Richard Streit Hamilton

Quick Info

10 January 1943
Cincinnati, Ohio, USA

Richard Hamilton is an American mathematician famed for his important contributions to proving the Poincaré Conjecture. He has been awarded prestigious prizes including the Oswald Veblen Prize, the Clay Research Award, the Leroy P Steele Prize and the Shaw Prize.


Richard Hamilton was the son of William Selden Hamilton (1906-1985) and Hester Streit (1910-1996). Selden Hamilton was born on 20 January 1906 in Franklin, Warren, Ohio. He graduated from Franklin High School in 1923, was awarded a B.A. by Yale in 1927 where, in addition to his studies, he was in the football squad and the lacrosse squad. He graduated Doctor of Medicine from Johns Hopkins in 1931, then served as intern at Miami Valley Hospital, Dayton before spending a year in the medical practice of his father Norval A Hamilton. He married Hester Streit, daughter of Oscar R Streit and June Bullock of Norwood, on 21 March 1933 at Oscar Streit's home in Norwood. Immediately after the wedding, the couple moved to Rochester, Minnesota, where Selden had a fellowship at the Mayo Clinic. Selden Hamilton registered for the World War II Draft in the Norwood Public Library on 16 October 1940. Selden and Hester Hamilton's first child, William Selden Hamilton Jr (1941-2021) was born in Cincinnati on 11 December 1941. We note that he was awarded a Ph.D. from Yale university and became a Professor of Slavic Languages and Linguistics at Wake Forest University. Their second son was Richard Streit Hamilton, the subject of this biography, born 10 January 1943. Selden Hamilton became a naval surgeon and spent the first two years of Richard's life in undertaking military service in Portsmouth, England. During these two years, Richard, his older brother William, and mother Hester lived with Hester's mother June Streit.

Richard attended Lotspeich Elementary School which had been founded by Helen Lotspeich in 1916. The school which began with Lotspeich as the only teacher, soon expanded adding more buildings and teachers. When Richard Hamilton studied there, it provided him with an excellent education. It was while at this school that his interest in mathematics took him to study at a level well above what was taught at the school [11]:-
In the fourth grade, out of curiosity I went to the library and took out a book on first year algebra. I taught myself first year algebra in a month, and went back for the second year algebra book.
From Lotspeich Elementary School, Hamilton progressed to Walnut Hills High School where his elder brother Billie was already a pupil. Walnut Hills High School was the third public High School in Cincinnati when it was established in 1895. By the time Richard and Billie Hamilton studied there, it had gained an excellent reputation and was the school where many of the brightest children in the district were educated. While at this school, both brothers were members of Scarab, a traditional secret society founded in 1909. In 1959 Billie Hamilton was President of Scarab and Richard Hamilton was Treasurer:-
The fraternity participated in all three interfraternity sports competitions - football, basketball, and baseball, as well as the interfraternity sing and the scholastic achievement contest. ... Scarab helped the Bank of Hope during its collection drive.
Billie Hamilton graduated from Walnut Hills High School in 1959 and Richard, who was one year behind his brother, decided to skip his final year at the school. Both brothers left the school in 1959 and began their university education at Yale University. This meant that Richard Hamilton was only sixteen year old when he began his university studies in Trumbull College, Yale University. One would expect that he found the mathematics classes the most interesting but the wide range of his interests is illustrated by the fact that he writes in [11]:-
The most interesting classes were in ancient Greek, where we read the tragedies, the comedies and the great orators (in the original), and my philosophy classes with Brand Blanshard, a wonderful old scholar who had not changed his philosophical ideas since before WWI.
He graduated summa cum laude from Yale with a B.A. in 1963 and then went to Princeton University to undertake graduate work. His thesis advisor was Robert Clifford Gunning (born 1931). Gunning had himself studied for a Ph.D. at Princeton University, his advisor being Salomon Bochner. He was on the staff at Princeton from 1957 and was the author of several excellent books on complex analysis and Riemann surfaces. While Hamilton was studying for his doctorate, Gunning was writing three books, Analytic functions of several complex variables, Lectures on Riemann Surfaces, and Lectures on Vector Bundles over Riemann Surfaces. Given the interests of his thesis advisor, it is not surprising that the topic of Hamilton's thesis is Riemann surfaces. He graduated with a Ph.D. in 1966 having written the 196-page thesis Variation of Structure of Riemann Surfaces.

While he was a graduate student, Hamilton married Sally Harper Swigert (1944-2022) in the Seventh Presbyterian Church on Saturday, 11 September 1965. Sally was the daughter of James Mack Swigert (known as Mack) and Alice Harrower. The reception was held in the Cincinnati Country Club. Richard and Sally Hamilton had one son, Andrew Streit Hamilton, born on 8 October 1966. Richard and Sally Hamilton were divorced several years later.

Describing Hamilton's outstanding contributions requires technical terms which may not be familiar to those reading this biography. Before we proceed, therefore, we will give Hamilton's own Abstract for the lecture The Ricci Flow and the Poincaré Conjecture which he gave to a general audience at Lehigh University on Monday 15 April 2019 [25]:-
A manifold is a topological space which is locally like a Euclidean space of some dimension. A two-dimensional manifold is called a surface. A manifold that does not go off to infinity but closes back on itself is called compact. We understand all the possible shapes of compact surfaces. The oriented ones are the sphere, the donut, or donuts with several holes.

Our world space is three dimensional (not counting time). But according to Einstein it is not flat but curved, with curvature appearing as gravity. Of course we have limited experience of only a small part of it, and hence no idea of the total shape. Like the earth, it may curve back on itself, or have wormholes connecting one part to a distant part. So topologists naturally asked what are all the possible shapes of compact three-dimensional manifolds?

On the compact surfaces, the sphere is the only one where every curve bounds a disk; on the others, a curve going around a hole does not. Such a manifold is called simply connected. In 1904, Poincaré conjectured the only compact simply connected three-manifold is the sphere. Famously, a proof of this conjecture was given by Dr Perelman in 2002-2003, using the Ricci flow.

A manifold can carry a metric, measuring the distances between points. In differential geometry, the metric is given as a quadratic function on tangent vectors, measuring the square of their lengths. The length of a path is given by an integral of the lengths of tangent vectors.

The heat equation on a surface (or manifold) is a partial differential evolution equation for a function whose solution describes how the initial given heat will spread by diffusion around the surface (or manifold) to approach a constant equal distribution of heat. The Ricci flow is a partial differential evolution equation for a metric which tries to spread the curvature to approach a constant equal distribution. The idea for the Poincaré conjecture is that if the metric converges to constant positive curvature, the manifold must be the sphere.
After graduating with his Ph.D., Hamilton was appointed to Cornell University. There he became a colleague of James Eells who was working on global analysis [9]:-
It is tempting to describe global analysis as a holistic approach to mathematics. In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas. Non-linearity, especially that caused by curvature, is a prevalent aspect. A prime example is Eells's most famous article, "Harmonic Mappings of Riemannian Manifolds", published in the American Journal of Mathematics in 1964. Written with J H Sampson of Johns Hopkins University, it founded the theory of "harmonic maps" and the "non-linear heat flow".
Hamilton was influenced by Eells, particularly the "Harmonic Mappings of Riemannian Manifolds" paper. He said [16]:-
This was the first example of using a nonlinear parabolic flow to solve an elliptic equation in geometry, and was my inspiration for creating the Ricci flow. ... By the mid seventies I had begun work on the Ricci flow, and published the first result in 1982 on the case of three dimensional manifolds with positive Ricci curvature.
Hamilton's 1982 paper, Three-manifolds with positive Ricci curvature, was an exceptionally important one, proving even more important in the fullness of time than was realised when it was published. Jerry Kazdan writes in a review [12]:-
While most of the arguments are quite elementary, using only the classical maximum principle to estimate various quantities, the convergence arguments are quite delicate and ingenious, and are special to the 3-dimensional case. One key observation is that the geometric entities all satisfy various heat equations. As an added bonus, this beautiful paper is nicely written and should be accessible to many geometers who would like to see a good application of analysis to geometry.
Sylvia Nasar and David Gruber write in [18]:-
That year [1982], Richard Hamilton, a mathematician at Cornell, published a paper on an equation called the Ricci flow, which he suspected could be relevant for solving Thurston's conjecture and thus the Poincaré conjecture. Like a heat equation, which describes how heat distributes itself evenly through a substance - flowing from hotter to cooler parts of a metal sheet, for example - to create a more uniform temperature, the Ricci flow, by smoothing out irregularities, gives manifolds a more uniform geometry.
This 1982 paper on the Ricci flow was, in fact, the first of 29 papers by Hamilton on the Ricci flow. His work in this area did indeed lead to the solution of the Poincaré Conjecture; we shall say more about this below.

This was not the only paper Hamilton published in 1982, for he also published The inverse function theorem of Nash and Moser. This 158-page paper was reviewed by Peter Michor who began his review as follows:-
This paper is a careful exposition of the Nash-Moser inverse function theorem in a setting that allows one to apply the theorem without the need to prove a new version of it for each new kind of application. Along with this comes much useful extra equipment (a priori estimates) fitting perfectly into the setting. Convincing examples show the theorem with its different "outfits" at work.
Although we have looked at Hamilton's 1982 papers, we should mention before continuing his book Harmonic maps of manifolds with boundary which he published in 1975. Jean-Claude Mitteau writes in the review [15]:-
This book on harmonic maps between Riemannian manifolds essentially concerns the examination of the case where the manifolds have a boundary. The method of demonstrating the existence of harmonic applications by the convergence, for tt \rightarrow ∞, of the solutions of the heat equation naturally associated with this problem, is extended in this new case, thus making it possible to obtain the existence of harmonic maps in homotopy classes of boundary varieties, subject to conditions on curvatures. We will also find in this book an introduction to specialised functional spaces for solving this type of problem.
Hamilton had many hobbies including riding horses and windsurfing. While at Cornell, he introduced his son Andrew to some of his sporting loves. He wrote in the autobiography [16]:-
My son Andrew would visit and we could snow ski in winter, and water ski and scuba dive in the summer.
Cornell, however, was not the right place for Hamilton as he explained in [16]:-
I really wanted to get out of Cornell. It was lovely in the summer, but summer was short. In the fall it rained all the time, and in the winter it snowed all the time, and in the spring it rained all the time. And I have allergies to mould spores, and that was not a healthy place for me to be, so to speak - so motive to leave ...
The Mathematical Sciences Research Institute (MSRI), at the University of California, Berkeley, was opened in temporary accommodation in September 1982. [To avoid confusion, let us note that the MSRI was renamed the Simons Laufer Mathematical Sciences Institute (SLMath) in 2022.] Shiing-Shen Chern was the founding director of the Institute and Calvin Moore was the founding deputy director. Hamilton's paper on the Ricci flow was seen to be highly significant, and he was invited to the MSRI in the first year it opened. Shing-Tung Yau and Richard Schoen were also invited to the MSRI in this first year. It was immediately clear that Hamilton, Yau and Schoen were working on similar problems in geometric analysis and that the interaction between them was vey fruitful.

In 1984 Hamilton, Yau and Schoen all moved to a position at the University of California at San Diego. The authors of [18] write:-
Yau was especially impressed by Hamilton, as much for his swagger as for his imagination. "I can have fun with Hamilton," Yau told us during the string-theory conference in Beijing. "I can go swimming with him. I go out with him and his girlfriends and all that." Yau was convinced that Hamilton could use the Ricci-flow equation to solve the Poincaré and Thurston conjectures, and he urged him to focus on the problems. "Meeting Yau changed his mathematical life," a friend of both mathematicians said of Hamilton. "This was the first time he had been on to something extremely big. Talking to Yau gave him courage and direction."
In 1995, Hamilton published the survey paper The formation of singularities in the Ricci flow. Man Chun Leung writes in the review [14]:-
In this paper the author surveys some of the basic geometrical properties of the Ricci flow with a view to considering what kind of singularities might form. The Ricci flow was introduced by the author in his celebrated work on deformation of metrics the by Ricci flow on compact 3-manifolds with positive Ricci curvature. It has been found to be a useful tool in the study of Riemannian geometry, both for compact and for open manifolds. ... The paper is well written and delightful to read. Many examples are worked out in detail to illustrate the idea. It gives a rather complete account and up-to-date references on current research on the Ricci flow.
For a longer quote from Man Chun Leung's review and other material on the Ricci flow and the Poincaré conjecture, see THIS LINK.

The 1996 Oswald Veblen Prize in Geometry was awarded at the Joint Mathematics Meetings in Orlando in January 1996 to Richard Hamilton of the University of California, San Diego, and to Gang Tian of the Massachusetts Institute of Technology. This was the first major award to Hamilton and the Citation begins [1]:-
Richard Hamilton is cited for his continuing study of the Ricci flow and related parabolic equations for a Riemannian metric and he is cited in particular for his analysis of the singularities which develop along these flows.
For the full Citation and other material relating to this award, see THIS LINK.

In 1998 Hamilton left the University of California at San Diego when he was appointed Davies Professor of Mathematics at Columbia University. While at Columbia, Hamilton received three further major awards. The first was the 2003 Clay Research Award [7]:-
The 2003 Clay Research Award was made to Richard Hamilton for his discovery of the Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis. Hamilton conceived of his work as a way to approach both the Poincaré Conjecture and the Thurston Geometrization Conjecture.
For other material relating to this award, see THIS LINK.

In 2009 he was awarded the Leroy P Steele Prize for Seminal Contribution to research. The award was for the paper he had published in 1982, over a quarter of a century earlier [2]:-
The 2009 Leroy P Steele Prize for Seminal Contribution to Research is awarded to Richard Hamilton for his paper "Three-manifolds with positive Ricci curvature", Journal of Differential Geometry 17 (1982), 255-306.
For other material relating to this award, see THIS LINK.

The 2011 Shaw Prize in Mathematical Sciences was awarded to Demetrios Christodoulou and Richard S Hamilton:-
... for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.
For other material relating to this award, see THIS LINK.

Other honours given to Hamilton include membership of the National Academy of Sciences in 1999 and membership of the American Academy of Arts & Sciences in 2003.

In 2022 Hamilton, then aged 79, joined the University of Hawaii [5]:-
After a more than 20-year effort, one of the greatest mathematicians of this century has landed at the University of Hawaii's flagship campus. Richard Hamilton, a top scholar in the field of geometry, has joined the University of Hawaii Mānoa faculty as an adjunct professor in the Department of Mathematics.

Hamilton's main contributions are focused around geometric analysis and partial differential equations (found in math-centred scientific fields such as physics and engineering). He is most well-known for creating the "Ricci Flow," a ground-breaking partial differential equation in geometry, which is similar to the diffusion of heat and the heat equation. Many other mathematicians have built upon Hamilton's methods to expand our knowledge of the field of geometry.

"Richard Hamilton is one of the most original and influential mathematicians currently working," said Department of Mathematics Professor and Chair Rufus Willett. "We are honoured to have him join our department, and look forward to his having many fruitful interactions with our students and faculty."

Hamilton is also a member of the Academy of Sciences, and received some of the most prestigious mathematics prizes. This includes the $1 million Shaw Prize split equally between Hamilton and Demetrios Christodoulou for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

'Wonderful' first visit to the islands

Hamilton first came to Hawaii in 1988 after Professor Emeritus Joel Weiner invited him to visit University of Hawaii Mānoa's mathematics department for an academic year.

"I fell in love with the islands the first morning I woke up here," Hamilton said. "The stay was wonderful and the department treated me great. I have in particular developed a strong relationship with Professor George Wilkens (researcher in the field of differential geometry) and more recently with Professor Monique Chyba, with whom I am collaborating on a textbook introducing the basics of differential geometry, as well as how they can be applied to many areas in mathematics, physics and engineering."

Wilkens added, "It took many years to formalise this collaboration and count Dr Hamilton as a member of our faculty. I cannot be more happy and delighted for our department."

References (show)

  1. 1996 Oswald Veblen Prize, Notices of the American Mathematical Society 43 (3) (1996), 325-327.
  2. 2009 Steele Prizes, Notices of the American Mathematical Society 56 (4) (2009), 488-491.
  3. 2011 Shaw Prize Mathematical Sciences, The Shaw Prize.
  4. An Essay on the Prize, 2011 Shaw Prize Mathematical Sciences, The Shaw Prize (28 September 2011).
  5. M Arakaki, World-renowned mathematician joins UH Mānoa faculty, University of Hawaii News (28 February 2022).
  6. Christodoulou and Hamilton Awarded Shaw Prize, Notices of the American Mathematical Society 58 (9) (2011), 1293.
  7. Clay Research Award 2003: Richard Hamilton, Clay Mathematics Institute (2003).
  8. Couple weds in noon-day service, The Drayton Herald (Wednesday 22 March 1933).
  9. D Elworthy, James Eells: Innovative mathematician, The Independent (17 April 2007).
  10. Hamilton and Tao Receive Clay Awards, Notices of the American Mathematical Society 51 (2) (2004), 234.
  11. R S Hamilton, Autobiography of Richard S Hamilton, The Shaw Prize (28 September 2011).
  12. J L Kazdan, Review: Three-manifolds with positive Ricci curvature, Mathematical Reviews MR0664497 (84a:53050).
  13. Leroy P Steele Prize for Seminal Contribution to Research (1993 - present), American Mathematical Society (2024).
  14. M C Leung, Review: The formation of singularities in the Ricci flow, Mathematical Reviews MR1375255 (97e:53075).
  15. J-C Mitteau, Review: Harmonic maps of manifolds with boundary, Mathematical Reviews MR0482822 (58 #2872).
  16. J Morgan, And Quiet Goes the Ricci Flow: A Conversation with Richard Hamilton. Interview by John Morgan, Simons Center for Geometry and Physics (3 August 2022).
  17. J Morgan and G Tian, Ricci Flow and the Poincaré Conjecture (American Mathematical Society, Clay Mathematics Institute, 2007).
  18. S Nasar and D Gruber, Manifold Destiny, The New Yorker (28 August 2006).
  19. Professor Richard Hamilton Wins Shaw Prize for Mathematics, Columbia News (8 June 2011).
  20. S Lojasiewicz Lecture 2011: Richard Hamilton, Institute of Mathematics of the Jagiellonian University (Monday 30 May 2011).
  21. S Lojasiewicz Lecturer: Richard Hamilton, Institute of Mathematics of the Jagiellonian University (2024).
  22. H V Rao, Unraveling the Mystery: The Amazing Proof of the Poincaré Conjecture, The STEM Post (3 April 2023).
  23. Richard Hamilton, Department of Mathematics, University of California San Diego (2024).
  24. The 2003 Clay Research Awards, The Year 2003, Clay Mathematics Institute (2003), 3.
  25. 2019 A Everett Pitcher Lecture Series: Richard S Hamilton, Columbia University, Lehigh University (15 April 2019).
  26. Richard S Hamilton, National Academy of Sciences.
  27. Richard Hamilton, American Academy of Arts & Sciences.
  28. LME Annual General Meeting Friday 17 November 2006, London Mathematical Society Newsletter 355 (2007), 25-26.

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Written by J J O'Connor and E F Robertson
Last Update March 2024