Remembering Leon Ehrenpreis: 1930-2010 Hershel Farkas, Takahiro Kawai, Peter Kuchment, Todd Eric Quinto, , Shlomo Sternberg, Daniele C. Struppa, B. Alan Taylor 1 Leon Ehrenpreis: a note on his mathematical work, by Daniele C. Struppa Leon Ehrenpreis passed away on August 16, 2010, at the age of eighty, after a life enriched by his passion for mathematics, music, running, and the scriptures. I first met Leon in 1978, when I was a doctoral student under Carlos Berenstein (himself a student of Leon), at a time when Carlos had invited him to lecture at the complex analysis seminar at the University of Maryland. I had just begun to attempt my first (of many) readings of his work on the Fundamental Principle, [11], and I was in awe of his presence. And yet, I discovered a gentle human being, with a passion for mathematics, and a genuine desire to help the newcomer (me, in that case) to understand its mysteries. What follows in this section is a quick overview of Ehrenpreis' work, mostly biased by my own interest and work on the Fundamental Principle. It is of course impossible to pay full tribute to the complexity of Ehrenpreis' work in such a short note. The subsequent sections are short remembrances, both mathematical and personal, of Leon's life, from some of his friends and collaborators. Leon Ehrenpreis' mathematical career started with his 1953 dissertation [4], which he wrote while doing his doctoral work at Columbia University, under the guidance of Chevalley. The dissertation, whose main results later appeared in [8], already shows some of Ehrenpreis' most characteristic skill in his willingness to tackle bold generalizations, as he extends distribution theory to the case in which Euclidean spaces are replaced by locally compact Hausdorff spaces denumerable at infinity, and the derivations @=@xi are replaced by a countable family of local, closed operators. He then plunged immediately in the topics that would become the main drivers for his Fun- damental Principle. In an impressive series of papers from 1954 to 1960, Ehrenpreis addressed the problem of division in a variety of contexts. Specifically, he investigated in [5] the issue of 0 surjectivity of partial differential operators with constant coefficients in the space DF of distri- butions of finite order, and in the space E 0 of compactly supported distributions. These were the heroic years of the application of the theory of distributions to the study of the solvability of linear constant coefficients partial differential equations, and Ehrenpreis' results joined those of H¨ormander[23], and Malgrange [27], to help develop the modern form of this theory. In [6] Ehrenpreis tackled the case of some special convolution equations, and extended the results of L.Schwartz on exponential solutions to such equations [30], an analysis that he continued later on in [7], where he considered a larger class of spaces. All of this culminated with his striking announcement [9] in 1960 of what became known as the Fundamental Principle of Ehrenpreis-Palamodov (this result was in fact also independently and concurrently discovered by Palamodov, see [28] and the Russian references therein), and whose complete proof appeared in his most important monograph [11]. The Fundamental Principle can be seen as a far reaching generalization of the well known representation theorem for solutions of ordinary, constant coefficients linear differential equa- tions, that we teach in any introductory class on differential equations. In order to state it (at least in a particular case), let us consider the space E of infinitely dif- n ferentiable functions on R , and let P1(D); ··· ;Pr(D) be r differential operators with constant P~ coefficients, whose symbols are the polynomials P1; ··· ;Pn: Let now E denote the space of in- finitely differentiable functions which are solutions of the system P1(D)f = ··· = Pr(D)f = 0. The Fundamental Principle roughly states that it is possible to obtain a topological isomor- phism between the space E P~ and the dual of a space of holomorphic functions satisfying suitable growth conditions on the union of certain varieties Vk, where fVkg is a finite sequence of al- gebraic subvarieties of V = fz : P1(z) = ··· = Pr(z) = 0g: The proof of the theorem is quite complicated, and even decades after Ehrenpreis' original announcement, it is still the subject of analysis and reinterpretations (see for example the chapters dedicated to the Fundamental Principle in [2] and in the third edition of H¨ormander'smonograph on several complex vari- ables [25]). The crux of the proof, however, is the understanding of the nature of the growth conditions that must be satisfied by holomorphic functions on the variety V , for the duality to take place. Such growth conditions must be imposed not just on the functions, but also on suitable derivatives that somehow incorporate the multiplicities related to the polynomials Pj: Maybe the most important consequence of this result is the representation theorem which states that every function f in E P~ can be represented as t X Z f(x) = Qk(x) exp(iz; x) dµk(z) k=1 Vk 0 0 where the Qks are polynomials and µks are bounded measures supported by Vk. It is important to point out that the Fundamental Principle actually extends to rectangular systems of differential equations, and that it holds for a very large class of spaces (what Ehren- preis called Analytically Uniform Spaces), a category which includes Schwartz's distributions, Beurling spaces, holomorphic functions, etc. (when the objects in these spaces are generalized functions, the representation theorem stated above has to be suitably interpreted, of course). One of the common themes in most of Ehrenpreis' work in those years was a sort of philo- sophical principle that entailed that one could extend most structural results which hold for holomorphic functions to spaces of solutions of systems of linear constant coefficient differential equations satisfying suitable conditions. Maybe the most striking such result was his beautiful and elegant proof of the classical Hartogs' theorem on the removability of compact singularities for holomorphic functions. By essentially using the arguments he had developed in the course of his study of division problems, Ehrenpreis demonstrated [10] that the Hartogs' phenomenon is not a peculiarity of holomorphic functions in several variables, but in fact holds for more general spaces of functions which are solutions of systems of differential equations (or even convolution equations), satisfying some specific algebraic properties. The paper is beautiful and elegant, and shows Ehrenpreis' talent 2 for simplicity, and anticipates the kind of algebraic treatment of systems of differential equations that became so important in years to come. The interest of Ehrenpreis in removability of singularity phenomena is also apparent in another beautiful series of papers, [12], [13], [14], and [15], dealing with elegant and surprising variations on the edge-of-the-edge theme. I am obviously unable to touch upon all the important work of Ehrenpreis, but it would be impossible not to mention his interest in the Radon transform, that absorbed much of his mathematical efforts in the last ten or so years. His work on this topic culminated in his second monograph [16], where Ehrenpreis introduces a generalized definition of the Radon transform in terms of very general geometric objects defined on a manifold (what he calls spreads). Just like his previous monograph [11], this more recent work brings a new and invigorating perspective to his subject matter. And like in the case of [11], this work will provide mathematicians with ideas and challenges that should keep them busy for a long time to come. As Ehrenpreis' former student Carlos Berenstein states in his review [1] of [16] "[this is] a book that is worth studying, although mining may be a more appropriate word, as the reader may find the clues to the keys he's searching for to open up subjects that are seemingly unrelated to this book. Thus, one finds at the end that the title is justified." 2 A remembrance from Hershel Farkas Leon Ehrenpreis was one of the leading mathematical figures of the past century. In many areas of mathematics, specifically, differential equations, Fourier analysis, number theory, and geometric analysis his name is a household word. He was a person that could discuss mathe- matical problems from a very wide variety of subfields and directly and indirectly had a great influence and impact on mathematicians and mathematics. He had the ability of getting to the core of a problem and usually the ability to give suggestions for possible solutions which he sometimes left for others, worked jointly with other people toward the solution, or just solved the problem himself. Ehrenpreis' diversity extended way beyond mathematics. He was a pi- anist, a marathon runner, a talmudic scholar and above all a fine and gentle soul. My personal association with Leon Ehrenpreis goes back over 50 years. We began as teacher-student when I was still doing my undergraduate degree. I heard about a famous, brilliant, young mathe- matician who was going to give a course entitled Mathematics and the Talmud and I decided to register for it. After the semester I asked him whether he would be willing to write a letter supporting my application to graduate school and he told me that I would be better off asking a person more familiar with my mathematical abilities. After that I did not have much personal contact with Leon until after I finished my graduate work and our paths crossed again. I was working on problems related to compact Riemann surfaces and we would meet now and then. Leon was also very much interested in this subject (Ehrenpreis conjecture, Schottky problem) and always expressed an interest in what I was doing.