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William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois & Gallot (1996) gave the simplest available proof.
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.
In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.
In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G.
James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
In geometry, a pleated surface is roughly a surface that may have simple folds but is not crumpled in more complicated ways. More precisely, a pleated surface is an isometry from a complete hyperbolic surface S to a hyperbolic 3-fold such that every point of S is in the interior of a geodesic that is mapped to a geodesic. They were introduced by Thurston, where they were called uncrumpled surfaces.
In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by William Thurston.
Yair Nathan Minsky is an Israeli-American mathematician whose research concerns three-dimensional topology, differential geometry, group theory and holomorphic dynamics. He is a professor at Yale University. He is known for having proved Thurston's ending lamination conjecture and as a student of curve complex geometry.
Francis Bonahon is a French mathematician, specializing in low-dimensional topology.
Albert Marden is an American mathematician, specializing in complex analysis and hyperbolic geometry.
References
- Canary, R. D.; Epstein, D. B. A.; Green, P. (2006) [1987], "Notes on notes of Thurston", in Canary, Richard D.; Epstein, David; Marden, Albert (eds.), Fundamentals of hyperbolic geometry: selected expositions, London Mathematical Society Lecture Note Series, vol. 328, Cambridge University Press, doi:10.1017/CBO9781139106986, ISBN 978-0-521-61558-7, MR 0903850
- Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes (original notes)
- Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes (TeX version)
- Thurston, William P. (1997), Levy, Silvio (ed.), Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, doi:10.1515/9781400865321, ISBN 978-0-691-08304-9, MR 1435975
- Thurston, William P. (2022), Farb, Benson; Gabai, David; Kerckhoff, Steven P. (eds.), The geometry and topology of three-manifolds, American Mathematical Society, ISBN 978-1-4704-6391-5
- ↑ Thurston 2022 pg. xi