Harold Shapiro's Research Interests

My earliest research interest (still dominant today) was complex analysis, which was the subject of both my Masters' and PhD dissertations. The former, a study of various extremal problems for polynomials, was most noteworthy for the recursive construction of a sequence of polynomials with coefficients restricted to the set {-1,1} with small maximum modulus on the unit circle. Only much later did I learn that Marcel J. E. Golay, an electrical engineer, had somewhat earlier made an equivalent discovery (in a different form, namely pairs of sequences whose autocorrelations are related in a certain way). The polynomials were (somewhat inaccurately) christened "Rudin-Shapiro polynomials" by J.-P. Kahane and R. Salem in their monograph, a designation which seems to have gained general acceptance (a better case could be made for "Golay-Shapiro polynomials"). My Masters' thesis is available online here, and you'll find the part introducing these polynomials here on page 35, page 36 and page 37.

In my doctoral thesis, I studied a class of linear extremal problems problems in Hardy spaces of analytic functions in the unit disk. This work can be seen as a pioneering effort in the area (later to become much more popular) of Banach spaces of analytic functions, studied with the aid of abstract tools (Hahn-Banach theorem, Riesz representation theorem). Here again most of the results I found turned out to have been obtained earlier, in this instance by the Moscow mathematician S. Ya. Khavinson. Moreover, Werner Rogosinski in England (also unaware of Khavinson's prior work, a circumstance in part explainable by the sluggish flow of scientific information from the Soviet Union during the cold war) had made essentially the same discoveries as I, and on the basis of his generous offer to a fledgling mathematician we published our work jointly.

Some Personal Reflections on Mathematics

In March 2003, a conference was held in Santa Barbara, California to coincide with my 75th birthday. The following is a slightly expanded version of my remarks following the banquet for the participants.
       First of all, let me express my warmest thanks to everyone. Especially, the organizers; and to each and every one of you who is attending and participating in this event. Of course, I feel greatly honored that my mathematical work was considered worthy to be the focal point of an international conference. And let me collectively thank tonight's previous speakers for all kind words and benevolent exaggerations spoken on my behalf, your friendship is very much appreciated and reciprocated. Now, in my turn, I would like to share with you some reflections from more than 50 years as a mathematician.
       In preparing, I first thought I would present my remarks in the form of answering a rhetorical question from a hypothetical interviewer: "What have you contributed to mathematics?" But this didn't work, the question seemed pretentious ...it is, after all for others to say what, if anything, I have contributed to mathematics. So, let me turn it around and address instead the question: "What has mathematics contributed to you?".
       The short answer is, I guess "A life." Of course, there are obvious things: a career, an income, a house in a nice suburb of Stockholm. And many friends and acquaintances, wonderful people whom surely I never would have met but for common interest in mathematics. But there is more.
       If this were a lay audience, I would now try to explain with analogies from music and poetry that mathematics has an aesthetic dimension, that one can seek truth and beauty here - and consolation sometimes. But you know all this! So instead, let me concentrate on some more elusive aspects of mathematics, which for want of a better word I will group under the label "universality".
       At the level of knowledge, universality signifies the tendency of mathematical notions to turn up in unexpected places. In the context of our conference, let me cite the (to me very exciting and unexpected) circumstance that quadrature domains, originally discovered in connection with an extremal problem in conformal mapping, turned out to relate to fluid mechanics and inverse problems of gravitation. Moreover, investigations in operator theory involving aspects of subnormal and hyponormal operators have also led, surprisingly, to quadrature domains.
       And, at the social level, universality signifies that shared mathematical interests recognize no barriers of nationality, language, race, gender, or age. I am happy and proud that the circles of my own collaborators transcend all of these divisions. I would even like to think that mathematics here has a nice little lesson for mankind in general. Well, perhaps it is utopian to hope that all strident factions will someday be united by a shared love of complex variables ...but it is a nice thought.
       "The world is too much with us." I remember when I got my doctorate degree in 1952: The cold war was at its height, the Korean war was raging, and so was McCarthyism. A year later America detonated a hydrogen bomb in the Marshall Islands. The pessimism of those times got to me. I felt that mathematics was just a game played with symbols, a beautiful one but somehow gloriously irrelevant to the issues of the day. During most of the 'fifties I did little mathematics. I fell in with a little group of radical social thinkers who published the magazine "Contemporary Issues" (and a German affiliate "Dinge der Zeit") and, under the nom de plume Jules Laurents I wrote articles calling for the cessation of atmospheric testing of nuclear weapons.
       Somehow, by the end of the 'fifties mathematics began to haunt me again. Gradually it became - and remained - a focal point of my life. It didn't seem just a gloriously irrelevant game. Maybe I became a "true believer" - but now somehow I was committed, and never afterwards regretted this choice. Through mathematics I came to feel I am participating in some enterprise that is mysterious and noble and universal . Even if I am just a very tiny part, I am a part, for all that. This, above all, mathematics contributed to me.

At Donald Newman's Retirement Ceremony

Speech at the Donald J. Newman Conference in Philadelphia, March 1996

       I appreciate very much the opportunity to participate in this conference honoring Donald J. Newman, and to say a few words about him and his influence on me.
       I have always regarded Donald as my teacher. Even if my thesis adviser officially was Norman Levinson, to whom I was attracted by his book "Gap and Density Theorems", I never got a problem, nor any help from him. When I told him I was reading that book he simply replied "Oh, don't read that." Levinson was then no longer interested in harmonic analysis, but in ordinary differential equations, which (at that time) I had no interest in. But, I did get a problem - from Donald! It was a typical Newman problem: an inequality to prove something was bounded by Sqrt[n], when the conventional wisdom only sufficed to give the bound Sqrt[n+1]. I became fascinated with this problem, which was the leitmotif for my Master's dissertation in 1951 (with the so-called Rudin-Shapiro polynomials as spinoff) and also stimulated my interest in function theoretic extremal problems, which was the subject of my doctoral dissertation and has remained a lifelong interest. So, to all intents and purposes Donald functioned as my thesis adviser.
       But, he was my teacher in a deeper sense than that! To explain what I mean, I would like to quote a story I enjoyed from Martin Buber's book "Tales of the Hasidim". Rabbi Leib, son of Sarah (he was a famous mystic) attached himself for some time to the house of study of a renowned holy man (Dov Baer, the Maggid of Mezritch, also known as the Great Maggid). "Ah", said a friend later, "you went to hear Torah from the Great Maggid". "No", replied Rabbi Leib, "I did not go to the Maggid to hear Torah, but only to watch how he laced and unlaced his felt boots."
       The meaning of the story (and there are many like it in the lore of mysticism) is, of course, that the Maggid did not have to say Torah ... he was Torah! Also, that the most important things one learnes from a teacher are not this or that fact, but are more in the realm of the intangible: attitudes, a certain style. One of my students expressed this very well in the phrase "the body language of mahematicians". The beginner becomes wiser as he ponders his teacher's remark "In proving this inequality there is no loss of generality in assuming the right hand side equals 1". Later in life, he cannot remember when he learned this idea.
       Donald's "house of study" was, for me, the lunchroom at City College in the years 1947-48. I remember the scene well: he would be seated at the table reading Landau's "Vorlesungen über Zahlentheorie". He didn't know much German, but could understand Landau's crisp prose, carried along by the formulas. From time to time some "disciple" would come by and say something like "I see how to get that estimate", and Donald would reply something like "OK, now try to get it without the log term. To change the metaphor, he was like a chessplayer playing twenty games simultaneously. He knew perfectly the position on each board, and could tell you all the previous moves. And I was taking all this in, understanding almost nothing, and yet learning. He didn't have to speak mathematics to me - he was mathematics.
       So, what did I learn as I watched him opening and closing his copy of Landau? I suppose the most important thing was the obvious joy he derived from all things mathematical. The elation at seeing a new theorem (whether one's own or another's, it didn't matter) or an elegant solution to a problem struck a responsive chord in me. This was far removed from the dispassionate "professionalism" I was later to encounter in graduate school. I came from a background (on my father's side) where Torah was studied in this spirit, but my disposition was not suited to this kind of scholarship. Mathematics was a new world, without bounds...it was already clear that it was so rich that the smallest of its problems could engage one for a lifetime. Mathematics was an art form. To Donald mathematics and music were inseparable, and through him I learned to love classical music.
       The fundamental unit of mathematics, and the measure of all things, was the problem. Most conversations began with "Here's a problem". A problem could be something whose answer was unknown to the poser, or an old chestnut making the rounds; or, a tantalizing special case of a general theorem, it didn't matter. As long as it was a good problem, which meant: it was easy to state, and had some "point", like possessing an elegant or surprising solution, or an air of paradox. Most of what I have learned in mathematics to this day is by way of problems in this sense.
       I would like to mention two very specific features of Donald as a mathematician. One was what I would call his democratic attitude. No one was too humble to be taken seriously (and especially, if he came with a good problem, or a good solution). Pedigree didn't matter, if Donald had any bias at all it was against "establishment" representatives. This also applied to the mathematics itself. To illustrate how precious this was to me, I can contrast it with what was said to me by an established mathematician who came to give a Colloquium talk at M.I.T. when I was a graduate student. In the Common Room, at coffee before his talk, he asked me what I was working on. Flattered that such a person should take note of my existence, I told him my thesis problem. "Oh", he said, "I had no idea anyone is still interested in that sort of thing. That's 1870 mathematics." Those were tough words, but I had my "network" and could withstand them. To Donald 1870 mathematics wasn't bad, although 1770 (Eulerian) mathematics was better.
       A second characteristic of Donald was his concrete approach to mathematics. It was (and still is) fashionable amond the graduate students to generalize everything. So if, say, you had found a cute theorem on polygons and showed it to someone, that person would say something like "Can you generalize this to polyhedra in R^n? I guess what you really are looking for is a theorem on convex sets in topological vector spaces, etc., etc.". Donald would take note of the theorem and say "What does this say in case the polygon is a triangle? What if it's an equilateral triangle?" This, I learned from him and I feel it has stood me in good stead.
       Well, those are some reminiscences of Donald in the years 1947-52. I restricted myself to that period, because afterwards we met infrequently, and also I don't know if I could do justice to his more recent ideas, like probabilistic truth. But thanks, Donald, for all you gave me in those formative years!

For Ambjörn Naeve's Doctoral Dissertation

       One can distinguish two types of scholar which I may call: Guardian of the conventional wisdom, and Dreamer of the impossible dream. I would say roughly that I am the former type and Ambjörn the latter. I am paid to teach the basics, like integration theory according to Rudin's book, and do that as best I can. The other type is much more interesting. These individuals turn up throughout history, often obsessed with a strange idea such as turning base metals into gold, constructing machines that fly, devices that enable us to see distant stars, etc.
       Actually, what is the conventional wisdom, other than the impossible dreams of 300 years ago? Who could have foretold 300 years ago that we would be sitting in a room tonight illuminated by the force of water falling in the north of Sweden, that some people arrived here in flying machines and others in machines rolling in tunnels blasted out of granite by a chemical mixture one can hold in one's hand?
       I remember my first meetings with Ambjörn, it was culture shock to learn his interest in magic, alchemy and like things. Now I don't regard these things as so strange. The alchemists didn't make gold from base metals but ultimately they (and their followers) achieved something far more amazing...modern chemistry. And even some apparent magic is explainable by long chains of logical causal steps.
       Ambjörn long ago developed an interest bordering on obsession for ray congruences and focal figures. Strangely, my own mathematical interests have been focussing there despite a totally different starting point. His dissertation is partly borne along on the thought that this branch of geometry, perhaps just because of its sheer magnificence, has an important role to play in the computational aspects of surfaces, perception, etc. How this is to happen is as yet not obvious, but one could, not long ago, say the same of rooms illuminated by the force of falling water, refrigerators driven by the sun's rays, etc. His faith is great, and it is hard not to share his enthusiasm. Maybe his "worst fears" will be realized and his ideas will succeed so well that they are transformed, in his own time, into conventional wisdom. Then we may see the spectacle of Sing Sing packed to capacity with teknologer taking their exams on focal surfaces. Qui vivra verra!
       I propose a toast, not to Ambjörn alone, but to all those past, present and future dreamers of the impossible dream. And for personal reasons I would include not only those who succeeded, but the others...those whose dream really was impossible, those that fell by the wayside, and those whose names we don't know because their good ideas were stolen or ursurped by those richer, more powerful or more unscrupulous.

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