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Selected Papers of V. S. Varadarajan: Volumes 2 and 3

Donald G. Babbitt, Ramesh Gangolli, K. R. Parthasarathy, Enrico G. Beltrametti, Gianni Cassinell, Rita Fioresi, and Anatoly N. Kochubeii
Hindustan Book Agency
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[Reviewed by
Michael Berg
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It is an honor for me to be able to write this review of the Selected Works of Veeravalli S. Varadarajan, long-time professor at UCLA, where I did my undergraduate work in the 1970s. It was my great good fortune to take a number of formal courses from Professor Varadarajan early on, and to be entirely captivated by his way of doing mathematics in front of our very eyes. I don’t think Varadarajan would mind my saying that his presentation of the material in question, with a dominant emphasis on deep connections and architecture, was not everyone’s cup of tea, given that most instructors’ pedagogical styles at that level are, to use Freeman Dyson’s marvelous characterization, tailored to frogs. By contrast, Varadarajan is a bird, and I was immediately captivated. To explain this metaphor, suffice it to say, à la Dyson, that frogs are mathematicians who love to explore their own chosen lily pads with infinite patience and love, while birds are mathematicians who prefer to soar over mathematical landscapes and delineate paths, bridges, and connecting ranges between different territories, indeed between different mathematical geologies. So it was, for example, that Varadarajan’s approach to undergraduate differential equations was very unusual indeed: he used an idiosyncratic text by Arnol’d (who of course never wrote anything that wasn’t idiosyncratic), and proposed a (for us) very unusual way of looking at it all, with vector fields and flows staring everything off with gusto. In other words, to my fledgling eyes (pun certainly intended), it was a psychological homecoming of sorts, to see this master in action, and I quickly gravitated to him in the hope of learning marvels couched in his way of thinking. Happily I was given the opportunity to do a lot of work with him.

There are three specific examples of Varadarajan’s mathematical style that I want to discuss, briefly, and with which I have had personal experience, so as to form an introduction of sorts to his Selecta. First, when I felt I needed more work in analysis (for which I had had very little ambition before, resulting in spotty performances) and I approached Varadarajan for a reading course, he not only agreed, but completely changed my life. Warning me that what he was about to propose was a bit on the austere side, he suggested that if I was willing to work hard, he would supervise me in a reading course using André Weil’s brand new Elliptic Functions According to Eisenstein and Kronecker. Well, I seized the opportunity and what ensued was what, for me, became the single most formative mathematical episode of my undergraduate life. Every week (late Wednesday mornings, as I recall) I was to meet with Varadarajan in his office to discuss my reading, which was defined as including working out everything that I didn’t immediately understand and doing every implicit exercise that Weil hinted at: “it is easily seen that …,” “it can be shown that …,” and so on.

I well recall doing hours’ worth of serious mathematical labor to get more light unto the matter, all the time being supervised by Varadarajan. His weekly scrutiny of my work was demoralizing only insofar as he did the right thing: it wasn’t that me proofs and sketches were wrong, just that they were generally myopic or wrongheaded, too froggy, not sufficiently avian. Of course he also corrected my mistakes, but it was always the case that what he showed to me as the right approach pointed toward an overarching structure into which what Weil was saying could be fitted. I was enthralled, for example, to find out that the doubly periodic functions Eisenstein proposed in such an unusual way, with what Weil called “Eisenstein summation” front and center, and which were attached to period parallelograms in the complex plane, were natural habitués of tori and in due course algebraic topologically meaningful invariants crept up out of the foliage. Varadarajan’s presentation was magical to me, and left me wanting much more.

Well, the next opportunity to arise (now that VSV had started me on a path toward a different kind of number theory than what I had seen previously: more algebra and topology, less combinatorics, to oversimplify things a bit) was a graduate seminar of his on class field theory. This was a tour de force of mathematical teaching at a very high level: the course was ostensibly aimed at advanced graduate students (of which I certainly was not one), but was also attended by a number of other very senior mathematicians, the late Robert Steinberg among them. A funny episode arose in this context: at one point in the proceedings Steinberg interrupted Varadarajan with a question about something he didn’t understand, and a discussion started up trying to get Steinberg to see it; eventually someone said something like “it’s categorical,” Varadarajan seconded that, and Steinberg immediately withdrew his objections. I guess this says something about what VSV was doing along the lines of what, certainly to me, was enchanting and radical — something like the French approach to corps des classes. And, indeed, Chevalley’s idèles and Weil’s adèles made a lot of appearances. I have kept Varadarajan’s handouts all these years: they make up a textbook in themselves.

For my third example of Varadarajan’s approach, let me recount what happened in the middle 1990s, when I did my first sabbatical with him. I had by this time developed a great (and still abiding) interest in the analytic approach to higher reciprocity laws and, sure enough, when I brought this to VSV’s attention he pointed me toward the question of generalizing the Gaussian kernel along lines suggested by Laumon in an algebraic geometric context. This is indeed one of the deepest open questions in this part of mathematics, i.e. the interface between Fourier analysis and number theory (with Hecke, Weil, Kubota figuring prominently), and is in fact intimately connected to any number of other things, e.g. the difficult business of functional and phase integrals that arise also in connections with quantum physics. Not only did Varadarajan suggest a fascinating new approach, but his suggestions pointed toward exploring the method of stationary phase, which is to be sure, central to the Feynman approach to quantum field theory. Varadarajan has very deep insights, as well as great breadth.

And this takes me to the Selecta. Varadarajan’s remarkably wide sweep is already clear from the latter two books’ subtitles: Part II is devoted to differential equations and representation theory, while Part III concerns physics, analysis and probability, and contains “reflections and reviews.” Part I, published earlier by the AMS (Parts II and III are just out, from the Hindustan Book Agency), presents a spectrum of articles, dealing with e.g. measures on topological spaces, Gaussian processes, Lie groups and Lie algebras (recall that VSV is the author of the seminal Lie Groups, Lie Algebras, and Their Representations), oscillatory integrals, differential equations, and quantum physics, including revolutionary work on non-archimedean Quantum Mechanics.

The papers are of course very well-crafted: Varadarajan is a famously clear and deep expositor, and very thorough. I would suggest that reading any of them, including the survey parts, suffices to inspire the reader to go at VSVs books, including, besides his wonderful text of Lie theory mentioned above, the early book on quantum theory, Geometry of Quantum Theory. Perhaps it is right to say that these two books encapsulate the view of mathematics and physics embraced by Varadarajan, in that his work is characterized by a focus on very deep themes and, at the same time, a very creative and careful sculpting of methodologies to address the questions he is driving at in his investigations.

The Introduction to Part I, by Varadarajan himself, notes that it was Yau and Cheng who suggested that a Selecta of his papers be published. He goes on to note that he owes his own mathematical personality, at least to a degree, to, among others, Valentine Bargmann (an Einstein connection), Pierre Deligne (a representative par excellence of the modern French school, although Deligne would probably object, in the style of Hercule Poirot, that he is Belgian), Harish-Chandra (where Lie and Dirac meet, so to speak), George Mackey (who wrote so many wonderful things on representation theory as well as quantum mechanics), and the aforementioned Robert Steinberg. About the latter, I once came to Varadarajan for a chat when I was still at UCLA, and he asked my about the courses I was signed up for. When he found out that I had not signed up for Steinberg’s course on algebraic number theory, Varadajan chided me immediately: as I recall he said that Steinberg was a great mathematician and I shouldn’t pass up the opportunity to take his course. So I did, or, as I recall, I at least attended it. Of course Varadarajan was right: it was fantastic.

Well, to close this long review, again, it is my great honor to be able to review these works by this wonderful and deep scholar to whom I owe so very much. His work, as presented in these three books, speaks for itself, both as regards its breadth and its depth. It is of the highest quality.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Contents for Volume 2: Differential Equations and Representation Theory
Local Reduction Theory and Moduli

  • Donald G. Babbitt, Varadarajan and Differential Equations
  • with Donald G. Babbitt, Formal reduction theory of meromorphic differential equations: A group theoretic view, Pacific J. Math. 109 (1983), 1-80
  • with Donald G. Babbitt, Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations, Mem. Amer. Math. Soc. 55 (1985), 1-147
  • with Donald G. Babbitt, Some remarks on the asymptotic existence theorem for meromorphic differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 247-262
  • with Donald G. Babbitt, Local moduli for meromorphic differential equations, Astérisque No. 169-170, Société Mathématique de France, (1989), 1-217
  • Letter from Pierre Deligne, Documents Mathematiques 5 (2007), 47-51

Representation Theory

  • V. S Varadarajan and Representation Theory, by Ramesh Gangolli
  • with Johan A. C. Kolk, On the transverse symbol of vectorial distributions and some applications to harmonic analysis, Indag. Math. 7 (1996), 67-96
  • Harish-Chandra, his work, and its legacy, in The Mathematical Legacy of Harish-Chandra (Robert S. Doran and V. S. Varadarajan, Editors), Proc. Sympos. Pure Math. 68, Amer. Math. Soc. 2000, 1-36
  • George Mackey and his work on representation theory and foundations of physics, in Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey (Robert S. Doran, Calvin C. Moore, and Robert J. Zimmer, Editors), Contemp. Math. 449, Amer. Math. Soc., 2008, 417-446
  • with Rahul N. Fernandez, Matrix Airy functions for compact Lie groups, Internat. J. Math. 20 (2009), 945-977
  • with Jonathan Rogawski, Some comments on Weyl's complete reducibility theorem, Pacific J. Math. 260 (2012), 687-694

Contents for Volume 3: Physics/Analysis/Reviews and Reflections
Foundations of Quantum Theory

  • Enrico G. Beltrametti and Gianni Cassinelli, Varadarajan and the Department of Physics of the University of Genoa
  • with P. Truini, Quantization of reductive Lie algebras: Construction and universality, Rev. Math. Phys. 5 (1993), 363-415
  • Quantum theory and geometry: Sixty years after von Neumann, Internat. J. Theoret. Phys. 32 (1993), 1815-1834
  • with Richard Arens, On the concept of Einstein-Podolsky-Rosen states and their structure, J. Math. Phys. 41 (2000), 638-651
  • with E. G. Beltrametti and S. Bugajski, Extensions of convexity models, J. Math. Phys. 41 (2000), 2500-2514
  • Vector bundles and connections in physics and mathematics: Some historical remarks, in A Tribute to C. S. Seshadri (V. Laksmibai, V. Balaji, V. B. Mehta, K. R. Nagarajan, and K. Paranjpe, Editors), Hindustan Book Agency, 2003, 502-541


  • V. S. Varadarajan and Supersymmetry by Rita Fioresi
  • with R. D'Auria, S. Ferrara, and M. A. Lledó, Spinor algebras, J. Geom. Phys. 40 (2001), 101-129
  • with C. Carmeli, G. Cassinelli, and A. Toigo, Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 263 (2006), 217-258
  • with C. Carmeli, G. Cassinelli, and A. Toigo, Erratum to: Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 307 (2011), 565-566
  • with R. Fioresi and M. A. Lledó, The Minkowski and conformal superspaces, J. Math. Phys. 48 (2007), 1-27
  • Introduction, in Supersymmetry in Mathematics and Physics, Lecture Notes in Math. 2027, Springer, 2011, 1-15

P-adic Physics

  • V. S. Varadarajan and p-Adic mathematical physics, by Anatoly N. Kochubei
  • Non-archimedean models for space-time, Modern Phys. Lett. A16 (2001), 387-395
  • with Trond Digernes and D. E. Weisbart, Schrödinger operators on local fields: Self-adjointness and path integral representations for propagators, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), 495-512
  • with Rahul N. Fernandez and David Weisbart, Airy functions over local fields, Lett. Math. Phys. 88 (2009), 187-206
  • Multipliers for the symmetry groups of p-adic spacetime, p-Adic Numbers Ultrametric Anal. Appl. 1 (2009), 69-78
  • with Jukka Virtanen, Structure, classification, and conformal symmetry, of elementary particles over non-Archimedean space-time, p-Adic Numbers Ultrametric Anal. Appl. 2 (2010), 157-174
  • Some problems involving Airy functions, Commun. Stoch. Anal. 6 (2012), 65-68


  • V. S. Varadarajan, The papers in Analysis/Probability
  • with J. J. Duistermaat and J. A. C. Kolk, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math. 49 (1983), 309-398
  • The method of stationary phase and applications to geometry and analysis on Lie groups, in Algebraic and Analytic Methods in Representation Theory, Perspect. Math. 17, Academic Press, 1997, 167-242
  • Some remarks on the analytic proof of the prime number theorem, Nieuw. Arch. Wisk. (4) 16 (1998), 153-160
  • Some remarks on the continuity of random fields on Lipschitz spaces, Expo. Math. 18 (2000), 177-196
  • with David Weisbart, Convergence of quantum systems on grids, J. Math. Anal. Appl. 336 (2007), 608-624.
  • Euler and his work on infinite series, Bull. Amer. Math. Soc. 44 (2007), 515-539
  • with Trond. Digernes, Notes on Euler's work on divergent factorial series and their associated continued fractions, Indian J. Pure Appl. Math. 41 (2010), 39-66

Reviews and Reflections

  • Henry Abel Dye, Pacific J. Math. 137 (1989), iii-v
  • Richard Friederich Arens (1919-2000), Pacific J. Math. 195 (2000), 259-260
  • Harish-Chandra memorial talk, in The Mathematical Legacy of Harish-Chandra, Proc. Sympos. Pure Math. 68 (Robert S. Doran and V. S. Varadarajan, Editors), Amer. Math. Soc., 2000, 55-57
  • Niels Vigand Pedersen (1949-1996), Notices Amer. Math. Soc. 45 (1998), 74-78
  • Moshe Flato Remembered (1937-1998), in Reflections on Quanta, Symmetries, and Supersymmetries, Springer, 2011, 53-55
  • Book Review: Andri Weil: The apprenticeship of a mathematician [Birkhuser, Basel, 1992]. Notices Amer. Math. Soc. 46 (1999), 448-456
  • Some mathematical reminiscences, Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. Methods Appl. Anal. 9 (2002), no. 3, v-xviii
  • Mackey, Harish-Chandra, and Representation theory, in Reflections on quanta, symmetries, and supersymmetries, Springer, 2011, 207-236