To the Latimer-Macduffee theorem and beyond!

In this paper we discuss some of the mathematical contributions made by Olga Taussky Todd. We begin with fairly general considerations, commenting on her role as a thesis advisor and giving an overview of her research contributions. Then we consider a specific work of hers, Another look at a matrix of Mark Kac, her last published work, which was written jointly with John Todd. We make a series of observations that explain certain numerical phenomena that they prove in this paper and end with a generalization of their work. © 1998 Elsevier Science Inc. All rights reserved. Kewvords: Matrix; Markov chain; Ehrenfest Urn Model 1. Olga as mathematician and mentor On 13 Apr i l 1996 the ma thema t i c s d e p a r t m e n t at Cal tech sponsored The Olga D a y Celebra t ion , a day o f ta lks in t r ibute to the ma thema t i ca l con t r ibut ions o f Olga Taussky Todd . I was hono red to be one o f the speakers a long with Dick Gross , Bob Gura ln ik , Ken Ribe t and Helene Shapi ro . This art icle is an edi ted account o f the r emarks tha t I made at the Olga D a y Celebra t ion . W h e n I cons idered wha t things I should say in t r ibute to Olga ‘ s work, my init ial inst inct was discuss her work on the L a t i m e r M a c d u f f e e Theorem. This This work is partially supported by the NSF. 0024-3795/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. P I I : S 0 0 2 4 3 7 9 5 ( 9 8 ) 1 0 0 0 6 X 22 P. Hanlon / Linear Algebra and its Applications 280 (1998) 21-37 theorem gives a correspondence between classes of integral matrices (classes under unimodular similarity) and elements of the class groups of algebraic number fields. This result always held a special fascination for Olga. She studied the Latimer-Macduffee Correspondence over an extended period and in a series of papers ([11-17,19,20,23-28,30]) she established a number of its more interesting properties. Moreover, her work on the Latimer-Macduffee Theorem was closely connected to my thesis work and so seemed the ideal topic for this tribute. In preparing for this talk, I had occasion to look through many of O1ga’s publications and I became more aware than before that her work on the Latimer-Macduffee Theorem was only a small piece of her total mathematical contribution. I decided to look beyond the Latimer-Macduffee Theorem to get a more global view of her contribution to the mathematical sciences. Three significant ways in which a mathematician can contribute a lasting legacy are: 1. their influence on the profession; 2. their students; 3. the body of their research. Of these three, probably Olga’s influence on the mathematical profession has been most often cited. Indeed, one cannot underestimate the leadership role she played by achieving great success in a professional climate that was unfriendly, if not hostile, to women. What accounts for Olga’s willingness to persist in a career track littered with gender-related obstacles? No doubt a significant contributing factor was her true devotion to mathematics. She was fascinated by and looked for significance in the most humble mathematical facts. She was particularly taken by numbers. She wore clothes decorated with numbers and wrote poems about numbers. Olga was interested in any mathematical theorem about numbers. Those who met her could not help but be struck and inspired by her devotion to our subject. She was someone who entered the mathematical profession out of pure love for the subject. What contributions did Olga make as a thesis advisor? Certainly she was productive, having an average of almost one student per year graduate during her years at Caltech. Below is a list of her 17 Ph.D. students with their current affiliations. Ed Bender, University of California at San Diego Daniel Davis, Monterey Bay Aquarium Research Institute Lorraine Foster, California State University at Northridge Fergus Gaines, University College Dublin Phil Hanlon, University of Michigan Charles Hobby, University of Alabama Charles Johnson, College of William and Mary Raphael Loewy, The Technion Don Maurer, John Hopkins University Joseph Parker. Optivision, Inc. P. Hanlon I Linear Algebra and its Applications 280 (1998) 21-37 23 Helene Shapiro, Swarthmore College Rober t C. Thompson, University of California at Santa Barbara Frank Uhlig, Auburn University Olga’s advising style was marked by her attentiveness and care. When I worked with Olga, we met at least once a week. She had material prepared to discuss with me at each meeting. It was not always mathematics that was directly related to my research. Many times she had prepared a presentation of a theorem or result which she thought was important to my mathematical training. It is difficult to quantify something like attentiveness to give a concrete measure of how much care Olga took for her students as an advisor. I had an idea a way to tell if her care and attentiveness were as strong as I remember. After graduating from Caltech, I took a post-doc at MIT. In the excitement of a new job and new location, I left Caltech without ever submitting my thesis for publication. Being a thesis advisor myself now, I realize how frustrated I would be if one of my students failed to submit his/her thesis for publication. So, if Olga was as attentive as I remember, this same frustration might show up in the correspondence I had with Olga during the period just following my graduation from Caltech in June of 1981. Indeed, in letters that Olga wrote in the subsequent eighteen months I found the following comments: I would like to know how you spent your summer here, whether you are writing up parts of your thesis . . . . August 1981 How about publishing parts of your thesis! September 1981. I look forward to hearing about your thesis work and how the write-up is progressing . . . . February 1982. I am anxious to discuss your thesis publication with you,. , but I do not dare to advise you April 1982. And finally, when all other pleas seem to be failing on deaf ears, Olga tried a different approach: We send you our very best wishes and hope you will not overwork yourself, even if your results will appear a little later. December 1982. Regrettably, I never did return to my thesis work. Despite this, Olga’s influence has been very much evident in my work. In more than half of my research publications the main result has been about matrices and in most of those, about spectral and algebraic properties of matrices. Matrix Theory was a subject that Olga emphasized in my training and I can directly relate her influence to much of the focus and success of my later research. 24 P. Hanlon / Linear Algebra and its Applications 280 (1998) 21-37 Certainly the most tangible contribution that Olga left us is her outstanding research. Scholarship was an important focus in Olga’s.!ife and she took great pride in her published work. Her first publication “Uber eine Verschiirfung des Hauptidelasatzes fur algebraische Zahlki~rper” (“Concerning a sharpening of the principal ideal theorem for algebraic numbers”) appeared in 1931 in the Journal fur die reine und angewandte Mathematik [10]. Her final publication Another Look at a Matrix of Mark Kac (joint with Jack Todd) appeared in LAA in 1991. In between these two, came over 170 research publications on a wide and diverse set of mathematical topics. The breadth of the work in these papers can be realized by noting that among them, her publications list 17 primary classifications from the current MR classifications scheme: 01 History and Biography 05 Combinatorics 10 Number Theory 12 Field Theory and Polynomials 14 Algebraic Geometry 15 Linear and Multilinear Algebra including matrix theory 16 Associative Rings and Algebras 18 Category Theory and Homological Algebra 20 Group Theory and generalizations 22 Topological Groups, Lie Groups 26 Real Functions 30 Functions of a complex variable 34 Ordinary differential equations 52 Convex and discrete geometry 65 Numerical analysis 68 Computer science 76 Fluid Mechanics Which of these was Olga’s favorite topic? The chart below divides up her publications by primary classification. Much of this information comes directly from the Math Reviews database. However, there are two complications. The first is that the MR Classification Scheme has changed several times since the start of MR. So, for publications that pre-date the current classification, I read through the abstracts and introductions and chose a classification that I thought most was most appropriate. Publications prior to 1940 were even more difficult. For these, I had to refer to the [37,5] to identify the publications. Then I again read the abstracts and introductions and assigned classifications that I felt were appropriate. My thanks to Jane Kister of Mathematical Reviews for her help with this project. I should also note that I have not included book reviews, research problems and reviews for MR (of which there are 177 included in the MR database!). Those who knew Olga well will not be surprised to see that Number Theory and Linear Algebra (which includes Matrix Theory) are by far the largest P. Hanlon / Linear Algebra and its Applications 280 (1998) 21 37 25 slices.