# William Browder

### Quick Info

New York City, New York, USA

**William Browder**is an American mathematician, specialising in algebraic topology, differential topology and differential geometry.

### Biography

**William Browder**'s parents were Earl Browder and Raissa Berkmann. William was the youngest of his parents' three children; the two older are Felix Browder (who has a biography in this archive) and Andrew Browder. All three boys have become leading mathematicians and so we should begin by giving some details of their parents Earl and Raissa Browder.

Earl Browder (1891-1973), the eighth of ten children, was educated at home in Wichita by his unemployed schoolteacher father William Browder and from him received his first Socialist teachings. Earl, who joined the Socialist party at age 16, worked as an accountant and took a correspondence school law course. He opposed World War I and was jailed twice between 1917 and 1920. The first time was for conspiring to defeat the operation of the draft law (a two year conviction) and for non-registration (a one-year conviction), and the second time for conspiracy. In the 1920s he often visited Russia as a representative of the Communist Trade Unions in the United States. He met Raissa Berkmann at a Soviet social function in Moscow on one of these visits.

Raissa Berkmann, born in St Petersburg in 1897, graduated with a law degree from the University of St Petersburg in 1917. Since she was Jewish, being admitted to university had not proved easy and it was difficult for anyone Jewish to practise law in Russia. After the Russian Revolution she taught at Moscow University and at the Lenin Institute. Earl Browder had married in 1911 but was separated from his wife. They were divorced in April 1926 and Earl and Raissa were married on 15 September 1926. The first two of their three children were born in Moscow; Felix in 1928 and Andrew in 1932. Earl Browder returned to the United States in 1929 but Raissa remained in Moscow until 1933. She joined her husband in the United States, crossing the Canadian border without a visa. The two boys, having an American father, did not require a visa, but [8]:-

Although [Raissa] joined her husband and lived in New York permanently thereafter, federal officials did not proceed against her until 1939. For the next sixteen years the Browders waged a legal battle to prevent her deportation.In 1934 their third son, William Browder (the subject of this biography), was born in New York City. Earl became General Secretary of the Communist Party of the United States in the year that William was born, holding that position for the next eleven years [7]:-

... he provided for his second family "an urban home of [culturally] superior circumstances." His relationship with his three sons was "very close." Ironically, the children were always fond of the sciences. Although they experienced some discrimination because of their father's beliefs, they all became mathematics professors at major American universities.During his time at elementary school, William Browder thought about what he wanted to be when he grew up [1]:-

I progressed through architect (based on Lincoln Logs and Tinkertoys), mechanical engineer (Erector sets), chemist, and finally physicist, after the excitement of the atomic bomb. It was in the pages of the 'New York Times' in August 1945 that I first read the description of the atom and nuclear fission and learned the atomic number and weight of the isotopes of uranium. I had read some wild science fiction before that, but this was a wilder reality.High school only served to strengthen his idea that he wanted to become a physicist. He enjoyed both physics and chemistry courses but he was bored in the mathematics classes. The mathematics he was taught convinced him that it was a boring computational subject devoid of beauty, with the one exception that he saw Euclidean geometry as beautiful. He graduated from the high school and entered the Massachusetts Institute of Technology intent on majoring in physics.

His first year at MIT was not quite what he had expected. First there was the realisation that, unlike at school, he was not necessarily the best student in every class. More significantly, he realised that he did not have the manual dexterity of many of the other students when it came to performing experiments in the physics and chemistry laboratories. However, he continued with his science based courses in his second year of study and made another discovery about himself in a mathematical physics course [1]:-

Some of the students had something called "physical intuition," which enabled them to give strange and wonderful answers to questions, which were greeted by the professor with delight but made no sense to me at all. At the same time the professor was making a computational hash of some mathematics that was explained in a very beautiful and motivated way in a mathematics course I was taking.This convinced Browder that his talents were in mathematics rather than physics so, after graduating with a B.S. from MIT in 1954, he began graduate studies in mathematics at Princeton University. He was soon enthralled with algebraic topology and worked towards a Ph.D. with John Coleman Moore as his advisor. Browder was appointed as an Instructor in Mathematics at the University of Rochester in 1957. By this time he was still quite a long way from completing work on this thesis. John Moore had given him a problem to work on but he had not made much progress [1]:-

I went off to Rochester with only one small result towards a thesis, to a department where I would have no one in my field to talk to. This in retrospect was a blessing of great proportions. After six months of depression and buoyed by the possibility of a much more interesting job at Cornell, I decided to simply sit down and try and write the simplest things I could deduce about the problem. Suddenly I saw a whole new aspect of the situation and, with energy and enthusiasm, slashed away at a thesis.Browder received a Ph.D. from Princeton in 1958 for his thesis

*Homology of Loop Spaces*. He was appointed as an Instructor at Cornell University in 1958 and, a year later, was promoted to Assistant Professor. He spent the academic year 1959-60 as an NSF Postdoctoral Fellow at the University of Chicago, and at the University of Oxford. His first papers were

*The cohomology of covering spaces of H-spaces*(1959),

*Homology operations and loop spaces*(1960),

*Loop spaces of H-spaces*(1960) and

*Homology and homotopy of H-spaces*(1960). Frank Adams calls this last mentioned paper "a remarkable study in the homology of H-spaces." Browder was promoted from Assistant Professor to Associate Professor at Cornell University in 1961. After spending the academic year 1963-64 as a member of the Institute for Advanced Study at Princeton, he was appointed as a Professor in the Mathematics Department of Princeton University in 1964. He was an invited speaker at the International Congress of Mathematicians in Moscow in 1966, giving the 30 minute address

*Embedding smooth manifolds*. He was a plenary speaker at the next International Congress of Mathematicians in Nice in 1970 giving the lecture

*Manifolds and Homotopy Theory*.

Browder was one of the inventors of surgery theory, which unifies methods and techniques from several branches of topology and applies them to the classification of high-dimensional manifolds. Peter Kahn writes:-

This procedure has wide-ranging, deep applications in every area of the topology of smooth manifolds, including transformation groups, classification of manifolds, and imbedding and immersion theory. Analogous techniques and applications hold for PL and topological manifolds.Others who made independent contributions to this important method are Sergi Novikov, Dennis Sullivan and Terry Wall. Browder published the book

*Surgery on simply-connected manifolds*in 1972. Peter Kahn writes in a review that the:-

... book gives a comprehensive introduction to the theory of 1-connected surgery as well as some of the most important applications. It has a number of features that make it especially useful for anyone wishing to learn surgery theory. First, the author takes special pains with the basic algebraic topology and algebra involved in 1-connected surgery ... . Secondly, the book collects a great deal of diverse background information from the literature, giving relatively complete expositions ... . Finally, and perhaps most important, is the way in which the author has organized the theory. After a chapter on preliminaries, he lists seven basic results of surgery theory and then proceeds to derive many important applications of the theory from these. The remainder of the book is devoted to a proof of the seven basic results. This procedure allows the student to arrive at interesting applications before becoming engulfed in the many technical details of the main proofs. It also focuses attention properly on the most significant tools provided by the theory for the study of 1-connected manifolds.His outstanding mathematical achievements and important contributions to the mathematical community naturally led to attempts to persuade him to leave Princeton and chair departments looking to raise their profile. In a letter he wrote in 1995, he gives some details of one such attempt:-

I was offered the chairmanship of [the University of Rochester Mathematics Department] in 1986, and I had many interesting and impressive conversations about it with many people at Rochester. Though I decided in the end not to leave Princeton, I found the attitudes of the Rochester administration and the Mathematics Department very laudable, and I tried to use my influence in the mathematical community to help in the building effort. I expended considerable effort in influencing people toward going there ...Browder has given remarkable service to the American Mathematical Society over many years. He was a Member at Large of the Council during 1967-69 and 1972-74. He was the American Mathematical Society Colloquium lecturer in 1977 giving the lectures

*Differential topology of higher dimensional manifolds*at St Louis in January of that year. He was elected Vice President of the Society in 1977-78 having served on many of the Society's committees starting with the Proceedings Editorial Committee in 1963. He was President of the Society in 1989-91. Allyn Jackson, in [4], recalled one difficult debate which took place during his time as President:-

The AMS membership was not unanimously against military funding for mathematics. Many thought the Society had no business telling mathematicians who they should and should not take funding from. I remember a heated Council debate in which William Browder, then AMS president, expressed his strong opposition to blanket condemnations of research grants from the military. He likened such condemnations to a "witch hunt" against those who opted to take such grants. (Browder, whose father Earl was persecuted because of his leadership of the American Communist Party, is not one to use the term "witch hunt" lightly.)After his term as President, Browder gave his retiring presidential address

*In search of symmetry*at the AMS meeting in Baltimore, Maryland, in January 1992. In the lecture he [3]:-

... combined short remarks about his presidency with a mathematical lecture - preceded by an informal interview in which he discussed a range of topics, including public awareness of mathematics and his interest in music. The lecture discussed the action of finite groups on manifolds, exploring the question of how large a finite group can effectively act on a given manifold (here, "effectively" means that there is no subgroup that fixes everything). A related question is, what kind of spaces have the given manifold as a covering space? Beginning with the historical roots of these questions, Browder concentrated on familiar examples such as the sphere, the $n$-sphere, or a product of spheres of different dimensions.In addition to his work for the American Mathematical Society, Browder has contributed to the mathematical scene in many other ways. For example he was editor of the

*Annals of Mathematics*from 1969 to 1980 and he was chair of the Office of Mathematical Sciences, National Research Council, from 1979 to 1983.

Browder has received (and continues to receive) honours for his outstanding contributions. He was elected to the National Academy of Sciences (1980), the American Academy of Arts and Sciences (1984), and the Finnish Academy of Science and Letters (1990). He was a plenary speaker at the British Mathematical Colloquium in 1979, giving the lecture

*Topologies of varieties*. The American Mathematical Society held a Special Session on Homotopy Theory at their Spring Meeting in Lawrenceville in April 2004 in honour of Browder's 70th birthday. Let us end this biography by quoting Browder's Statement when standing for the Nominating Committee of the American Mathematical Society in 1998. It gives a clear picture of how he feels mathematicians should contribute:-

I would look for candidates who, while having a strong research orientation, have the political skills to contribute to both the smooth running of the organization and to the ongoing efforts to communicate the importance of mathematics to policymakers and to the general public. The building of such a group of people is important to the long-term health of our subject. They should be representative of the diverse interests of the membership, both in mathematical questions and in political and social viewpoints relevant to our community.Let us note that he was elected to the Nominating Committee.

### References (show)

- M Cook,
*Mathematicians : an outer view of an inner world*(Princeton University Press, Princeton-Oxford, 2009). - J G Ryan,
*Earl Browder: The Failure of American Communism*(University Alabama Press, 2005). - W Browder, In search of symmetry. AMS retiring presidential address presented in Baltimore, Maryland, January 1992,
*Selected Lectures in Mathematics*(American Mathematical Society, Providence, RI, 1994). - A Jackson, DARPA and Hilbert,
*Notices Amer. Math. Soc.***55**(4) (2008), 445. - Mathematical descendants of William Browder, in
*Prospects in topology, Princeton, New Jersey, 1994*(Princeton University Press, Princeton, NJ, 1995), xi-xiii. - Nominating Committee, William Browder,
*Notices Amer. Math. Soc.***45**(8) (1998), 1007-1018. - J G Ryan, The Making of a Native Marxist: The Early Career of Earl Browder,
*The Review of Politics***39**(3) (1977), 332-362.

### Additional Resources (show)

Other websites about William Browder:

### Honours (show)

Honours awarded to William Browder

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update July 2011

Last Update July 2011