematical Economics. Our Econometrica paper was also jointly authored with my doctoral student, John H. Boyd III. I told Ciprian about the growing impact of our papers during what turned out to be our last conversation. He was pleased to learn that I had continued working on our model and there were many others now pursuing problems in Ramsey equilibrium theory.

The opportunity to work with Ciprian was extraordinarily rewarding on many fronts. He taught me many of his mathematical modeling strategies, which I continue to apply in my research and teaching. He distinguished between elementary and nontrivial arguments. He regarded our research as belonging to those categories’ intersection. We usually worked after our Wednesday lunches. He preferred meeting in noisy public places in the student union building. Ciprian used color-coded paper to distinguish notes between his vast array of coauthors. He told me I shared blue paper with Roger Temam as there was no way to confuse the two projects! I could not help but notice how fast Ciprian produced insights about our problem’s mathematical structure. I was much slower! He told me it was not a problem. He said from the beginning we would adapt mathematical techniques to fit the model’s economic structure. This is a common feature in our papers. Ciprian’s generosity was a tremendous help to me on several fronts. He was as much a mentor to me as Professor Spulber. Working with Ciprian was, without doubt, the most important collaboration of my career.

Robert A. Becker is a professor in the department of economics at Indiana University Bloomington. His email address is becker@iu.edu.

Hari Bercovici

I first met Ciprian in 1971, when I was in my first year at the University of Bucharest. On the advice of his friends in the Romanian Mathematical Society, he conducted a problem seminar with a small group of first-year students. While our regular courses tended to present mathematics as a vast subject built to a high degree of perfection, Ciprian showed us that there are in fact many unanswered questions, as well as subtle flaws in this edifice. Indeed, he told us that this may be a subject to which even we may be able to contribute something, which was a very exciting prospect. He expected us to answer problems that required material, the essence of which he explained briefly, that went far beyond what was presented in our regular courses. Indeed some of the problems were open at the time. The solutions we presented were subject to lively criticism that continued sometimes for many hours. The seminar continued, with some interruptions, for more than two years, during which time the number of participants declined to three: Radu Gologan, Dan Timotin, and me. We were all encouraged to attend the operator theory seminar where we saw in real time how Ciprian and his collaborators solved Halmos’s problem on quasitriangularity: it was an involved plot where ideas and approaches evolved and changed drastically from week to week. At that point, Ciprian felt that it was time for us to do some serious work, and he presented us with three research problems that we had to choose from. For all three of us, these problems became the kernel of a research program that eventually developed into a PhD thesis, though Ciprian was not our formal doctoral advisor. The problem I chose was about dilation theory and it also led to a collaboration with Ciprian himself and Sz.-Nagy. Since that time, Ciprian treated me as a friend and helped me throughout my career, even though I was not always aware of his help at the time. He encouraged me indirectly to leave Romania, and extraordinary luck made it so that we were both in Paris for several months. He and his recently reunited family made me feel very welcome, and Ciprian even arranged for me to give some paid lectures in Paris and Bordeaux. Our mathematical collaboration continued throughout the years. When I was a student in Ann Arbor, Ciprian, Carl Pearcy, and I met often in Fort Wayne and worked together for a weekend. The discussions were as contentious as our old problem seminar and the best ideas often came to us during the drive back. Ciprian was certainly instrumental in obtaining my appointment in Bloomington (and he was rather upset that there had been two votes against!) That was an emotionally difficult time which his friendly support helped me overcome.

Figure 3.

Ciprian Foias, Israel Gohberg, Billy Rhodes, Hari Bercovici, and Vinay Deodhar, October 13, 1989, in Bloomington.

My life would certainly have been much different and less interesting had I not met Ciprian. With the exception of my parents, he was the person who contributed most to my well-being. I will always miss his advice, his criticism, and the lively disputes we had, mathematical or not.

Animikh Biswas

When I received the invitation to contribute an article in memory of Ciprian, it brought back a flood of memories as well as the acute sense of loss that I felt on his passing. My association with Ciprian (to me, Professor Foias then) goes back a little over twenty-five years when in spring 1996, with much trepidation, I approached him to accept me as a student. I had taken only one course from him until then; but to my surprise, he accepted me as his student. Thus started my mathematical journey with him as my teacher, which continued till his passing in March 2020. To me, he never ceased to be my teacher.

I remember those good old days at Indiana University, taking many courses from Ciprian. Sometimes, our experience was humorous and inspirational at the same time. He would almost always go off course, take up the gap times of twenty minutes in between classes and then profusely apologize to the professor of the following class for encroaching on the class time. Sometimes, the reason would be that he would get stuck while proving a theorem. He had elaborate notes, which he would prepare ahead of time and distribute to the class. We learned later that often, these notes were completely original research which would then subsequently be published in journals. But he refused to glance at his own notes while he was proving these theorems in class impromptu, often in ways completely different from the notes. Soon, we would find ourselves chipping in to help him prove these theorems in class, initially because we wanted the class to end so that we could grab lunch, but also often because it was fun. I cannot say if it was by design, but I found that these were the classes where I learned the most.

Figure 4.

Ciprian with a group of his students: Olson, Grujic, Foias, Kukavica, Titi, Biswas, Cheskidov, and Dascaliuc (in front), April 13, 2014.

Ciprian “retired” from Indiana University in 2000 and joined Texas A&M University that year. I visited him frequently at College Station. I got to know him much more closely during these years; about his humility, grace, compassion, self-effacing manner, and of course, his passion for mathematics. His kindness and generosity towards his collaborators, particularly junior ones, and his utter honesty in ascribing due credit was unmatched. I learned first-hand about his inimitable style of doing mathematics; about how it was driven by a pure scientific quest. He often told me that we don’t have the time to wait for big inspirational ideas to hit us, we should just keep working, even if the problem seems insignificant. Sometimes, while we worked, it felt to me like an endless meandering of a quiet stream, not sure where we were headed but just flowing; but then, suddenly, after an arduous traverse, a splendid vista would emerge that would make the journey worthwhile.

We would work intensely until late evening. When we took breaks, we would either browse the bookshelves at Barnes & Noble or just talk. Ciprian had a vast repertoire of knowledge; about science, economics, religion, philosophy, and many other things. He was an avid mountain climber. He told me stories about his mountain climbing trips in Romania to the Carpathian mountains. He told me other stories too, like the one about his wanting to become a university professor. It seems when he was a young boy, he and his father, who was a doctor and a prominent person in the city administration, and whom Ciprian deeply revered, were taking a stroll. He saw a shabbily dressed, unkempt person walking towards them. His father bowed to that person in deference and hurriedly moved away from the sidewalk to make way. Ciprian was surprised because he only knew of people bowing to his father. His father later explained that person was his esteemed professor at the university. That was the catalyst of his desire to become a professor.

Ciprian started out being my adviser, and at some point during our journey, he transitioned to being a father figure. I would turn to him for advice and guidance, and often, during really difficult times in my life, he gave me solace. Of course, I miss working on exciting mathematical problems with him. I also miss just talking to him, over the phone, about a myriad of other things. Although I will not get that opportunity again, I will forever cherish those memories, and the shining examples that he has set through his mathematics and the integrity and humility with which he lived his life.

Animikh Biswas is a professor and the chair of the department of mathematics & statistics at the University of Maryland, Baltimore County. His email address is abiswas@umbc.edu.

3D Navier-Stokes Equations: The Dynamics of a Blow-Up

Alexey Cheskidov

Being Ciprian’s student it was impossible not to develop at least a curiosity for the regularity problem. It seemed all the other equations where just toy models that Ciprian studied to gain more insights and develop new tools to tackle the problem that he really cherished. He liked the simplest setting of the 3D torus, and the equations were often written as

$$\begin{equation} \frac{d}{dt} u + \nu Au +B(u,u) = g, \tag{1}{} \end{equation}$$

where the bilinear term and the Stokes operator are defined as

$$\begin{equation*} B(u,u)= P(u\cdot \nabla )u, \qquad Au = -\Delta u, \end{equation*}$$

and $P$ is the Leray projection onto the divergence free vector fields. The energy space of square integrable divergence free functions $H$ is equipped with the $L^2$ norm denoted by $|\cdot |$, and the enstrophy space $V$ is equipped with the norm $|\nabla \cdot |$, the double norm, denoted by $\|\cdot \|$. Then 1 can be thought of as an ODE in $V$ until the time of blow up of the double norm. This was what we called the Foias-Temam framework in Indiana. However, once Ciprian moved to Texas A&M University in 2002, he started teaching the Navier-Stokes equations in an elementary way that avoided the use of Functional Analysis. I remember he was given an alarm, so the classes did not run for hours as during his last years at Indiana, where we could spend days analyzing such things as the most general trajectory of a yo-yo (Ciprian’s interpretation of a problem suggested by Oscar Manley as a simple exercise for a continuum mechanics course). It was enlightening to relearn the basic NSE theory in a self-contained elementary way where the use of Sobolev inequalities was forbidden. A weak solution was defined as an $l^2$ valued function of Fourier coefficients satisfying

$$\begin{equation*} \begin{split} \hat{u}(k,t) - \hat{u}(k,0) &= \int _0^t \big \{ -\nu |k|^2 \hat{u}(k,\tau ) +\hat{g}(k,\tau ) \\ &\qquad {}- i\sum _{h \in \mathbb{Z}^3} \big [(k\cdot \hat{u}(h,\tau )) \hat{u}(k-h,\tau )\\ &\qquad {}- |k|^{-2} (k\cdot \hat{u}(h,\tau )) (k\cdot \hat{u}(k-h,\tau )) k \big ]\big \}\, d\tau , \end{split} \end{equation*}$$

and the goal was to recover basic existence, uniqueness, Leray structure results, using only Duhamel’s principle and undergraduate Real Analysis.

At one point when we started looking at weak global attractors, Ciprian and I discussed whether a strong global attractor, if it existed, had to coincide with the weak one. This happened to be an open problem, and while solving it, we developed a theory of evolutionary systems applicable to equations such as the 3D NSE where the regularity and uniqueness were in limbo, without making any assumption on the weak solutions. I think Ciprian believed in regularity of the 3D NSE, and he wanted to add a truly nontrivial application of the framework to the paper, which was otherwise finished. So back in Indiana I spent my final year as a PhD student analyzing Ciprian’s tridiagonal model for the NSE, a simplified toy model, similar to the dyadic model that recently gained popularity. A skeleton for the energy cascade, which provides insights into possible blow-up scenario for the NSE, the model is written as 1 with

$$\begin{equation*} \begin{split} (Au)_n &= n^\alpha u_n, \qquad \\ (B(u,u))_n &= - n^\beta u^2_{n-1} +(n+1)^\beta u_n u_{n+1}, \qquad n \in \mathbb{N}, \end{split} \end{equation*}$$

and $u_0=0$. The role of $H$ is played by $l^2$ with the usual inner product and norm $(u,v)=\sum u_n v_n$, $|u|= \sqrt {(u,u)}$. Also, as for the NSE, the double norm is defined as $\|u\|=\sqrt {(Au, u)}$. When $\alpha =2/3$ and $\beta =11/6$,

$$\begin{equation} |(B(u, u), Au)| \lesssim |Au|^{\frac{3}{2}} \|u\|^{\frac{3}{2}}, \tag{2}{} \end{equation}$$

which is exactly what Sobolev estimates give for the 3D NSE. In general, the choice $\alpha = 2/d$, $\beta = 3/2 + 1/d$ would correspond to the $d$-dimensional Navier-Stokes equations. While the global regularity holds for $d\leq 2$, proving a blow-up for large $d$ happened to be extremely challenging. We know it occurs when $2\beta - 3\alpha - 3 > 0$, but it is still an open question for any finite $d$. For the actual 3D NSE it is possible to show that 2 can be saturated by divergence free vector fields. For instance, interacting Dirichlet kernels produce such examples. Hence, a dynamical approach is needed to exclude a possible blow-up. The tridiagonal model for the NSE captures the energy cascade essential for a blow-up in one of the simplest possible ways. Even though the regularity is still in limbo, Ciprian’s approach provides insight into the regularity problem that is hard to see from the actual equations.

Alexey Cheskidov is a professor in the department of math statistics and computer science at The University of Illinois at Chicago. His email address is acheskid@uic.edu.

Peter Constantin

Ciprian Foias was a brilliant, powerful, and original mathematician. My first interaction with him was in my third year as an undergrad at the University of Bucharest, when he gave an elective course on Navier-Stokes equations and I took the course notes. It turns out that he must have liked them. Several years later, I think it was in 1981, while on active duty in the IDF, I managed to go to his lecture at the Toeplitz centenary meeting in Tel Aviv. I was standing in the back, uniform, M16, and all. He came to me and emphatically told me that I “have to” come to Indiana University for a postdoc. This was before I even applied for jobs. I was very flattered. My PhD with Agmon was on scattering theory for Schrödinger equations, linear PDE, very far from Navier-Stokes equations. But, after my military service ended and I got a few offers, I chose I.U. over more prestigious places because of Foias. It was the right decision, and a turning point in my life. I got to the US in 1982, and took the Greyhound bus from Chicago to Bloomington. After passing what seemed to me like interminable farming fields, the bus pulled in at a Taco Bell and the driver announced that we had arrived at Bloomington. I was hesitating to get off, but standing by the bus, there was Ciprian, waiting to pick me up.

Ciprian took me immediately under his wing. I spent in total a little over a year in Bloomington. The math department was so friendly and supportive, it has a warm place in my heart to this day. The collaboration with Ciprian was exciting. We worked on a method to prove finite dimensionality of attractors in dissipative PDE. The interactions were intense: we would work all day, go to our respective homes, work some more, and in the morning we would both have “news” for the other. Ciprian was competitive, and I am sort of the same. He called it “ping-pong.” I loved it.

Figure 5.

Igor Kukavica, Peter Constantin, and Ciprian, at a meeting for the 60th birthday of Peter in 2011.

Ciprian had several regular visitors with whom he had projects that continued for years. He would always want to polish results further and never gave up on his research themes. He pursued his early interest in finite dimensional determining modes initiated with Giovanni Prodi for many years and in many ways, with collaborators and students. This point of view is still used for proving uniqueness of stochastically forced evolution equations. In addition to his collaborator Roger Temam, regular visitors to Bloomington included Jean-Claude Saut, with whom Foias had a long standing project on a Poincaré-Dulac normal form for unforced 3D Navier-Stokes Equations.

Ciprian had broad knowledge and interests. Almost every day we would take breaks and walk to the bookstore. Ciprian would go to a book which he had thumbed through the day before, and consider it some more. He called that “flirting” with the book. After a couple of visits, he would often buy it. This could be a book on math, or science, or politics, or history, or science fiction. I remember him talking about science fiction books earnestly with my nine-year-old daughter some fourteen years later. It was charming, and she found him wonderful. Ciprian was also a serious alpinist. He used to say that when he was young he was able to pull himself up with a grip of only a few fingers. When I met him, he wasn’t doing technical climbs anymore, but we did hike, play tennis, and drive in his signature beat-up yellow Ford Pinto over the hilly countryside to local horse races in Kentucky.

After moving away from Bloomington, I continued to collaborate with Ciprian. We wrote a book on Navier-Stokes equations, based on a course he gave in Indiana and a course I gave the next year in Chicago. Ciprian wanted the book to be self-contained and as elementary as possible. Many of our students, and their students in turn, went through this little blue book. I remember fondly the times we were working on a book on inertial manifolds (with Basil Nicolaenko and Roger Temam). Basil lived in Santa Fe, and we rented a house and finished writing the book there. In total we had these two books, an AMS memoir (with Roger Temam), and twelve joint papers with Foias and collaborators, the last one in 1997. Although my direct collaboration ended, I continued to work with his students and collaborators, who became my friends and collaborators, including Jean-Claude Saut, Edriss Titi, Igor Kukavica, and A. Cheskidov.

Ciprian Foias was a teacher, a mentor and a friend, and much more than that, a unique human being.

Alp Eden

Ratip Berker and his exact solutions

Ciprian Foias’s fondness for everything related to Navier-Stokes equations is legendary. However I was surprised to find out that he had a high opinion of Ratip Berker, a Turkish applied mathematician specializing in fluid mechanics. Ciprian believed that the exact solutions that are described in Ratip Berker’s book that had originally appeared in the Handbuch der Physik (1963) should be studied carefully. While studying the works of Ratip Berker, I realized that some of the exact solutions found by the Berker’s transformation were also called turbulent solutions (Bass (1974) JMAA 47, 458–503). Ciprian did not spell out why he was interested in these exact solutions. However, I will venture to claim that it had something to do with his quest for the global attractor for his beloved equations. Ciprian’s positive attitude towards classical approaches to Navier-Stokes equations was different from that of Olga Ladyzhenskaya. In the preface to her famous book (Mathematical theory of viscous incompressible flow, English edition (1963)) Ladyzhenskaya criticized classical approaches, like Berker’s, for finding exact solutions to Navier-Stokes equations. She claimed that they were not useful in studying general initial value problems.

Call me Ciprian

I visited Ciprian a year after completing my PhD thesis. At the time I used to address him as Professor Foias. At some point, he smiled and told me that now I should start calling him by his name: Ciprian. It was an emotional moment for me. However, it took me a good while before I could do so.

High table

Our routine when we worked together was the same. Since he woke up very early and worked for hours, we would meet later in the morning, work a while and then go to lunch together. Ciprian knew some of the distinguished faculty members of Indiana University. It was very exciting to sit at the same table with them and listen to their views about world affairs. Even Ciprian would be listening most of the time.

Napkin theorems

After lunch we used to go to an ice cream parlor, because Ciprian loved ice cream. We would sit down at a table and talk mathematics for hours. It was there the “napkin theorems” were produced. Since we quickly ran out paper Ciprian would start writing on napkins! The next stop was always a bookshop, usually the Book Corner (sometimes IU Bookstore) that was located at the center of downtown Bloomington. We would recommend books to each other, and most of the time he would buy me a book that he had read recently. We also had the habit of giving stationery as gifts to each other, especially different types of pens and pencils.

Our shortest paper

One of our papers was on the best constant for Lieb-Thirring inequalities in one space dimension (1991). After countless revisions of the paper I started complaining. I was at the start of my career and I had to publish papers. He told me since we did not prove anything new and we had to make sure that it was perfect. Although it was one of our shorter papers it took us many revisions before we could send it to a journal.

I was back in Turkey and teaching at Bogazici University when a faculty member from the Physics Department told me that he was using our paper in his graduate course in Quantum Mechanics to illustrate the power of operator theory in quantum mechanics. After many years some people started taking notice of our paper. Ciprian was happy to hear about that.

An appetite for challenges

While listening to seminars and colloquia, Ciprian would take notes in a notebook where the page would be folded in two. He would write the claim or theorem on one side of the page, leaving the other side empty. When I inquired about it he told me that he would attempt to prove the results later without “cheating.”

Compassionate character

Ciprian’s compassion for mathematics is well known. He could spend literally hours trying to figure out a “glitch.” After he lectured more than three hours in a graduate class, I asked his permission to leave the classroom. He was surprised and asked me whether I had a previous engagement.

His compassion was not only reserved to mathematics, he was compassionate about every endeavour he took. He was also deeply caring and considerate towards his colleagues and students. His unique understanding of human nature and its frailty was masked under his sometimes naive appearance.

Alp Eden is a retired professor of mathematics at Bogazici University. His email address is alp.eden5@gmail.com.

In Memory of My Friend, Mentor, and Teacher

Art Frazho

It was with great sadness that I learned that Ciprian passed away. Ciprian was certainly one of the greatest men I ever knew and I shall forever be indebted to him for everything he taught me from mathematics to life.

His research encompassed a vast array of subject areas from mathematics, economics, and physics, to engineering. Ciprian always liked working in coffee shops and cafes. The walks back and forth to the coffee shops were always enjoyable. During our meeting he would fill reams of paper with equations. Many times it took a week or two to comprehend what he was doing and there was always a hidden jewel in the formulas that I initially missed. I remember he said, if you do not have an idea, start calculating and it always yields results. Since I am not a mathematician by training, many times I would ask him elementary questions and he never made me feel uncomfortable. He always answered these question and provided a unique insight into the problem.

Figure 6.

Bela Szokefalvi-Nagy and Ciprian, at the 1983 Wabash Seminar in Crawfordsville, Indiana.

Before I meet Ciprian I read his book with B. Sz.-Nagy (Harmonic analysis of operators on Hilbert space) in graduate school. I was perhaps the only engineer to read his book, and this changed my life. The book is simply filled with beautiful mathematics which gave me a much deeper understanding of engineering. This changed my approach to engineering problems forever. So I was very happy and impressed when I met him at the Wabash Conference in 1980. This was the beginning of a long friendship. Even though I studied his book very hard, he pointed out many important aspects I missed and this was extremely helpful.

I remember being in his office and he was talking about mountain climbing, and how he could hold on to a crevice in the rocks. He was a bit older then and surprised me by taking two fingers and lifting himself up by the 1/2 inch piece of wood on the top of the door trim.

I remember driving to Churchill Downs and talking mathematics and horse racing. He knew the lineage of all the great thoroughbreds. He enjoyed driving and knew all the back roads to the track. Rien Kaashoek was visiting Bloomington and we were working in a bookshop. Rien picked up a book on art history, covered up the painter’s name and Ciprian correctly named all the artists. I swear his memory was photographic and his knowledge was simply astounding.

He was a true Renaissance man who will be greatly missed.

Art Frazho is a professor in the school of aeronautics and astronautics at Purdue University. His email address is arthur.e.frazho.1@purdue.edu.

Some Things Ciprian Foias Taught Me

Michael Jolly

The mathematical topics Ciprian Foias taught me came in several flavors. First, there were approaches to finite dimensional behavior in infinite dimensional systems that in retrospect, at least, seem natural: global attractors, inertial manifolds, determining modes. Then there are some which, to this day, seem to come from nowhere: approximating the global attractor through Caratheodory and Nevalinna-Pick interpolation schemes, establishing sets of finite backward growth rates defined by Dirichlet quotients. Only when I consider Ciprian’s combined command of operator theory, partial differential equations, and dynamical systems, can I begin to understand why (not how) he came up with such ideas. Finally, there was the more physical. In asking me to look over the manuscript for his paper What do the Navier-Stokes equations tell us about turbulence?, Ciprian introduced me to how challenging it is to bring rigor to brilliant, yet largely heuristic and vague theories.

Ciprian taught me the value of examining toy problems. He found much to explore in the Lorenz system of three ordinary differential equations. He also was always excited to see what insights numerical computations could provide. He once proudly showed me some codes he had written in Basic to compute reflection coefficients. Ciprian had an expansive view of mathematics. He told me that the field needs people working at all levels, from those charting new routes into big open questions to those toiling to fill the cracks left behind after major advances. He taught me that one could judge the relative value of a result without dismissing it.

Coffee shop work sessions would include digressions into biology, history, religion, and politics. Ciprian was always cheerful when patching the holes in my mathematical training. He never became frustrated when I did not know something in operator theory or analysis. He would, however, lose patience when trying to convert me to his political beliefs. It was clear I could never match his store of information, nor the depth of his personal experiences. I learned that I should simply nod in surrender.

Ciprian was incredibly generous with his time for students. I saw him spend countless hours tutoring high school children of friends and staff. Some were destined for careers in mathematics, other simply needed some help. Even of the latter, he would speak in glowing terms about their native intelligence. In searching for another phrase to replace “native intelligence,” I happened upon “horse sense.” Ciprian spoke in admiration of the farmer in overalls at Churchill Downs who had an innate ability to pick a winner. He was very open-minded about the thought processes of different people.

Ciprian shared both his ideas and his students. He was happy when collaborations between them formed. I always considered myself to be one of his students, and benefited greatly from being part of that group. He was a family man, not just at home, but also at work. He dazzled us, amused us (stories abound), and nurtured us. I believe he still does.

Michael Jolly is a professor in the department of mathematics at Indiana University Bloomington. His email address is msjolly@indiana.edu.

Igor Kukavica

I feel very fortunate to have been Ciprian’s student. He was a caring and dedicated advisor whose unwavering enthusiasm for science and discovery was inspiring to me and to anyone who knew him. As an advisor, he did not set regular meeting times, his door was always open. Whenever I would come by and knock, he would stop what he was doing and greet me with an anticipating smile: “What’s the news?”

Ciprian was one of the pioneers of the mathematical theory of the Navier-Stokes equations. He was fascinated with the chaotic nature and finite dimensionality of fluid flows. One of his earliest influential results was on the determination of the flow by a finite number of modes FP67. The main theorem states that if a finite number of Fourier modes agree for two flows, then these flows have to agree. Much later, the Foias-Prodi approach turned out to be an essential tool for constructing invariant measures for stochastic flows. The Foias-Prodi Theorem provided a strong indication that the global attractor for the Navier-Stokes equations is finite-dimensional, a fact that was later proven rigorously in FT89. Much effort has been dedicated to finding a concrete upper bound for the attractor dimension, which was ultimately accomplished in CFT88CFT85.

Another approach to finite-dimensionality for the dissipative system is based on the very successful idea of inertial manifolds, introduced in FST88. These are forward-invariant manifolds with the exponential tracking property. The existence of the inertial manifolds has been subsequently shown for many interesting physical systems, but the existence of an inertial manifold for the Navier-Stokes system is still open. On the other hand, the exponential attractors, which are objects somewhere in between attractors and inertial manifolds, have been established for many systems which possess attractors. From Ciprian’s work, my two personal favorites are the Gevrey approach to the analyticity and backward uniqueness based on the Dirichlet quotients.

One of Ciprian’s notable traits was his unique teaching style. I had the privilege of being a student in two of his courses: his Ordinary Differential Equations course when I was a first-year graduate student and later a graduate course in $H^\infty$ control. Both classes were held on Tuesdays and Thursdays for eighty minutes each. Ciprian would always begin by distributing carefully typed notes on the material for that day. He would then start calmly explaining the material. His energy level would slowly build throughout the session, motivated by the mathematics he was presenting. He would often question himself why something was true and would never pass up an opportunity to prove the theorems he was using or citing. He did not like erasing the board, wanting the material to remain on the board for as long as possible. To accommodate this, Ciprian would search for blank areas on the board where he would add another irregularly shaped box to add a fact or a proof. His enthusiasm would always increase as the end of class approached, which naturally led to him going overtime. Once, after extending the class for an extra hour, he was still working through an involved proof and turned around to ask the class, “Does anybody need to go for lunch?”

Ciprian gave each of his students the same enthusiasm and attention that he gave mathematics. He would praise any student idea that he had not anticipated. Bonus problems were his way of inspiring students to prove themselves; the maximum score a student could achieve in his classes was 150% (even after he would state in the first class that “There will be no bonuses in this class!”). He is the only professor I know who gave out a grade of A++.

Ciprian’s love for mathematics may only have been surpassed by his passion for horse racing. Some of my favorite memories of him are of the times he would take me with him to conferences. Once, on a return flight from a conference in Austin, Texas, we had a chance to talk about mathematics and life. During the flight, we were looking at some relatively standard cloud formations. Ciprian turned to me and explained that Benard convection led to this pattern formation. As we were landing, the gentleman on his right asked us what we did for a living. After Ciprian asked him the same, he was shocked to discover that the gentleman was a horse owner involved in horse racing. Ciprian exclaimed “We could have talked all this time!”

It is difficult for me to say farewell to Ciprian. He was tremendously supportive of all of his students, colleagues, and friends. He was an incredible mathematician and an amazing person who inspired so many of us.

Igor Kukavica is a professor in the department of mathematics at the University of Southern California. His email address is kukavica@usc.edu.

Carl Pearcy

It was my good fortune to first meet Ciprian Foias at the 1966 International Congress of Mathematicians in Moscow, where he seemed to be the most popular participant due to the number of languages in which he was fluent: English, French, German, Romanian, and Russian, at least. Thus, he was almost always occupied translating some speaker’s lecture from one of those languages to another. It was quite funny to listen to one of his translations because the speaker frequently didn’t understand the language being translated into, and Ciprian would from time to time slip in something like “Professor X says Y but he really meant Z,” which, of course, the speaker wouldn’t understand a word of!

I had the good sense to personally visit Ciprian and his lovely family in Bucharest in 1974, during which time I met his father and mother, as well as his charming wife and daughters, Antonia and Dara. Since that time, I have had the great good fortune to be regarded as a member of his family, which has made me very happy, led to almost 50 years of mathematical collaboration with Ciprian, and to many family celebrations with his entire family.

Everyone knows that Ciprian was one of the best mathematicians of his generation, and he is among the few people whose names are attached to specific real numbers. But perhaps those of us who did not know him as well as I did were not aware of what an outstanding man he was. He was always modest, generous to a fault, a real “old-world gentleman,” and spent much of his time helping the careers of others whose mathematical prowess was somewhat less than his own.

He is now, and always will be, sorely missed by everyone who knew him well, and much of his mathematics will be pertinent for a very long time.

Carl Pearcy is a professor emeritus of mathematics at Texas A & M University and University of Michigan. His email address is cpearcy@math.tamu.edu.

Ricardo M. S. Rosa

I met Ciprian in August 1992, when I went to Indiana University to do my PhD with Roger Temam. Ciprian was teaching ordinary differential equations (ODE), but, of course, it was nothing ordinary. It was on Banach spaces, to say the least, with his own set of carefully tailored, minimally written lecture notes. In his words, just the bare minimum to help with the main steps of the proofs. That was a common aspect of his teaching notes. In class, however, he would unravel the notes down to every detail. I was amazed by his style of teaching. He would often sidetrack and give details of the proofs of secondary results, no matter how basic or advanced those were. It was like a jazz player improvising with great mastery, on a song he knew so well. I guess that style helped him keep so sharp on so many subjects. I took many classes with him. Not only because they were directly related to my research, but also because I loved his lectures and his stories.

Figure 7.

Ciprian, Oscar Manley, Ricardo Rosa, and Roger Temam, the four authors of the book [FMRT01] attending a meeting in Orsay, France, in April 2000.

During my PhD, I did not coauthor any work with him, but he did play a fundamental role early on. My first published paper was a direct result of that very first ODE class, on one of a few open problems he would often mention in class. It was only at the end of my PhD program that we began working on things that would eventually lead us to publish together.

It was in a reading course on statistical solutions of the Navier-Stokes equations (NSE), where I learned the essence and motivation for studying statistical solutions. A few years later, Roger Temam invited me to write the “Navier-Stokes Equations and Turbulence” book with the two of them and also Oscar Manley, what we would call the FMRT book. I learned so much from them, and I saw how their different styles could work so well together.

The theory of statistical solutions was one of the many contributions of Ciprian. It was a novel notion of solution associated with the NSE, with a random distribution of initial conditions. The twist is that the 3D NSE are not known to be well-posed. The aim was to develop a rigorous foundation for the study of turbulence, in particular for the notion of ensemble average, introduced in the early works on the conventional theory of turbulence. In a sense, statistical solutions are to ensemble averages what the Leray-Hopf weak solutions are to formal solutions of the Navier-Stokes equations. This became a new field of research, with the foundational works being published by Ciprian in 1972 and 1973, arising from joint research with Giovanni Prodi, a few years earlier. The theory is beautiful in terms of showcasing a range of deep results in analysis, with effective applications in turbulence. In a joint work by Ciprian and Roger, it was used to resolve an inconsistency in a work by Eberhard Hopf on self-similar “turbulent” solutions. It is also worth mentioning his efforts on a conjecture by Prodi, which is a weaker version of the problem of well-posedness for the 3D NSE, namely whether stationary statistical solutions are carried by sets of regular solutions, a sort of asymptotic regularization property.

I would visit Ciprian often. In College Station, we would walk to different coffee shops, to work in the morning and in the afternoon. Those were pleasant and memorable walks! On the way to one of the coffee shops, there was a park with some exercise equipment, and he would always stop there and do some pull-ups. Years passed and he would still do the pull-ups. He liked to show he was still in shape.

He would often start a meeting talking about climbing, politics (like the walking stick he would take with him “just in case,” during one of the elections), or, of course, horse racing. So much so that, while writing this note, I decided to take my kids to a horse race in Rio de Janeiro, on a famous track, with a view to Corcovado. The kids loved the place, and they helped me place a single bet. And we got lucky! Lucky that I’ve got this great opportunity to talk to them about this person that was so special to me. Lucky that we spent a wonderful evening honoring the great person and great mathematician he was.

Ricardo M. S. Rosa is a professor in the department of applied mathematics at the Instituto de Matemátical/Universidade Federal do Rio de Janeiro—IM/UFRJ. His email address is rrosa@im.ufrj.br.

Reminiscence of Ciprian Foias

Jean-Claude Saut

I first met Ciprian at the ICM in Helsinski in 1978, just before his extraordinary escape to Paris. We became close friends from the beginning of our collaboration on the normal form approach of the Navier-Stokes equations with potential forces. This collaboration lasted till our 2018 survey paper (with Luong Hoang) where we tried to explain how Poincaré and Dulac met Navier and Stokes…

As everybody knows, Ciprian was fascinated by the Navier-Stokes equation. At this time, it was usual to publish Compte Rendus notes in the French Academy of Sciences to quickly announce important results (with a limited number of five per year). Jacques-Louis Lions suggested keeping one free in case you solve a famous conjecture (such as the Riemann hypothesis) and Ciprian always kept one in case he solved Navier-Stokes in 3D!

Working with Ciprian in Bloomington was exciting, though a bit exhausting. The working days were productive, but very long. I still keep the many handwritten notes he wrote after our meetings, summarizing what we had done and adding new bright ideas. There were, however, welcome digressions and discussions on various extra mathematics topics; Ciprian had a vast historical knowledge. He also liked literature, especially science fiction, and he tried (unsuccessfully) to convince me to share his taste for it. I also discovered that Ciprian was a fan of horse races and I remember a long evening drive to a horse race field in the next state of Ohio (at that time there were apparently no horse races in Indiana) and his expert comments on the racing horses.

Ciprian was always full of original ideas in all domains, for instance he told me once “imagine what kind of super mathematics could be made in a civilization where the special functions would be as familiar as the trigonometric functions for us!”

Our last physical meeting was at the occasion of my visit to Texas A$\&$M University in 2015, and it was unforgettable. We did not meet at the university, but at the Barnes and Noble bookstore which Ciprian would walk to everyday. He was not far from 80 but still bright and enthusiastic, and we resumed our working sessions as in the old days, leading to our last research paper (also with Luong Hoang).

Ciprian was an extraordinarily kind and generous person, truly concerned about the life of his friends. I remember his relief at Penn State when he saw us (Phil Holmes, Edriss Titi, Gal Berkooz—one of Holmes’s students—and me) arrive very late from Cornell, having flown in a tiny private plane piloted by Gal. The first plane we boarded had a battery problem and a funny event occurred after boarding the second one. Ciprian’s comment on seeing us was “Oh, here you are, I was afraid you were dead!”

We all miss this great mathematician and man.

Jean-Claude Saut is professor emeritus in the département de mathématiques at Université Paris-Sud 11. His email address is jean-claude.saut@universite-paris-saclay.fr.

Allen Tannenbaum

I first met Ciprian when he was visiting the Weizmann Institute in 1981. He and his family had recently come out of Romania and he was settling in what was to become his new home, Bloomington, Indiana. Our close relationship, which began during that visit to Israel, had all the ingredients that continued over many years: mathematics, politics, Romanian culture and history, and some horse racing. Thinking about Ciprian, all of these aspects of his amazing mind and personality somehow have become blended in my own mind, and so this remembrance will reflect this.

Starting with the mathematics, the late 1970s and early 1980s were the beginning of robust control, i.e., the design of feedback systems in the presence of uncertainly. At the time, it was known that Nevanlinna-Pick interpolation was the key methodology for treating the special case of single input/single output systems, and one needed an extension to a version of Nevanlinna-Pick for matrix-valued rational functions. When Ciprian visited the Weizmann Institute, we struck up a friendship and discussed about various issues concerning Israel! In any event, he asked me what I was working on, and since I knew he was an analyst, I mentioned to him the matrix Nevanlinna-Pick problem. Immediately, he wrote the solution, and told me that I really needed to study the Commutant Lifting Theorem! Thus, it was Ciprian who provided the first complete solution to the so-called “multivariate H-infinity problem.”

Figure 8.

Ciprian Foias, Allen Tannenbaum, and Israel Gohberg with a guest.

Our relationship was renewed in Lincoln, Nebraska, at a conference in the fall of 1985. There began a long and very fruitful collaboration that lasted many years and soon involved Hari Bercovici as well, with visits to Bloomington several times a year. Ciprian invited me to the Romanian lunch held every Wednesday at the faculty club of Indiana University. There “we” Romanians would discuss the political issues of the day (mainly about the Soviet Union) under the benign supervision of Nick Spulber, a distinguished professor of economics, whom Ciprian held in the highest regard. Sometime in the late 1980s, Ciprian developed a passion for horse racing. He would carefully study the racing forms every evening, make his picks, and try to develop various mathematical models. Our typical day would be to work quite late on a given problem, and then go to our respective residences. I would work almost the entire night. If I came up with something interesting, I would be excited to show Ciprian the next morning. Many times, Ciprian had other plans, namely, he would want to check the racing results from the previous day, and I learned that I would have to wait until the subject was sufficiently completed before we could get back to work. In 1990, Ciprian spent much of the spring at the Technion. We were working with Hari Bercovici on various extensions of the Commutant Lifting Theorem with an eye to applications in control. Ciprian and I would exchange regular emails with Hari. Hari was “Spiderman,” Ciprian was “Eiger” (Ciprian was an accomplished mountain climber), and I was “Thor.” We proved a “structured” version generalizing the spectral Commutant Lifting Theorem, we had done earlier. It was a wonderful time. Ciprian also met my wife’s parents. Ciprian was an expert on the Romanian language and they would extensively discuss the etymology of various terms in the language. Of course the situation in Romania and the fall of Ceausescu was analyzed for hours as well as the future of Romania. Ciprian was a great patriot, and he loved his native country and feared for its future. In general, in addition to his logical mathematical side, he had a great passion for many things in life ranging from history to chess and of course horse racing.

The last time I saw Ciprian was in 2013 at Texas A&M. As usual, we had a wide-ranging conversation, this time discussing computer vision, the Dirac equation, and the situation in Iraq. In the end, Ciprian was my true mentor. Even though when we started our main collaboration, I was nine years out from my PhD, I learned much of my mathematics from Ciprian Foias of blessed memory. Frankly, he was inspirational, and personally with my family, an amazing friend. He will be sorely missed.

Roger Temam

I first met Ciprian during a CIME School in Varenna (Como, Italy), organized by Giovanni Prodi. The title of the School was “Problems in Nonlinear Analysis.” It took place on August 20–28, 1970, and it was a satellite conference of the ICM meeting held the week after in Nice, France, and to which Ciprian was an invited speaker in the Analysis section. I never met Ciprian before, but I knew of him of course, and because of our common mathematical interests, we immediately became connected.

We had many discussions, and since Ciprian was on sabbatical that year, I invited him to visit me at Université of Paris-Orsay during the fall of 1970. It was the beginning of a long collaboration and a long friendship. I visited Ciprian twice in Bucharest during the following years, then Ciprian was again an invited speaker at the ICM meeting in Helsinki in 1978, in the section of Real and Functional Analysis. After his lecture, Ciprian left the meeting and went to Paris.

Like many other universities, the University of Paris-Orsay offered Ciprian a professorship, but he chose to go to the Indiana University in Bloomington. He stayed about a year and a half in France, while waiting for his family to join him. With the strong support of the French mathematical community, it was understood that President Giscard d’Estaing of France who was supposed to visit Romania would raise the question of allowing Ciprian’s family to join him. It was a terrible disappointment for Ciprian when it appeared that the airplane of the President could not take off due to weather conditions. Finally, things happened, the family was reunited, and they eventually went to Bloomington after some time spent in France.

Figure 9.

Roger Temam and Ciprian discussing in the coffee shop of Borders bookstore in Bloomington, September 2005; picture taken by Ricardo Rosa who was taking part in the discussion.

After Ciprian left Paris, we met a countless number of times in Bloomington, at conferences, and in other places, mostly in the US. Ciprian very much liked to drive, and we had many excursions which took us to spectacular places, alone or with our spouses or with our children sometimes. Ciprian also liked hiking in the mountains and he induced me to enjoy hiking as well. Our first hiking excursion together with a group of colleagues and friends was to reach the top of Longs Peak located in the Rocky Mountain National Park, and one of the highest mountains in Colorado. On another occasion we hiked to the top of Bear Peak in Boulder, Colorado, at 8144 ft.

As I said, a collaboration immediately started, which eventually produced 57 joint articles (some in collaboration with others), a volume in the Memoirs of AMS and three books, one book together with Peter Constantin and Basil Nicolaenko, CFNT89aCFNT89b, one book together with Alp Eden and Basil Nicolaenko, EFNT94 and our last book together with Oscar Manley and Ricardo Rosa, FMRT01. This is not the place to describe the content of these 61 publications, but I can say a few words about the main directions of research that we covered. We started with the statistical solutions of the Navier-Stokes equations, a subject dear to Ciprian and on which he already wrote two major articles together with Giovanni Prodi. I got interested in a new report on the subject in which he was involved (Orsay report 1975). Then, at a time where dynamical systems were a popular subject, we got interested in the attractors for infinite dimensional dynamical systems, and their dimension; this included the incompressible Navier-Stokes equations and, later, the Kuramoto-Sivashinsky equation and many other related equations. We also introduced and studied the concepts of inertial manifolds and inertial sets, studied the Gevrey analyticity of the solutions of the Navier-Stokes equations, and the concepts of determining modes and determining nodes, generalizing a result of Ciprian and Prodi.

These last years we returned to the subject of our first collaboration, and studied the statistical solutions of the Navier-Stokes equations, and their relationship with the attractors, thus leading to the book with Oscar Manley and Ricardo Rosa, and several subsequent articles. Particularly memorable were the interactions with Oscar Manley which led to very vivid discussions, which got hot sometimes, especially in the summers when there was not yet air conditioning in the mathematics building at IU. Oscar Manley was a program director at the Department of Energy, a physicist who had an interest and a remarkable understanding of mathematics. He taught us some physics, and, in this way, we could write papers with a physical insight. Before I got a position in Bloomington, we would meet for about one month in the summer and discuss. Ciprian was very inventive and very hard working. He outpowered all the friends and colleagues that I knew.

Ciprian was very enthusiastic at work. He was also very kind and generous. He had extensive knowledge in many subjects beside mathematics. He was a very dear friend. All his students, friends, and collaborators will miss him very much.

Ciprian Foias, a Brilliant Mathematician, a Caring Friend, and a Family Man

Edriss Titi

When I came to Indiana University to do my PhD in August 1982, I was fortunate to take the graduate course on Partial Differential Equations from Professor Ciprian Foias. I was fascinated by how everyone in the department spoke highly of him, with great respect and admiration. Halfway through the semester, he approached me saying, “I am working on the Navier-Stokes equations, would you like to work with me?” Of course I was very flattered, to say the least, but I had no idea what the Navier-Stokes equations were. Fortunately, early in the semester I had become friends with Peter Constantin, who arrived at the same time to do a postdoc with Foias. When I consulted with Peter, he looked straight at me and said “and what are you waiting for?” I rushed to Ciprian telling him that I was honored and happy to work under his supervision. This was one of the best decisions in my life, and the best advice I have ever received from a friend. Right away Ciprian wanted to give me a problem to work on, but I wanted to take more advanced courses in nonlinear analysis. This worked perfectly well, Ciprian would teach an advanced topics course of mutual interest and I would have to recruit enough students to take it. Ciprian’s topics courses were in the style of College de France, teaching material from his current research interests. I was amazed by his brilliant ideas and attempts to improve his own theorems on the spot. He simply taught us how to think. Ciprian would often ask me what other courses I plan to take to see if there is anything being offered that could feed his hunger for mathematics. One time I told him that I was taking Eric Bedford’s course on Several Complex Variables. He asked Bedford if he could sit in on the course, which Bedford welcomed. Bedford also conducted discussion sessions in which he asked us to present our homework solutions on the blackboard. Ciprian literally considered himself as one of the students. He behaved modestly, and wanted to go to the blackboard to present his solutions with the other students. Of course, Bedford could not hide his amusement by this surprising request.

Figure 10.

Pascal Chossat, Basil Nicolaenko, Edriss Titi, Ciprian, and Oscar Manley, Dynamical Systems workshop at IMA, University of Minnesota, circa 1990.

In 1997–1998, we spent a whole year together at the Los Alamos National Laboratory, while Ciprian was the Stanislaw Ulam visiting scholar at the Center of Nonlinear Studies. Our original research plan was to revisit Ciprian’s seminal work on Statistical Solutions of the Navier-Stokes, but soon the plan was changed and we got excited about another project collaborating with Darryl Holm on the $\alpha$-models of turbulence, joined by two PhD students, Eric Olson and Shannon Wynne. This was one of my most productive and exciting scientific years, and a chance to get to know Ciprian closely. We lived in Santa Fe and carpooled every day to Los Alamos. We would arrive at the office at 8 AM and work continuously until at least 5 PM. We would often be so engaged that we would miss lunch. Ciprian would then offer to split his cookies to keep us going. Ciprian, who was also a mountaineer, was fascinated by the landscape, which we explored while hiking in the Los Alamos and Santa Fe area trails. Each hike, as well as each carpool ride, would be either a stimulating discussion about our ongoing research or Ciprian giving me a lesson in geology. During the weekends, we would meet at one of the coffee shops in Santa Fe to continue our work, but soon the discussion would divert to other topics. Ciprian enjoyed talking about politics, history, religion, science, science fiction, and, of course, horses. We often disagreed about politics. However, Ciprian was amazingly knowledgeable about historical facts, which made it impossible to win an argument with him.

Ciprian was not only a brilliant mathematician who generously shared his passion, ideas, and love of mathematics with others, but he was also a very caring advisor and sincere friend. One time, while I was a student working with him in his office, he got angry about something that I did. Five minutes later, he looked at me and said apologetically “I know I do the same, but it is wrong and you should not do it.” Then, he compassionately shared with me a personal story. He said that once, when he was young, his father shouted at him angrily for doing something wrong. However, when Ciprian protested to his father saying “but father you do the same” his father responded “I know I do the same, but it is wrong and you should not do it.”

I am very fortunate to have had Ciprian as a PhD advisor, teacher, collaborator, colleague, and a sincere friend who generous shared his thoughts, ideas, and advice. Ciprian not only had a great influence on my scientific life and career, but also taught me generosity, scientific integrity, and above all humility. I certainly miss him.

Dan Voiculescu

I began studying mathematics at the University of Bucharest in 1967 and began interacting with Ciprian during my second year as a student. I was doing advanced reading at the time, but this was not matched with research work. Ciprian attracted many young people to study Hilbert space operators. While in my last years as a student, I joined Constantin Apostol and Ciprian in work on quasitriangular operators. This class of operators, introduced by Paul Halmos can be described as compact perturbations of operators for which there exists an orthonormal basis indexed by $\mathbb{N}$ with respect to which their matrix is triangular. We were able to completely clarify which operators have this property by proving that in the absence of $T - \lambda I$ being a Fredholm operator of negative index for some $\lambda \in \mathbb{C}$ the operator $T$ is quasitriangular. The work led to several further results and technical developments on approximation of Hilbert space operators, though it did not have the significance for the invariant subspace problem Paul Halmos had hoped. I learned many things about research and approximation of operators from this collaboration. In particular, after so many years I am still very impressed by the way Ciprian made us stay focused on the problem of understanding this class of operators in the early stages, when the solution was not at all in sight.

I remember a research tip from Ciprian. I was trying to learn advanced representation theory of Lie groups. One day, Ciprian showed me a short paper by Alexander Kirillov on representations of certain infinite dimensional unitary groups and told me that I might also think about such problems which were only beginning to be studied. This paid off, it led me to work on representations of $U(\infty )$. I like to think of this as a lesson that investing one’s research energy can be like picking stocks.

Ciprian was a passionate mountain climber, which I very much admire, though I did not share this passion.

Figure 11.

Ciprian warming up before a discussion; picture taken at College Station by Ricardo Rosa.

Dan Voiculescu is a professor in the department of mathematics at the University of California, Berkeley. His email address is dvv@math.berkeley.edu.