by Margaret A.M. Murray The MIT Press, Cambridge, MA, 2000. 277 pages. $29.95 |

Margaret Murray is Associate Professor of Mathematics at Virginia Tech. Her original field of research was harmonic analysis; in fact she has co-authored a book entitled *Clifford Algebras and Dirac Analysis in Harmonic Analysis*^{1} (by no means her only publication in pure mathematics). She has her Ph.D. from Yale University, so one sees that she has excellent credentials in mathematics. This book proves that she has plenty of credentials in the humanities as well,^{2} not only because it so beautifully and clearly written but also because it is so carefully and painstakingly researched The subject matter is a sociological-historical study of 36 women mathematicians who received their Ph.D.'s between the years 1940 and 1959; Murray studies their lives and careers in an attempt to answer two basic questions, formulated in her *Preface: *"How then do women become mathematicians?" And "How do they find satisfying work and earn respect and remuneration in a field that is largely defined and dominated by men?" Other, subsidiary, questions are raised and studied, for example problems of "Interweaving a Career and a Life" (read "marriage and child-rearing"), Chapter 6; "Family backgrounds and Early Influences" (how do girls and young women cope with the societal/familial attitudes that mathematics is not "women's work?") Chapter 3; and "Dimensions of Personal and Professional Success" (was academic research abandoned for teaching and/or industry as a result of subtle--or not-so-subtle--pressure from the male establishment?) Chapter 8.

Murray has given a synopsis of her book in the *Preface*, which is worth quoting:

Chapter 2 provides an introduction to the women mathematics

Ph.D.'s of the forties and fifties: their background, education and

accomplishments. Special attention is given to a select group of

thirty-six who agreed to participate in intensive oral history inter-

viewing. The remaining [six] chapters are based on the lives and

experiences of the thirty-six interviewees, with a view to under-

standing how women become mathematicians in the leanest times,

when social and cultural force are least supportive of their ambitions.

It is in these times that women rely most heavily on their own

resources and inventiveness to persevere in work the love and care about.

Since 1980, according to Murray's statistics, women have earned 17.6 percent of the mathematics Ph.D.'s awarded in the U. S. During the period she studies in this book, the percentage was much lower, 6.2 percent in fact. (statistical tables are presented in Chapter 1). In our area, transport theory, it is almost zero judging from the attendance at the ICTT meetings--virtually all the women participants are *Italiane,* according to my own recollection. And this raises an interesting point. During my years of living in Italy when I was regularly attending UMI meetings I was struck by the high percentage of women mathematicians presenting papers at these *comvegni.* Italy of course is a country where one thinks of women as filling a more subservient role than in the US, but at least in mathematics circles this seems not to be the case. In reading this book I kept the question in mind "Why?" but I'm not sure I came up with a satisfactory answer.^{3}

In Appendix B she lists the 36 interviewees, giving vital statistics (dates of birth and, in a few cases, deaths; institution granting the Ph.D.; dates of study; and marital history, including number of children borne). Most of the women graduated from what today would be viewed as "elite institutions:" Chicago, Columbia, Pennsylvania, NYU, Stanford, MIT, Cornell, Yale, Harvard and the like. Other graduated from perhaps lesser but still highly respectable schools such as Illinois, Minnesota, Wisconsin, UNC, etc. So Murray is dealing with a group of very talented mathematicians--just to gain admittance to the graduate program in such schools was an intellectual achievement, particularly after 1945 when hordes of returning veterans were studying under the aegis of the GI Bill.

Some of the women making up the study are so famous that everybody will be familiar with their names, for example Cathleen Morawetz and Mary Ellen Rudin.^{ }Both wound up with prestigious professorships, Morawetz at Courant and Rudin at Wisconsin.^{ }However Rudin spent many years waiting for her appointment due to the anti-nepotism policies then in effect at the University of Wisconsin (where her husband was/is a professor of mathematics).^{4} Wisconsin was by no was alone in enforcing such medieval policies. Murray points out, in Chapter 1, that these rules "…were applied prejudicially and often peculiarly against women." Many of the women interviewed for this book suffered from anti-nepotism rules, which were finally done away with under pressure from the civil rights laws of the federal government. Murray describes a particularly blatant case in note 11 to Chapter 3: A tenured associate professor, Josephine Mitchell, was fired by the University of Illinois when she married an untenured assistant professor in the same department.^{5} Another probable victim of nepotism as he famous woman mathematician, Julia Robinson, who might well have been an interviewee, but she died in 1985.^{6 }Other women mathematicians described by Murray as having made "particularly substantial contributions" include Dorothy Maharam Stone, Alesandra Bellow, Domina Eberle Spencer, Josephine Mitchell (*vide supra*), Jane Cronin Scanlon, Esther Sieden, Wanda Szmielev, Marian Boykan-Pour-Els, Mary Bishop Weiss, Vera Pless, and Tilla (Klotz Milnor) Weinstein. Many of these names may be familiar to, especially, our older readers.

The indignities which women mathematicians had not only to suffer but, apparently, to suffer graciously included being barred from teaching while pregnant because "it was considered inappropriate and even obscene for an obviously pregnant woman to hold forth in front of a classroom, especially before a class of men." (Chapter 2).^{7 }And of course there were the continual suggestions or even outright statements that mathematics was not a profession for women. There were those conservative professors and parents who thought that women's place was only in the home and then the "liberals" who granted the women the right to an education and a career, but in "women's work" like stenography or, if teaching, only in the humanities.^{8}

Despite all this, of the 36 women interviewed, 28 did achieve tenure in math departments at one or more colleges or universities in the US, sixteen at major research universities. This is one of the bright spots of an otherwise depressing story. Another is the overt assistance tendered to women graduate students and your Ph.D.'s by such forward-looking advisers as Lipman Bers of Syracuse, NYU and Columbia, Wilhelm Magnus of NYU and Kenneth Wolfson of Rutgers. Murray describes Bers and Magnus as "especially friendly to women" and Wolfson as "forward looking." (Chapter 2)

I suppose another bright spot is that nowadays while anti-woman bias is still rampant it is usually hidden rather than being overt, and until current political imperatives deem otherwise the government *does* oppose it. And the number of women Ph.D.'s in mathematics *has* tripled since the era studied by Murrray. But we still have a long way to come; we need to take a lesson from the Italians. One looks forward to other studies, perhaps by Murray. Several questions raised by her book remain unanswered however:

- What of the women who might have taken mathematics but were discouraged from doing so by parental, teacher, and/or peer pressure?
- How has the situation evolved since 1959? The raw data are in Murray's book, but they need to be interpreted.
- The 36 mathematicians interviewed by Murray were largely chosen because they were successful in mathematics. What held back those who were those not so successful?
- As pointed out earlier, the 36 mathematicians chosen for the study were all highly talented. Was there/is there a place in mathematics for the less talented, or does a women have to be better than men to compete with them in the world of mathematics on an equal footing? Are all mathematicians equal, or are some "more equal than others?"

Raising these questions is in no way intended to downgrade the value of the book. In fact the mark of a good piece of scholarship should perhaps be that it raises more questions than it answers. In any case you should read book for yourself. It is important, well written and well researched (171 items in the bibliography). And then decide for yourself whether our society has really accepted women as full-fledged members and, if not, what you can personally do about it.

**Footnotes**- John E. Gilbert and Margaret A. Murray,
*Clifford Algebras and Dirac Operators in Harmonic Analysis.*Cambridge University Press, 1991. For readers who not be sure of the meaning of harmonic analysis, here is a brief description: One point of view toward harmonic analysis is that it is synonymous with Fourier analysis, the study of how a general time series (i.e. a function of time, be it discrete or continuous) can be written as the superposition of periodic time series. A familiar setting for this Fourier analysis is**R**but it works just as well in a Euclidean space of any dimension and, more abstractly, on a locally compact Abelian group (where the characters of the dual group play the role of the periodic functions used as the building blocks for the representation of a general function on the group. Harmonic analysis is often also viewed as the study of harmonic functions (solutions of Laplace's equation) presumably called "harmonic" because the trigonometric functions used in Fourier series are eigenfunctions of the Laplacian operator. The multivariable real Hardy spaces arise as the space of boundary values (on^{3},**R**^{n}) of certain harmonic functions on the upper half space**R**=_{+}^{n+1}**R**X^{n}**R**(here_{+}**R**denotes the set of nonnegative real numbers). The theory of these Hardy spaces is, in turn, intertwined with the theory of singular integral operators (e.g. Riesz transforms), the same objects familiar to transport theorists in more applied settings (see K.M. Case and P.F. Zweifel,_{+}*Linear Transport Theory*, Addison-Wesley, Reading, MA, 1967). There are generalizations of this type of harmonica analysis to more general domains with non smooth boundaries and more delicate types of singular integral operators; some other parts of Prof. Murray's work have been significant contributions in this area. As an aside, let me also mention that the currently fashionable wavelet theory amounts to a more flexible type of harmonic analysis especially adapted for modern applications exploiting the latest advances in digital computers, and that harmonic analysis plays a crucial role in music theory, but I will not discuss these anymore here. The prototypical Dirac operator arises in the factorization of the Schrödinger equation (a perturbation of the Laplacian) in ordinary quantum mechanics; it is a linear combination of first derivatives with coefficients in a Clifford algebra (the Dirac γ matrices). A more abstract version of these Dirac operators form the main theme of the book of Gilbert-Murray. I thank Prof. Joe Ball of the Virginia Tech mathematics department for supplying much of the information in this footnote. - Murray achieved tenure in the mathematics department at Virginia Tech based on her excellent research record in pure mathematics. However about five years ago she decided to devote herself full-time to the history of mathematics and has now been promoted to full professor on the basis of her work in that area. She may have been motivated towards historical research after teaching a course in the history of mathematics (just I was motivated to do musical research when I started teaching a course in the physics of music). This flexibility is one of the compensations for working at a university in a low-paid (but tenured) position.
- The situation in the U.S. may be changing. In her
*Preface*Murray tells us that "… by the mid-1990's, women were earning approximately one-quarter of the [mathematics] Ph.D.'s awarded each year. If these trends continue, women will rapidly leave behind the token status they have occupied in the mathematical community for over half a century." - Mary Ellen Rudin, a topologist, is married to another famous mathematician, the analyst Walter Rudin, I have frequently heard arguments as to which is a "better" mathematician, although it seems to be a case of comparing apples and oranges.

5. Today the university administration would have another potent tool at its

disposal. The tenured associate professor might easily be charged with sexual

harassment as soon as she started dating the untenured assistant professor.

Plus ça change, plus c'est la même chose!

6. The story goes that when Robinson was awarded a MacArthur fellowship in

1983 an officer of the Foundation attempted to contact her by calling the mathematics department at Berkeley (where her husband was on the faculty). The secretary who answered the phone didn't know who Julia Robinson was; another secretary advised her "That's probably Prof. Robinson's wife."

7. And more often than not women graduate students were required to prepare the

teas before colloquia and to clean up afterwards while the men hobnobbed with the speaker.

8. In case anybody thinks such attitudes are a thing of the past, let her/him attend

a concert of the Vienna Philharmonic and count the number of women in the orchestra. The last time I did this, a year or two ago, the number was zero. The Austrian government has threatened to withdraw its financial support from the orchestra if they don't mend their ways, but so far no mending as far as I know. Interestingly enough, the last time I saw the orchestra play was at a performance of *Die Meistersinger* at the Vienna State Opera in January 1999. The *conductor* was a woman, Simone Young, but I don't think the members of the orchestra were happy about it. In fact, they didn't even watch the conductor. It was a great performance anyway.