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Autobiography of Richard S Hamilton

Richard S Hamilton

Richard S Hamilton

I was born in Cincinnati, Ohio in 1943.  My father was a surgeon; he had recently finished his residency at the Mayo Clinic when the Japanese bombed Pearl Harbor.  He volunteered as a naval surgeon in the war, and was stationed in Portsmouth, England during my first two years of life, repairing wounded pilots.  My mother lived with my grandmother, together with me and my brother Billie who was only one year older than I, until my father returned.

I attended Lotspeich elementary school, where I received an excellent education.  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. Then I attended Walnut Hills High School, one of the best in the nation, which was a public high school taking the brightest kids from the whole city.  Skipping senior year, I went to Yale at 16, along with my brother.  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.

I did my graduate work at Princeton, writing my thesis at age 23 in 1966 on Riemann surfaces with Bob Gunning.  During that time I was married (and divorced several years later) and my only son Andrew was born.  My first academic position was at Cornell, where for several years I had the pleasure to work with Jim Eells Jr., who had just finished his groundbreaking paper with Joe Sampson on the harmonic map flow.  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.  My son Andrew would visit and we could snow ski in winter, and water ski and scuba dive in the summer.

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.  This got a lot of attention, and I was invited to visit the Mathematical Sciences Research Institute (MSRI), Berkeley, CA the first year it opened, along with S.T. Yau and Rick Schoen.  The next year Yau, Rick and I all moved to the University of San Diego (UCSD), CA and Gerhard Huisken came to visit also.  With all these excellent mathematicians around working on similar problems in geometric analysis, it was the ideal environment to develop the Ricci flow further.  Yau had already pointed out that the Ricci flow could pinch along necks, and that this could provide the first step in the proof, by breaking the manifold into simpler pieces which could sustain constant curvature geometries.

Later, I moved back to the East coast to Columbia University, where I am now Davies Professor.  In the Ricci flow I extended the results to four dimensions, derived the important Li-Yau type estimate for the curvature, developed the classification of singularities by ancient solutions and proved many properties of ancient solutions, showed that in three dimensions the curvature is pinched toward non-negative, developed the method of analytic surgery to bypass pinching necks and used it to classify four dimensional manifolds with positive isotropic curvature.  I also showed how one could complete the proof of the three dimensional Poincare conjecture provided one could also do similar surgeries in three dimensions, and explained the program to Grigory Perelman, pointing out the importance of avoiding collapses.  In a series of brilliant papers in 2003 Perelman did this with a very clever non-collapsing result derived from a novel Li-Yau type estimate for the adjoint heat equation.

Now I am looking at future possible developments in the Ricci flow, including possible applications to four dimensional topology, Kaehler geometry, and stationary solutions in Relativity.

28 September 2011, Hong Kong