These pathologies fall into two types: (a) bad behavior in characteristic p and (b) bad behavior in moduli spaces. In a sequence of four papers published in the American Journal of Mathematics between 19, Mumford explored pathological behavior in algebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples. Work on pathologies in algebraic geometry He also was one of the founders of the toroidal embedding theory and sought to apply the theory to Gröbner basis techniques, through students who worked in algebraic computation. He published some further books of lectures on the theory. This work on the equations defining abelian varieties appeared in 1966–7. Mumford's research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, and enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group. Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and a study of Goro Shimura's papers from the 1960s. They are now available as The Red Book of Varieties and Schemes (ISBN 3-X). His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction. Ramanujam) and Curves on an Algebraic Surface combined the old and new theories. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, and on algebraic surfaces. Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He married Erika, an author and poet, in 1959 and they had four children, Stephen, Peter, Jeremy, and Suchitra. He completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. At Harvard, he became a Putnam Fellow in 19. Mumford then went to Harvard University, where he became a student of Oscar Zariski. He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. His father William started an experimental school in Tanzania and worked for the then newly created United Nations. Mumford was born in Worth, West Sussex in England, of an English father and American mother. 3 Work on pathologies in algebraic geometry.He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University. In 2010 he was awarded the National Medal of Science. He won the Fields Medal and was a MacArthur Fellow. The link I gave in b) fortunately still works, and we should be grateful to the authors for this act of generosity.BBVA Foundation Frontiers of Knowledge Award (2012)ĭavid Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. Mumford and Oda have published in 2015 the rest of Mumford's course mentioned in b) as the book Algebraic Geometry II at the Hindustan Book Agency. The last chapter (chapter VI) is called Schemes and Functors and, as the name says, is dedicated to the point of view we are discussing.ĭ) Brian Osserman has a very pleasant short hand-out here on the use of the functorial point of view to illuminate the construction of the product of two schemes. However there is a set of notes (more than 300 pages) coauthored by Oda, corresponding to Chapters I to VIII, which can be found on-line here.Ĭ) Eisenbud and Harris wrote a book on schemes which came out of a common project with Mumford. The book has been reprinted by Springer.ī) Unfortunately Mumford didn't publish the rest of his course. The appeal is not only aesthetic, but also technical: it is with the functorial method that parameter spaces, like Grassmannians, Hilbert schemes,… are constructed.Ī) In Mumford's Introduction to Algebraic Geometry, affectionately called The Red Book,Ĭhapter Ii, §6: The functor of points of a prescheme. There is a real aesthetic appeal to the realization that concepts like closed or open immersions, tangent spaces,… can be expressed purely in terms of functors. The characterization is then relatively easy (once Grothendieck has shown us what to do!): the functor must be a sheaf in the Zariski topology and satisfy a condition which translates that a scheme is covered by affines. Given a scheme $T$, you can associate to it the contravariant functor $h_T: \mathcal$$ Since your question might interest other readers, allow me to expand it.
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