Archives of American Mathematics

Photo Caption: M Atiyah 29 Mar 69 

“The Atiyah-Singer index theorem was the toughest hurdle for me, but, somehow, we conquered it too. (To be sure, after it appeared in print, Singer told me that it didn’t come out quite right—the relation with the Riemann-Roch theorem was unclear or perhaps even misstated—but there it was, and I feel sure that my fellow ignoramuses and I learned something worth knowing that we hadn’t known before.)”–Paul R. Halmos, I Want to Be a Mathematician

Michael Francis Atiyah contributed to a wide range of topics in mathematics centering on the interaction between geometry and analysis. His work showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory. He was awarded the Fields Medal in 1966.

The ideas which led to Atiyah being awarded a Fields Medal were later seen to be relevant to gauge theories of elementary particles.

The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing. 

In addition to the Fields Medal, Atiyah received many honors during his career including the Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Benjamin Franklin Medal, and the Nehru Medal. In 2004, he and Isadore Singer were awarded the Neils Abel prize of £480 000 for their work on the Atiyah-Singer Index Theorem.

Michael Francis Atiyah Biography

Photo Caption: Anneli Lax 

“Anneli is an outstanding mathematical editor (look at any volume of the New Mathematical Library).” —Paul R. Halmos, I Have a Photographic Memory

Anneli Lax Cahn’s greatest contribution to mathematical literature was triggered by a very different sort of event. The launch of the Soviet satellite Sputnik in 1957 was a shock to the American scientific community, a shock felt at every level. Much thought was devoted to the education of a new generation who would accelerate the pace of American scientific productivity. It was at this point that Lax realized the major contribution that could be made in mathematics education. Out of this endeavor grew the New Mathematical Library. The idea was to make accessible to interested high school students deep results in mathematics described by research mathematicians.

Anneli Lax Cahn

Photo Caption: Mary Cartwright 3 June 68 Cambridge

“She became Mistress of Girton College (Cambridge), and, later, Dame Mary; she is just an outstanding complex analyst.” —Paul R. Halmos, I Have a Photographic Memory

Dame Mary Cartwright (1900-1998)
Mary Lucy Cartwright was the first woman mathematician elected to the Royal Society of London. While at Cambridge University, under the supervision of G.H. Hardy and E.C. Titschmarsh, her thesis on zeros of integral functions generated a series of papers and eventually led to her book on integral functions. Although she did important work with Dirichlet series, Abel summation, analytic functions regular on the unit circle, integral functions, and cluster sets, she is best known for her work with Littlewood on van der Pol’s equation and nonlinear oscillators. Cartwright served as Mistress of Girton College and as president of the British Mathematical Association and the London Mathematical Society. She was a recipient of the Sylvester Medal from the Royal Society and the De Morgan Medal from the London Mathematical Society. She authored nearly 100 articles and books. She was a very effective administrator at Cambridge University and ambassador for several mathematical and scientific organizations. In 1969, Queen Elizabeth II elevated her to Dame Mary Cartwright, the female equivalent of a knighthood.

Mary Cartwright

Photo Caption: McLaughlin, Ann Arbor, 1963 

“Jack is a hard-working and broadly informed algebraist, but the sort of algebra he writes papers about is not the sort I am fond of. Example: he discovered one of the notorious sporadic simple groups. I am a hard-working and quite knowledgeable operator-theorist, and the kind of operator theories that interest me leave Jack completely cold. It turned out, however, that there is a part of mathematics we both know and like. It is a small subject, not considered deep. It has, they say, no intrinsic importance; it is merely a useful tool and an occasional source of examples in other subjects. Both Jack and I tend to be shamefaced about admitting we like it; it is a little like admitting that you read westerns. The subject I am talking about is linear algebra.” —Paul R. Halmos, I Want to Be a Mathematician. 

Jack McLaughlin’s research ranged widely, encompassing several subfields of algebra—lattice theory, finite groups, and commutative algebra. He discovered one of the sporadic finite simple groups, that of order 898,128,000, which now bears his name. He also participated in the discovery of a module of finite projective dimension with a negative intersection multiplicity. His work on group cohomology, most of which passed on through the writings of his students, has had an important impact on the field. He was well respected by his colleagues in the field. Paul Halmos, his UM colleague, once said that there are a number of ways to tackle a mathematical problem, but when all else fails, ask McLaughlin.

“Jack McLaughlin (1923 – 2001)”, ContinuUM, The Newsletter of the Department of Mathematics at the University of Michigan (pdf)