Robert F. Riley (December 22, 1935–March 4, 2000) was an American mathematician. He is known for his work in

low-dimensional topology 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 theory of 3-manifolds and 4-manifolds, knot theory, ...

using computational tools and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, being one of the inspirations for

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) 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. Thurst ...

's later breakthroughs in

3-dimensional topology 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 (a tangent plane) to a small ...

Career

Riley earned a bachelor's degree in mathematics from

MIT The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...

in 1957; shortly thereafter he dropped out of the graduate program and went on to work in industry, eventually moving to Amsterdam in 1966. In 1968 he took a temporary position at the

University of Southampton The University of Southampton (abbreviated as ''Soton'' in post-nominal letters) is a public university, public research university in Southampton, England. Southampton is a founding member of the Russell Group of research-intensive universit ...

. He defended his Ph.D. at this institution in 1980, under the nominal direction of David Singerman. For the next two years he occupied a postdoctoral position in

Boulder In geology, a boulder (or rarely bowlder) is a rock fragment with size greater than in diameter. Smaller pieces are called cobbles and pebbles. While a boulder may be small enough to move or roll manually, others are extremely massive. In ...

where William Thurston was employed at the time, before moving on to Binghamton University as a professor.

Mathematical work

Riley's research was in

geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...

, especially in

knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, where he mostly studied

representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...

knot group In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Other conventions consi ...

s. Early on, following work of

Ralph Fox Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played ...

, he was interested in morphisms to finite groups. Later on in Southampton, considering

\mathrm_2(\mathbb C)

-representations sending peripheral elements to parabolics led him to discover the hyperbolic structure on the complement of the

figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under ...

and some others. This was one of the few examples of hyperbolic 3-manifolds that were available at the time, and as such it was one of the motivations which led to William Thurston's geometrisation conjecture, which includes as a particular case a criterion for a knot complement to support a hyperbolic structure. One notable feature of Riley's work is that it relied much on the assistance of a computer.

Selected publications

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Notes

References

* * {{DEFAULTSORT:Riley, Robert 2000 deaths American topologists 20th-century American mathematicians