Clifford Taubes
Encyclopedia
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University
and works in gauge field theory, differential geometry, and low-dimensional topology
.
in physics in 1980 under the direction of Arthur Jaffe
, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex
equations and the Bogomol'nyi monopole
equations.
Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space
of solutions to the Yang-Mills equations was used by Simon Donaldson
in his proof of Donaldson's theorem
. He proved in that R4 has an uncountable number of smooth structure
s (see also exotic R4
), and (with Raoul Bott
in ) proved Witten's rigidity theorem on the elliptic genus.
is equal to an invariant which enumerates certain pseudoholomorphic curves and is now known as Taubes's Gromov invariant
. This fact has transformed mathematicians' understanding of the topology of symplectic four-manifolds.
More recently (in ), by using Seiberg–Witten Floer homology
as developed by Peter Kronheimer and Tomasz Mrowka
together with some new estimates on the spectral flow of Dirac operator
s and some methods from ), Taubes proved the longstanding Weinstein conjecture
for all three-dimensional contact manifolds, thus establishing that the Reeb vector field on such a manifold always has a closed orbit. Expanding both on this and on the equivalence of the Seiberg–Witten and Gromov invariants, Taubes has also proven (in a long series of preprints, beginning with ) that a contact 3-manifold's embedded contact homology is isomorphic to a version of its Seiberg–Witten Floer cohomology. More recently, Taubes, in collaboration with Kutluhan and Lee, have announced and begun publishing a proof that embedded contact homology is isomorphic to Heegaard Floer homology.
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
and works in gauge field theory, differential geometry, and low-dimensional topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Early career
Taubes received his Ph.D.Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in physics in 1980 under the direction of Arthur Jaffe
Arthur Jaffe
Arthur Jaffe is an American mathematical physicist and a professor at Harvard University. Born on December 22, 1937 he attended Princeton University as an undergraduate obtaining a degree in chemistry, and later Clare College, Cambridge, as a Marshall Scholar, obtaining a degree in mathematics...
, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
equations and the Bogomol'nyi monopole
Monopole (mathematics)
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. The connection and Higgs field should satisfy the Bogomolnyi equation and be of finite action....
equations.
Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of solutions to the Yang-Mills equations was used by Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
in his proof of Donaldson's theorem
Donaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
. He proved in that R4 has an uncountable number of smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
s (see also exotic R4
Exotic R4
In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...
), and (with Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
in ) proved Witten's rigidity theorem on the elliptic genus.
Work based on Seiberg–Witten theory
In a series of four long papers in the 1990s (collected in ), Taubes proved that, on a closed symplectic four-manifold, the (gauge-theoretic) Seiberg–Witten invariantSeiberg–Witten invariant
In mathematics, Seiberg–Witten invariants are invariants of compact smooth 4-manifolds introduced by , using the Seiberg–Witten theory studied by during their investigations of Seiberg–Witten gauge theory....
is equal to an invariant which enumerates certain pseudoholomorphic curves and is now known as Taubes's Gromov invariant
Taubes's Gromov invariant
In mathematics, the Gromov invariant of Clifford Taubes counts embedded pseudoholomorphic curves in a symplectic 4-manifold...
. This fact has transformed mathematicians' understanding of the topology of symplectic four-manifolds.
More recently (in ), by using Seiberg–Witten Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
as developed by Peter Kronheimer and Tomasz Mrowka
Tomasz Mrowka
Tomasz Mrowka is a Polish American mathematician. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology since 2010. A graduate of MIT, he received the Ph.D. from University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby...
together with some new estimates on the spectral flow of Dirac operator
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
s and some methods from ), Taubes proved the longstanding Weinstein conjecture
Weinstein conjecture
In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the current understanding is that a regular compact contact type level set of a Hamiltonian on a symplectic manifold should carry at least one...
for all three-dimensional contact manifolds, thus establishing that the Reeb vector field on such a manifold always has a closed orbit. Expanding both on this and on the equivalence of the Seiberg–Witten and Gromov invariants, Taubes has also proven (in a long series of preprints, beginning with ) that a contact 3-manifold's embedded contact homology is isomorphic to a version of its Seiberg–Witten Floer cohomology. More recently, Taubes, in collaboration with Kutluhan and Lee, have announced and begun publishing a proof that embedded contact homology is isomorphic to Heegaard Floer homology.
Honors and awards
- Four-time speaker at International Congress of MathematiciansInternational Congress of MathematiciansThe International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
(1986, 1994 (plenary), 1998, 2010 (plenary)) - Veblen Prize (AMS) (1991)
- Elie Cartan Prize (Académie des Sciences) (1993)
- Elected as a fellow of the American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
in 1995. - Elected to the National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
in 1996. - Clay Research AwardClay Research AwardThe Clay Research Award is given annually by the Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research...
(2008) - NAS Award in MathematicsNAS Award in MathematicsThe NAS Award in Mathematics is awarded by the U.S. National Academy of Sciences "for excellence of research in the mathematical sciences published within the past ten years." It has been awarded every four years since 1988....
(2008) from the National Academy of Sciences. - Shaw PrizeShaw PrizeThe Shaw Prize is an annual award first presented by the Shaw Prize Foundation in 2004. Established in 2002 in Hong Kong, it honours living "individuals, regardless of race, nationality and religious belief, who have achieved significant breakthrough in academic and scientific research or...
in Mathematics (2009) jointly with Simon DonaldsonSimon DonaldsonSimon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
Books
- Modeling Differential Equations in Biology ISBN 0-13-017325-8
- The L Squared Moduli Spaces on Four Manifold With Cylindrical Ends (Monographs in Geometry and Topology)ISBN 1-57146-007-1
- Metrics, Connections and Gluing Theorems (CBMS Regional Conference Series in Mathematics) ISBN 0-8218-0323-9