Albert Marden | |
---|---|
Born | |
Nationality | American |
Alma mater | Harvard University |
Scientific career | |
Fields | Mathematics |
Institutions | University of Minnesota |
Doctoral advisor | Lars Ahlfors |
Albert Marden (born 18 November 1934) is an American mathematician, specializing in complex analysis and hyperbolic geometry.
Marden received his PhD in 1962 from Harvard University with thesis advisor Lars Ahlfors. [1] Marden has been a professor at the University of Minnesota since the 1970s, where he is now professor emeritus. He was a member of the Institute for Advanced Study (IAS) in the academic year 1969–70, Fall 1978, and Fall 1987. [2]
His research deals with Riemann surfaces, quadratic differentials, Teichmüller spaces, hyperbolic geometry of surfaces and 3-manifolds, Fuchsian groups, Kleinian groups, complex dynamics, and low-dimensional geometric analysis.
Concerning properties of hyperbolic 3-manifolds, Marden formulated in 1974 the tameness conjecture, [3] which was proved in 2004 by Ian Agol and independently by a collaborative effort of Danny Calegari and David Gabai. [4]
In 1962, he gave a talk (as an approved speaker but not an invited speaker) on A sufficient condition for the bilinear relation on open Riemann surfaces at the International Congress of Mathematicians in Stockholm. In 2012 he was elected a Fellow of the American Mathematical Society. His doctoral students include Howard Masur.
William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal for his contributions to the study of 3-manifolds in 1982.
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
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 act.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional 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 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 Kleinian group is a discrete subgroup of the group of the Möbius transformations. This group, denoted PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by its center, which consists of the identity matrix and its product by −1. The group PSL(2, C) has several natural representations: as conformal transformations of the Riemann sphere; as orientation-preserving isometries of the 3-dimensional hyperbolic space H3; and as orientation-preserving conformal transformations of the open unit ball B3 in R3. 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 (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) 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 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 topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover that is a Haken manifold.
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky (1877).
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
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.
David Bernard Alper Epstein FRS is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded the University of Warwick mathematics department with Christopher Zeeman and is founding editor of the journal Experimental Mathematics.
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston (1982), 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.
The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks.
In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0.
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
Francis Bonahon is a French mathematician, specializing in low-dimensional topology.
Michael Kapovich is a Russian-American mathematician.
Godfrey Peter Scott, known as Peter Scott, is a British mathematician, known for the Scott core theorem.