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.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds,defined as smooth manifolds with a Riemannian metric. This gives,in particular,local notions of angle,length of curves,surface area and volume. From those,some other global quantities can be derived by integrating local contributions.
In mathematics,Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces,which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions,it is not always possible to assign a single geometry to a whole topological space. Instead,the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982),and implies several other conjectures,such as the Poincaréconjecture and Thurston's elliptization conjecture.
In mathematics,a building is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds,finite projective planes,and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
In the mathematical disciplines of topology and geometry,an orbifold is a generalization of a manifold. Roughly speaking,an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
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.
Richard Streit Hamilton is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. Hamilton is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaréconjecture and geometrization conjecture in the field of geometric topology. Grigori Perelman built upon Hamilton's results to prove the conjectures,and was awarded a Millennium Prize for his work. However,Perelman declined the award,regarding Hamilton's contribution as being equal to his own.
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 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.
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words,it is a -bundle over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces,and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
In mathematics,a manifold is a topological space that locally resembles Euclidean space near each point. More precisely,an -dimensional manifold,or -manifold for short,is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In mathematics,Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry,he is a leading researcher in those subjects.
Tian Gang is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry,Gromov-Witten theory,and geometric analysis.
In topology,a branch of mathematics,an aspherical space is a topological space with all homotopy groups equal to 0 when .
In mathematics,spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds,one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces,where one can use comparison triangles to quantify the curvature of a space;in this context,non-positively curved spaces are known as (locally) CAT(0) spaces.
Francis Thomas Farrell is an American mathematician who has made contributions in the area of topology and differential geometry. Farrell is a distinguished professor emeritus of mathematics at Binghamton University. He also holds a position at the Yau Mathematical Sciences Center,Tsinghua University.
Ralph Martin Kaufmann is a German mathematician working in the United States.
Michael Kapovich is a Russian-American mathematician.
