Concept

Hyperbolic link

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. As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds. Borromean rings are hyperbolic. Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.

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https://en.wikipedia.org/wiki/Hyperbolic_link

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Nicolas Monod

Jan Sickmann Hesthaven

Prof. Hesthaven received an M.Sc. in computational physics from the Technical University of Denmark (DTU) in August 1991. During the studies, the last 6 months of 1989 was spend at JET, the european fusion laboratory in Culham, UK. Following graduation, he was awarded a 3 year fellowship to begin work towards a Ph.D. at Riso National Laboratory in the Department of Optics and Fluid Dynamics. During the 3 years of study, the academic year of 1993-1994 was spend in the Division of Applied Mathematics at Brown University and three 3 months during the summer of 1994 in Department of Mathematics and Statistics at University of New Mexico. In August 1995, he recieved a Ph.D. in Numerical Analysis from the Institute of Mathematical Modelling (DTU). Following graduation in August 1995, he was awarded an NSF Postdoctoral Fellowship in Advanced Scientific Computing and was approinted Visiting Assistant Professor in the Division of Applied Mathematics at Brown University. In December of 1996, he was appointed consultant to the Institute of Computer Applications in Science and Engineering(ICASE) at NASA Langley Research Center (NASA LaRC). As of July 1999, he was appointed Assistant Professor of Applied Mathematics, in September 2000 he was awarded an Alfred P. Sloan Fellowship, as of July 2001 he was awarded a Manning Assistant Professorship, and in March 2002, he was awarded an NSF Career Award. In January 2003, he was promoted to Associate Professor of Applied Mathematics with tenure and in May 2004 he was awarded Philip J. Bray Award for Excellence in Teaching in the Sciences (the highest award given for teaching excellence in all sciences at Brown University). He was promoted to Professor of Applied Mathematics as of July 2005. From October 2006 to June 2013, he was the Founding Director of the Center for Computation and Visualization (CCV) at Brown University. As of October 2007, he holds the (honorary) title of Professor (Adjunct) at the Technical University of Denmark. In November 2009, he successfully defended his dr.techn thesis at the Technical University of Denmark and was rewarded the degree of Doctor Technices -- the highest academic distinction awarded based on ... substantial and lasting contributions that has helped to move the research area forward and penetrated into applications. As grant Co-PI he served from Aug 2010 to June 2013 as Deputy Director of the Institute of Computational and Experimental Research in Mathematics (ICERM), the newest NSF Mathematical Sciences Research Institute. After having spend his entire academic career at Brown University, Prof Hesthaven decided to pursue new challenges and joined the Mathematics Institute of Computational Science and Engineering (MATHICSE) at Ecole Polytechnique Fédérale de Lausanne (EPFL) in Switzerland in July 2013. In March 2014 he was elected SIAM Fellow for contributions to high-order methods for partial differential equations.

Related concepts (4)

Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

Hyperbolic volume

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.

Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

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