# Bhama Srinivasan

### Quick Info

Born
22 April 1935

Summary
Bhama Srinivasan is an Indian-born mathematician who has worked in England and America and is known for her work in the representation theory of finite groups.

### Biography

Bhama Srinivasan was born and brought up in Madras, (now Chennai) India. Her father, V K Rangaswami (born 15 June 1908 in Srirangam), attended P S High School, Mylapore, and then went to England where he studied for an M.A. at Oxford University. Returning to India, he worked for the Bengal Nagpur Railways from 1933, and in the following year he married Indira (born 31 October 1916 in Allahabad), the daughter of N V Raghavan. Bhama Srinivasan was their first child, with Viji Srinivasan and Prabha Kannan being younger children. In [3] Bhama describes her background:-
I grew up in Madras, India, in a liberal and progressive family where books and education were taken for granted. There were precedents in my family for higher education in the West. My father and uncle had studied at Oxford, and a cousin did brilliantly at Cambridge, where she later became a don at Newnham College. Her brother became a radio astronomer and worked at Cambridge, Stanford, and Sydney. (A by-product of this environment was that I learned English as a child and grew up bilingual.) However, behind the expectations of doing well at school was an unspoken assumption: education for a woman was not intended to lead to a career. Rather, the highest fulfilment for a woman came through marriage and children; her education was intended to help her be an intelligent partner to her husband and well-informed mother to her children. I had a grandfather who was an amateur practitioner of mathematics and I was supposed to have taken after him.
Srinivasan attended an all-girls' High School in Madras and soon mathematics became her favourite subject. She had a good teacher who let her appreciate the beauty of Euclidean geometry and also gave her a good grounding in understanding the concept of proof. After graduating from the High School, she entered a co-educational college in Madras to take a mathematics degree. The course at the College proved a disappointment and failed to inspire her [3]:-
The curriculum was old-fashioned and the textbooks were those that had been used in England at least 30 years earlier.
After graduating with a B.A. she entered the University of Madras to undertake postgraduate studies for a Master's Degree. This soon proved to be a remarkable improvement on her College course and she was able to attend lecture courses on topics of current mathematical interest. It was at this time that she encountered for the first time the ideas of 'modern algebra' as set out by Bartel van der Waerden in his 1930 two-volume masterpiece [3]:-
An important presence in the mathematical scene at Madras was a Jesuit priest, Father Racine, who headed the Mathematics Department at Loyola College. He was acquainted with the latest mathematical developments in Europe. Several of his undergraduate students later went on to do research at the prestigious Tata Institute of Fundamental Research in Bombay. However, Loyola College did not admit women and thus women students were denied the opportunity of studying under and being noticed by Father Racine. The first piece of good luck I had was that Father Racine gave a course on abstract algebra at the University of Madras, using the great text by van der Waerden based on lectures by Emmy Noether. I also had courses on topology and other subjects from two other excellent professors. Thus I was suddenly thrust into the twentieth century, and this was an exciting experience for me. However, I did not have any ambitions to be a researcher in mathematics at this stage, or, for that matter, to pursue any serious career at all.
After the award of an M.Sc. from the University of Madras, Srinivasan married a mechanical engineer. Certainly marriage was totally expected in the culture of the day and when her husband moved to Manchester, England, to undertake practical instruction to improve his skills in mechanical engineering, she went with him. In Manchester there was an exceptional school of mathematics, particularly strong in algebra, at this time. Max Newman was head of the Department and he had appointed two outstanding young algebraists, Sandy Green and Bernhard Neumann. Srinivasan now had an opportunity to continue her mathematical development by registering as a Ph.D. student at Manchester. This she did and worked towards her doctorate advised by Sandy Green. Her husband was fully supportive of her pursuing a career in mathematics and at times he had to argue her case against other family members who were unhappy that she should be anything other than a 'good wife'. Srinivasan was awarded a Ph.D. by the University of Manchester in 1960 after submitting her thesis Problems on Modular Representations of Finite Groups. Also in 1960 she published her first paper On the indecomposable representations of a certain class of groups which appeared in the Proceedings of the London Mathematical Society. She explains in the introduction to the paper what 'certain class' of groups she studied:-
Let G be a finite group. If p is a prime which, divides the order of G, there exists in general an infinite number of indecomposable representations of G over a field of characteristic p. We are concerned here with the indecomposable representations of a group G which is an extension of a p-nilpotent group by a cyclic group of order prime to p.
After the award of her doctorate, Srinivasan remained in England where she was appointed to the University of Keele, Staffordshire. Her second paper, On the modular characters of the special linear group SL(2, pn ), was published in 1964. This paper contained results she had obtained while working for her Ph.D. and carried the following acknowledgement:-
The work which led to this paper was done under the supervision of Dr J A Green, and I would like to thank him for his generous help and constant encouragement.
Srinivasan was awarded a Postdoctoral Fellowship by the National Research Council of Canada which funded her visit to the University of British Columbia in 1965-66. There she met Rimhak Ree and the two had "many valuable and stimulating discussions". While at the University of British Columbia she undertook research for her paper The characters of the finite symplectic group $Sp(4, q)$ which she submitted for publication in September 1966 although it was 1968 before it appeared in print. After spending the year in Canada, she returned to India where she worked at the Ramanujan Institute at the University of Madras. She writes in [3] about the reaction of those around her in India:-
Though I did not experience overt discrimination, it was quite common for people to say to me "Your husband has a good job; why should you work?" or "Aren't you taking away a job from a breadwinner?" and so on. One well-meaning family friend said "It is a pity you don't have children; but isn't it wonderful that you have something to keep you occupied?".
After splitting up with her husband, Srinivasan went to the United States in 1970 when she was appointed as an associate professor at Clark University in Worcester, Massachusetts. An important year for her research was 1976-77 which she spent at Institute for Advanced Study at Princeton. Also in 1977 she became a U.S. citizen. Increasingly she became involved in the mathematical life in the United States, particularly with the Association for Women in Mathematics and the American Mathematical Society. In January 1979 she gave an invited address at the American Mathematical Society Joint Mathematics Meeting in Biloxi, Mississippi. Also in 1979 her book Representations of finite Chevalley groups appeared in the Springer-Verlag Lecture Notes in Mathematics Series. G I Lehrer writes in a review:-
... the subject matter of the book under review is very much alive, dynamic and incomplete. In view of this, the book makes no claim that it presents a complete theory. However the author is to be congratulated for the service which she has provided for those who wish to feel more comfortable with the formidable tools which are being used today in this subject. The book would be suitable for study by advanced graduate students. Although there is no explicit list of unsolved problems, these do abound, and are mentioned in various places throughout the text.
In 1980 Srinivasan moved to Chicago when she was appointed as a professor at the University of Illinois at Chicago. In the following year she became President of the Association for Women in Mathematics serving the Association in this role during 1981-1983. During these years the Association organised its first major international conference, the Noether Symposium at Bryn Mawr College to honour Emmy Noether's 100th birthday. Many years later, in conversation with Lenore Blum, she talked about the years of her presidency of the Association for Women in Mathematics [1]:-
Bhama and I talked more about the Association for Women in Mathematics. She reminded me of the tensions that had begun to surface during the early 1980s: Were we an organization of research mathematicians or did we represent the interests of all women in mathematics, particularly in education? Now that we were not as preoccupied with political issues as in the early years, it seemed we were having an identity crisis! Bhama recalls, "I was concerned about how to balance our various (and sometimes conflicting) constituencies and interests. So I set up a number of new committees [including the Committee on Mathematics Education ... and the Maternity Committee ...] to address these issues and involve many more women in the workings of the Association for Women in Mathematics.
In 1990 Srinivasan was invited to give the prestigious Noether lecture to the Association for Women in Mathematics. Her lecture The invasion of geometry into finite group theory was delivered at Louisville, Kentucky. She served the American Mathematical Society as an editor of the Proceedings of the American Mathematical Society (1983-1987) and as a member of the Society's Editorial Boards Committee (1991-1994). She has also served as an editor of Communications in Algebra (1978-1984), and Mathematical Surveys and Monographs (1991-1993).

We now quote her own description of her research contributions:-
My research is in the area of Representation Theory of Finite Groups. Since the structure of an abstract finite group is often difficult to understand, one tries to represent it by a group of matrices over some field. The theory of finite group representations has had a rich history over the last 100 years .... In this century [20th century] a central figure was Richard Brauer who founded the theory of modular representations of finite groups. In particular I work with finite reductive groups, which are analogues of Lie groups over finite fields. A big breakthrough in the representation theory of finite reductive groups occurred with the work of George Lusztig in the late 1970's and the 1980's. He introduced tools such as l-adic cohomology and intersection cohomology into the theory, which was then changed for ever. My work since the early 1980's, some of it with my colleague Paul Fong, has been the study of l-modular representations of finite classical groups. Our work has led to further work in this direction in Aachen, Kassel and Paris. An exposition of our work appears in a recent research monograph, "Representations of finite reductive groups" by M Cabanes and M Enguehard (Cambridge, 2004). The representation theory of finite classical groups also has rich connections with Combinatorics. Combinatorial objects such as Young tableaux and symmetric functions such as Hall-Littlewood functions arise in a natural way. I am also interested in these symmetric functions.
Although some students find her lectures challenging, many find her an outstanding lecturer. Here are two comments from students who have studied with her at the University of Illinois at Chicago:-
1. She's a very good professor, one of the nicest people I've met at the University of Illinois at Chicago, I think she is somewhat of a legend now ... she really knows math, I would definitely recommend her for any advanced math course.
2. She was one of my favourite teachers, if you do not recognize her for her talents, than you do not know a good mathematician when you see one.
Let us end this biography by quoting Srinivasan's approach to mathematics [2]:-
My own view of mathematics through the years has been that 'truth and beauty are enough' ... Aren't truth and beauty enough? In fact, I have often reminded my students that the best mathematical achievements took place when the question, 'What is it for?' was not asked. However, the beautiful interactions with physics which have been going on in recent years have made an impact on me too. The modular representations of finite groups of Lie type have now been linked to quantum groups by the work of Lusztig, and this is very exciting.

### References (show)

1. L Blum, A Brief History of the Association for Women in Mathematics: The Presidents' Perspectives, Notices Amer. Math. Soc. 38 (7) (1991), 738-774.
2. Notice for Bhama Srinivasan's 1990 Noether Lecture, The Invasion of Geometry into Finite Group Theory, Association for Women in Mathematics (2005).
http://www.awm-math.org/noetherbrochure/Srinivasan90.html
3. B Srinivasan, In Her Own Words, Notices Amer. Math. Soc. 38 (7) (1991), 702-706.