# Dennis Parnell Sullivan

### Quick Info

Born
12 February 1941
Port Huron, Michigan, USA

Summary
Dennis Sullivan is an American mathematician known for his work on topology, both algebraic and geometric, and on dynamical systems.

### Biography

Although Dennis Sullivan was born in Port Huron, Michigan, he was brought up in Houston, Texas, and has always considered himself a Texan. He attended school in Houston before entering Rice University, also in Houston, to study chemistry. He switched from chemistry to mathematics when he found that the mathematics courses he took were the most exciting of all the science courses, and he was awarded a B.A. by Rice in 1963. He then went to Princeton University to undertake graduate studies in mathematics. At Princeton his thesis advisor was William Browder and Sullivan was awarded a Ph.D. in 1966 for his thesis Triangulating Homotopy Equivalences. Anthony Phillips writes [10]:-
William Browder's students at that time were mining surgery theory for topological gold. Dennis' thesis was in this vein and led to his work on the Hauptvermutung (1967).
It was this work which led to Sullivan receiving the Oswald Veblen Prize in Geometry in 1971 from the American Mathematical Society. The sixth award of this prize was made:-
To Dennis P Sullivan for his work on the Hauptvermutung summarized in the paper 'On the Hauptvermutung for manifolds', Bulletin of the American Mathematical Society, volume 73 (1967) ...
After the award of his doctorate Sullivan was appointed to a NATO Fellowship at Warwick University in England. I [EFR] was a research student at Warwick at this time and I remember many evenings when I played bridge with fellow research students while Dennis worked at an adjacent table writing mathematics papers. After holding the fellowship at Warwick, Sullivan held a Miller Fellowship at the University of California at Berkeley working on the Adams conjecture, $K$-theory, and étale homotopy. In 1969 he went to the Massachusetts Institute of Technology as a Sloan Fellow of Mathematics where [9]:-
... his work focussed on what he named geometric topology (in particular the study of Galois symmetry) and on the construction of minimal models for the rational homotopy type of manifolds, using differential forms.
In 1970 he produced a set of notes entitled Geometric topology: localization, periodicity and Galois symmetry. In the same year he was an invited speaker at the International Congress of Mathematicians in Nice, France, where he gave the lecture Galois symmetry in manifold theory at the primes. The importance of this work can be clearly seen from the fact that in 2005, thirty-five years after they were written, both the lecture notes and Sullivan's lecture to the Nice Congress were published by Springer-Verlag. John McCleary writes in a review of the 2005 publication:-
In 1970, Sullivan circulated a set of notes, the "MIT notes", introducing localization and completion of topological spaces to homotopy theory, and other important concepts and constructions that have had a major influence on the development of topology. A version of the notes appeared as "Genetics of homotopy theory and the Adams conjecture" (1974). Although it has been a long time since 1970, their publication now is more than an historical exercise. ... C T C Wall wrote of the MIT notes, "it is difficult to summarise Sullivan's work so briefly: the full philosophical exposition in (the notes) should be read." The exposition in the notes focuses on epistemological questions - in particular, what is the underlying algebraic nature of a manifold, and how can we know it? ... My old copy of the MIT notes included a photo of the author, barely recognizable after so much photocopying. The photos in the new book of the author and his children together with the Postscript give a rare insight into the development of deep mathematical ideas. The notes remain worth reading for the boldness of their ideas, the clear mastery of available structure they command, and the fresh picture they provide for geometric topology. The editor [Andrew Ranicki] must be thanked for making the notes available to another generation of topologists.
At Berkeley a new building, Evans Hall, was completed in 1971 to house the Mathematics Department. Many thought it a decidedly unattractive design and the postgraduates decided to paint some walls to brighten up their environment. Sullivan wrote to Lee Mosher in 2002 explaining the part he played [9]:-
In 1971 I was a guest of the University of California giving lectures in the Math Dept. At the same time there was a confrontation between the trustees and the graduate students et al. The latter planned to continue decorating the walls of the department by painting attractive murals and the trustees forbade it. At tea some students came up and invited me to join their painting the next day. I became enthusiastic when one bearded fellow [Bill Thurston] showed me an incredible drawing of an embedded curve in the triply punctured disk and asked if I thought this would be interesting to paint. I said, 'You bet,' and the next day we spent all afternoon doing it. As we transferred the figure to the wall it was natural and automatic to do it in terms of bunches of strands at a time - as an approximate foliation - and then connect them up at the end as long as the numbers worked out. Thus some years later in '76 when Bill gave an impromptu 3-hour lecture about his theory of surface transformations I absorbed it painlessly at a heuristic level after the experience of several hours of painting in '71.
Sullivan spent the academic year 1973-74 in France visiting the University of Paris-Orsay. He was invited to become a permanent professor at the Institut des Hautes Études Scientifique outside Paris and, he took up this position in 1974. At the IHÉS, Sullivan [5]:-
... was a master at orchestrating activity and interest among visitors and was especially effective with young people. Recent Fields Medalist Curtis McMullen is a good example of Sullivan's influence: Although McMullen received his Ph.D. from Harvard, he was really Sullivan's student, and it was while visiting the IHÉS that McMullen got the idea for his thesis problem. Another example is Gromov: It was Sullivan's invitation that first brought Gromov to the IHÉS as a visitor in 1977, three years after Gromov had gotten out of the Soviet Union.
In 1981 Sullivan was appointed to the Albert Einstein Chair in Science at the Graduate Center of the City University of New York but continued to hold his professorship on a part-time basis at the Institut des Hautes Études Scientifique [4]:-
During the 1980s the resources of the [Albert Einstein] Chair allowed the founding of a regular seminar in geometry and chaos theory that brought first-rank international scholars to CUNY [the City University of New York] and New York City. Subsequently, the seminar has been supported by The Graduate Center, pursuing the connections between topology and the mathematical models of nature provided by quantum field theory and fluid mechanics.
Writing about this seminar, Sullivan explained where the strong relationship between algebraic topology and quantum field theory arises [4]:-
I am particularly interested in the method of algebraic topology which associates linear objects (homology groups) to nonlinear objects with points ( manifolds...) just like quantum theory associates linear spaces of states to classical systems with points. The main character in algebraic topology is the nilpotent operator or boundary operator while in quantum field theory an important role is played by the nilpotent operators called Q and "delta" which encode whatever symmetry is present in the action of the particular theory and measure the obstruction to invariantly assign meaning to the integral over all paths. In algebraic topology there is a powerful idea, due first to Stasheff but going beyond his famous and elegant concept of an infinitely homotopy associative algebra, which allows one to live with slightly false algebraic identities in a new world where they become effectively true. In quantum field theory the necessity to regularize or cutoff which sometimes destroys, but only slightly, identities expressing various symmetries and structures may provide an opportunity to use this powerful idea from algebraic topology. Finally algebraic and geometric topology has always directed it efforts towards understanding in an algebraic way geometric objects like manifolds which are the classical models of spacetime, while quantum field theory often begins its specification of a particular theory with the classical action defined on the classical fields spread over spacetime and then proceeds to its algebraic algorithms.
In 1996 Sullivan resigned from his professorship in Paris to take up a professorship in mathematics at the State University of New York at Stony Brook, continuing to hold his part-time position at the Graduate Center of the City University of New York. In session 1998-99 SUNY promoted Sullivan to Distinguished Professor. Their announcement reads as follows [1]:-
Dennis Parnell Sullivan, Department of Mathematics, Stony Brook, is one of the great mathematicians of our time and one of the most important topologists of the last 100 years. He has contributed to several diverse areas of mathematics including topology, geometry and dynamics and complex analysis.
Sullivan has received other major honours. In addition to the 1971 Oswald Veblen Prize in Geometry mentioned above, he received the 1981 Prix Élie Cartan from the French Academy of Sciences, the 1994 King Faisal International Prize for Science (mathematics), and the Ordem Scientifico Nacional by the Brazilian Academy of Sciences in 1998. He received the New York City Mayor's Award for Excellence in Science and Technology in 1997. He was awarded the 2004 National Medal of Science by President George W Bush at a ceremony in the White House:-
For his achievements in mathematics, including solving some of the most difficult problems and creating entirely new areas of activity, and for uncovering striking, unexpected connections between seemingly unrelated fields.
The description of his contributions leading to the award of the Medal are described in [7]:-
Sullivan's early work was in homotopy theory and surgery, to which he brought a new, geometric point of view. His geometric insights led to many important results on the topology of manifolds. His theory of real and rational homotopy types, based on differential forms, has had profound applications, for example, to the topology of complex algebraic varieties. Sullivan has made important contributions to the study of foliations and dynamical systems. He has also proved foundational results on quasiconformal and Lipschitz manifolds, categories that are intermediate between the topological and smooth ones. During the 1980s and 1990s, he was responsible for the emergence of the field of conformal dynamics as a lively and important branch of mathematics straddling the traditional borders between pure and applied areas. In recent years, he launched the field of string topology.
In 2006 Sullivan received the Leroy P Steele Prize for Lifetime Achievement from the American Mathematical Society. The citation states:-
Dennis Sullivan has made fundamental contributions to many branches of mathematics. Sullivan's theory of localization and Galois symmetry, propagated in his famous 1970 MIT [Massachusetts Institute of Technology] notes, has been at the heart of many subsequent developments in homotopy theory. Sullivan used it to solve the Adams Conjecture and the Hauptvermutung for combinatorial manifolds. Later Sullivan developed and applied rational homotopy theory to problems about closed geodesics, the automorphism group of a finite complex, the topology of Kähler manifolds, and the classification of smooth manifolds. He has reinvented himself several times, playing major or dominant roles in dynamical systems, Kleinian groups, and low dimensional topology. These brief remarks do not do justice to the scope of Sullivan's ideas and influence. Beyond the specific theories he has developed and the problems he has solved - and there are many significant ones not mentioned here - his uniform vision of mathematics permeates his work and has inspired those around him. For many years he was at the center of the mathematical conversation at IHÉS [Institut des Hautes Études Scientifiques]. Later he moved to New York where his weekly seminar remains an important feature of mathematical life in the City.
He received the Wolf Prize in Mathematics in 2010 for his contributions to algebraic topology and conformal dynamics [11]:-
Dennis Sullivan has made fundamental contributions in many areas, especially in algebraic topology and dynamical systems. His early work helped lay the foundations for the surgery theory approach to the classification of higher dimensional manifolds, most particularly providing a complete classification of simply connected manifolds within a given homotopy type. He developed the notions of localization and completion in homotopy theory and used this theory to prove the Adams conjecture (also proved independently by Quillen). Sullivan and Quillen introduced the rational homotopy type of space. Sullivan showed that it can be computed using a minimal model of an associated differential graded algebra. Sullivan's ideas have had far-reaching influence and applications in algebraic topology. One of Sullivan's most important contributions was to forge the new mathematical techniques needed to rigorously establish the predictions of Feigenbaum's renormalization as an explanation of the phenomenon of universality in dynamical systems. Sullivan's "no wandering domains" theorem settled the classification of dynamics for iterated rational maps of the Riemann sphere, solving a sixty-year-old conjecture by Fatou and Julia. His work generated a surge of activity by introducing quasiconformal methods to the field and establishing an inspiring dictionary between rational maps and Kleinian groups of continuing interest. His rigidity theorem for Kleinian groups has important applications in Teichmüller theory and for Thurston's geometrization program for 3-manifolds. His recent work on topological field theories and the formalism of string theory can be viewed as a by-product of his quest for an ultimate understanding of the nature of space and how it can be encoded in strange algebraic structures. Sullivan's work has been consistently innovative and inspirational. Beyond the solution of difficult outstanding problems, his work has generated important and active areas of research pursued by many mathematicians.
He was elected a fellow of the American Academy of Arts and Sciences (1991), a member of the National Academy of Sciences (1983), and of the Brazilian National Academy of Sciences (1984), and he is a member of the New York Academy of Sciences. He has served the American Mathematical Society as vice-president (1990-93). He received honorary degrees from the University of Warwick (1983) and the École Normale Supérieure de Lyon (2001). Another honour was the conference held at the CUNY Graduate Center in September 2002 to celebrate the 20th anniversary of his appointment as the Albert Einstein Chair in Science at The City University of New York.

Sullivan is married to a mathematician also on the faculty at Stony Brook. He has three daughters and three sons.

### References (show)

1. 1998-99 Distinguished Professor Recipients SUNY, On Course, May/June 1999. http://www.suny.edu/provost/mayjune.cfm
2. 2006 Steele Prizes, Notices Amer. Math. Soc. 53 (4) (2006), 464-470.
3. Dennis P Sullivan Receives 2006 AMS Steele Prize for Lifetime Achievement, American Mathematical Society News, Events and Announcements (Friday 13 January 2006). http://www.ams.org/news?news_id=540
4. Einstein Chair Mathematics Seminar, The Graduate Center, SUNY. http://math.gc.cuny.edu/seminars/einsteinchair.html
5. A Jackson, The IHÉS at Forty, Notices Amer. Math. Soc. 46 (3) (1999), 329-337.
6. A Jackson, Major Gift Launches New Geometry and Physics Center, Notices Amer. Math. Soc. 53 (3) (2006), 685-686.
7. A Jackson, Sullivan Receives 2004 National Medal of Science, Notices Amer. Math. Soc. 52 (3) (2005), 346.
8. R Kirby, Review: Mathematics at Berkeley : A history, Notices Amer. Math. Soc. 55 (10) (2008), 1237-1240.
9. L Mosher, What is ... A Train Track?, Notices Amer. Math. Soc. 50 (3) (2003), 354-356.
10. A Phillips, (2005), Dennis Sullivan - A short history, in Graphs and patterns in mathematics and theoretical physics (American Mathematical Society, Providence RI, 2005), xiii.
11. Yau and Sullivan Awarded 2010 Wolf Prize, Notices Amer. Math. Soc. 57 (6) (2010), 748-749.