.DATE: 11/24/07 TIME: 21:44:06 Subject: To djlewis@umich.edu: on [Storer's obituary, a related mathematical point, and other observations on the Mathematics Newsletter] Don, A copy of the Continuum, the Department's Newsletter was sent to me. I noticed a number of things. 1. There was an obituary for Tom Storer, my main reason for writing this message. 2. You were not listed among the emeritii, or anywhere else in the list of department people that I could see. 3. You still have Bloch as chair. 4. Milne was listed as an emeritus. bout #1. My paper "The field of definition of function fields and a problem in the reducibility of polynomials in two variables," Illinois Journal of Math. 17, (1973), 128--146 connects historically with various developments of which many have been a part. Tom Storer played an important part in one of them. The paper contains the first version of I call the branch cycle criterion (lemma) for a cover to be defined over Q in terms of a branch cycle description of the cover. The main application in the paper is to proving that if two polynomials f and g are defined over Q are indecomposable, and they have exactly the same ranges over almost all residue class fields, then f(ax+b)=g(x) for some constants a and b. (In all but obvious cases, a and b are also in Q.) A second result is that if you change Q to a number field K, and you make a group theory assumption -- I proved that assumption as a consequence of the classification of finite simple groups in a section of my Schinzel volume paper -- then there are exceptional degrees to the statement over Q, but we know exactly what they are (appeared in the Schinzel volume paper). Many people have revisited this territory -- most profoundly Peter Muller -- and everyone who has, has marveled that the first result is pure arithmetic. The Branch Cycle Lemma is now very well known, and well-used. Still, to do that result I had to give a good reason why f and g satisfying the "range" hypothesis were not defined over Q. After a week of conversations with Storer, he showed in general what I was able to only prove by example: That the polynomials were not defined over the reals. This was a cyclotomy application. That is, my paper shows the situation produces a difference set of a doubly transitive group with an n-cycle. The result I needed was that -1 was not a multiplier of the difference set. Also, the second result had a certain profundity to it. From it I formulated a conjecture that was a strong version of what became known as the Genus 0 problem. John (Thompson) formulated that when I told him what came of this paper. In the end my original guess turned out to be correct, but that was many years later. That is, the monodromy group of an indecomposable degree n rational function x -> f(x) (over the complexes) is always closely related to A_n, or S_n or a cycle or dihedral group, with only finitely many exceptional n, of which most of the interesting exceptional cases are those that came up in the Davenport problem above. The groups in those case were projective linear groups. Here is why I'm writing. First it is probably surprising that Storer and I communicated. I knew from Whiteman his expertise in cyclotomy. This happened when I visited Ann Arbor one summer at the end of the time I was the first time at the Institute. It would have been nicer to have seen something about Storer's precise mathematical contributions in the obituary. What was said about him was not a great representation of this guy who was full of considerable anguish toward mathematics, though he did make precise contributions. He had an interesting philosophy that I couldn't have agreed with less. Second: My paper was written long before there were electronic tools. So, I don't have an electronic copy on my web site. How can I get an electronic copy of it from the Illinois Journal? Do you know someone who would know? Happy holidays, Mike djlewis@umich.edu Subject: Re: Storer's obituary, a related mathematical point, and other observations on the Mathematics Newsletter Date: November 26, 2007 2:23:47 PM MST To: mfri4@aol.com Mike, For some reason, the editor chose to use the obituary for Tom writen by a daughter, with some small remarks about math. Why don't you write a note on Tom"s contribution to math and his philosophy to be included in the next ContinuUM? As for an electronic version of your Illinois paper, I would check with the UM Science Library to see if the Illinois Jnl has been Googleized. In time the entire UM collection will be digitized in collaboration with Google. Also, You can probably scan the paper into your computer and that way get an electronic copy. It was good to hear from you. I hope all is going well for you. Carolyn is not doing well--in continue pain. I finally got her to go see a specialist in post-polio syndrome. There is no known cure for it, but a specialist can usually come up with ways to alleviate the pain. Her internal medicine man said it was a wast of time to see a polio specialist. To my surprise I have heard from three of my old students for the first time in a decade, I hope that you have a great and healthy new year. Don From: llscott@cstone.net Subject: Feit memory? Date: September 21, 2004 8:20:51 AM MDT To: mfried@uci.edu Cc: lls2l@virginia.edu, llscott@cstone.net I need your help in saying a few words about Walter Feit's Galois theory interests. Although I am organizing a Notices Memorial Article on Walter, I do not intend any subarticle on Galois Theory per se, since there is so much space that needs to be devoted to Walter's earlier work, especially the Odd Order paper. But I do not want to omit Galois theory entirely. If there is some aspect of Walter's work or contribution to the area, or some aspect of your personal interaction with him, that stands out in your memory, then I would appreciate your writing it down. I would quote from your remarks with attribution. To get started, do you think it is true that you were the one who got Walter interested in Galois theory? Or was it the fact that Thompson had become interested in it? Best, Len Dear Len, My relation with Walter came through the intercession of Jim Ax, at the beginning of my second year out of graduate school when I was at the Institute for Advanced Study. Ax knew Walter from Cornell, something you probably knew well. Instead of going to U. of Chicago with tenure in 1969, I went to Stony Brook. That was a bad decision. Yet, the mathematical topics that came from Ax's putting us together had a serious life that involved John, Walter and I repeatedly in ways the public wouldn't know. The mathematical problem was Davenport's Problem and the appearance of symmetric block designs. The precise topic had two parts: 1. That polynomial covers with monodromy groups having such designs have limited degrees. 2. That NONE of those covers could have definition field Q. Item #1 was part of my Santa Cruz talk in the late 1970s, and also of Walter's. The effect of Santa Cruz underlies much of the rest of this message, and still continuing mathematical developments. Item #2was done with pure arithmetic and was the opening salvo in the braid group approach to the Inverse Galois problem. The start was through a simple, though everywhere useful, result I called the Branch Cycle Lemma. So, item #2 needed no classification results, and it solved Davenport's problem in its original form over Q. Item #1, however, required the classification of doubly transitive groups. That eventually gave the conjecture I made that 31 was the upper bound for the degrees in #1. The conclusion from that was that Davenport's problem was false over a general number field, though in a bounded way. (I phrased it to Ax as akin to the Ax-Kochen solution of Artin's Conjecture.) Walter knew both #1 and #2. Between us, the conjecture for #1 was established. #2 was surprising because it took nothing from the classification. It was a short surprising argument at the end of my Illinois journal paper in the early 1970s. Walter was among the first to use the Branch Cycle Lemma in practical cases of the IGP. John Thompson's role: John learned of my early 1970s paper on the braid approach to the Inverse Galois Problem from Walter. That started a correspondence between John and me when I was on a Fulbright to Finland in the early 1980s. That led to my being at Florida for three years with John. A most significant influence on John's papers was from The Davenport problem and related topics. What caused John to consider the genus 0 problem happened during a walk to lunch one day early in my time at Florida. So, #1 had everything to do with the genus 0 problem (on monodromy groups of genus 0 covers). I documented some of this in the paper I wrote for John's birthday volume, with some statements on Walter's role. So, Walter's influence there really went though a separate stage between John and myself. What I've just done is to give an offhand outline of how these problems involved John, Walter and myself. Bob Guralnick's role in this happened because John wanted me to immediately start working with him on his final formulation of the genus 0 problem. I said it was not the kind of theoretical work I liked best. Also, Bob knew so much more about using the classification than I. The title of my paper on John is: "The relation between two genus 0 problems of John Thompson." The other genus 0 problem was about those genus 0 (nearly) modular curves that relate to the Monster. The paper shows there is a relation between them in which the Monster type topic has a practical generalization: Santa Cruz again. Walter makes a significant appearance in that paper. Since these are offhand remarks, I could probably do better with thought. Naturally you will get a different picture of the genus 0 problem from Bob Guralnick. Bob, too, makes significant appearances in my papers reminding of the source of problems. Thank you for asking, Mike Tuesday, September 21, 2004 Dear Mike, Much, much thanks for your remarks. Can you give me a more precise reference for the Thompson volume article? I couldn't spot it on MathSciNet. I would like to have a look and get back with you. (Maybe you have a web site preprint version?) Your comments suggest there is more here than I had thought before. Best, Len Dear Len, I refer to the paper as Thompson-genus0, and it is on my web site. We are supposed to get reprints in October. I'll attach a pdf file with this message. Of course, this paper was about John. So, \S7.2 (p. 20-25) is about my mathematical relation to John -- outside the Inverse Galois Problem. You also see Peter Mueller and Bob Guralnick there, and something fairly grand, the relation of Serre's Open Image Theorem (this is probably Serre's most famous theorem to number theorists) to this whole set of ideas. Walter appears in \S 2.3, though less intimately. This paper is companion to another paper called "Extension of constants series and towers of exceptional covers." This was the ambitious paper. Walter and a generalization of Serre's Open Image Theorem appear there, too. Sincerely, Mike SUMMARY: You did give me a chance to explain what is going on with applications that use some serious group theory. Like typical geometry related people from Harvard, applications that require such group theory aren't yet a part of your thinking. Not much I can do about that, except to point out my expertise there -- as I did when Nigel Boston's student was asking questions at the Oberwolfach last May. I don't really like giving the anecdote I'm about to give. Still, I want you to know that people at Harvard once knew about serious old applications. When I was 26 I lectured on my proof of the Schur Conjecture -- complete description of the polynomial maps f over a number field K for which f gives a one-one map on infinitely many residue classes of K. Irving Kaplansky was the advisor of Robert MacRae, who was an assistant professor at Univ. of Mich. when I was a graduate student. MacRae wrote up two sections of my thesis for publication (with my name on it) and when later I (two years at the IAS) solved the Schur Conjecture he told Kaplansky about me. There was almost no audience when I gave my talk at University of Chicago, two guys -- one clearly quite old -- in the far back of what looked like bleachers, and Paul Sally, Kaplansky and a couple of their graduate students in the front, with a blizzard going on outside. I was disheartened by the small audience, and gave a not-so-great talk. Still, when my talk was done, the guys in the back came down to talk to me, with the older man doing all the talking. He said: "I was very impressed. You probably didn't know it, but I wrote papers on the Schur Conjecture, and Schur was my advisor. I'll see you at the Institute next week." I said I didn't know that, thanked him for his enthusiasm, and responded I would look for him at the Institute. When he left, I turned to Paul Sally and asked: "Who was that kindly old gentleman?" Sally's response: "That was Richard Brauer!" The next week Professor Brauer offered me -- at 26 -- tenure at University of Chicago, and here's what he said. There was a tradition once where group theorists knew about the serious applications of group theory. You aren't really a group theorist but you appreciate and know how to use groups. It is a great tradition, and I hope you keep it up. Well, I have, and it has worked, and yet I've still made no dent on the great gap between the technical world of group theorists, and the technical world of geometers, even though I understand them both.