Robert Riley (mathematician)

Robert Riley (December 22, 1935–March 4, 2000[1]) was an american mathematician. He is known for his work in low-dimensional topology using computational tools and hyperbolic geometry, being one of the inspirations for William Thurston's later breakthroughs in 3-dimensional topologys[2].

Career

Riley earned a bachelor's degree in mathematics from MIT in 1957; shortly thereafter he dropped out of the graduate program and went on to work in the industry, eventually moving to Amsterdam in 1966. In 1968 he took a temporary position at the university of Southampton[3]. He defended his PhD at this institution in 1980, under the nominal direction of David Singerman[4]. For the next two years he occupied a postdoctoral position in Boulder where William Thurston was employed at the time, before moving on to Binghamton University as a professor[3].

Mathematical work

Riley's research was in geometric topology, especially in knot theory, where he mostly studied representations of knot groups. Early on, following work of Ralph Fox, he was interested in morphisms to finite groups. Later on in Southampton, considering -representations sending peripheral elements to parabolics led him to discover the hyperbolic structure on the complement of the figure-eight knot and some others[5][6]. 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)[7]. One notable feature of Riley's work is that it relied much on the assistance of a computer[8].

Selected publications
    R75a.Riley, Robert (1975a). "Discrete parabolic representations of link groups". Mathematika. 22 (2): 141–150. MR 0425946.
      R75b.Riley, Robert (1975b). "A quadratic parabolic group". Math. Proc. Cambridge Philos. Soc. 77: 281–288. MR 0412416.
        R13.Riley, Robert (2013). "A personal account of the discovery of hyperbolic structures on some knot complements". Expositiones Mathematicae. 31 (2): 104–115. arXiv:1301.4601. MR 3057120.

        Notes
        1. "Deaths–Robert Freed Riley" (PDF). notices of the AMS. 47 (6): 679. 2000.
        2. Thurston 1982, p. 360.
        3. Brin, Jones & Singerman 2013.
        4. Robert Riley at the Mathematics Genealogy Project
        5. Riley 1975a.
        6. Riley 1975b.
        7. Thurston 1982.
        8. Riley 2013.
        gollark: Probably. The main issue I can see is that you would have to rewrite the entire metadata block on changes, because start/end in XTMF are offsets from the metadata region's end.
        gollark: I thought about that, but:- strings in a binary format will be about the same length- integers will have some space saving, but I don't think it's very significant- it would, in a custom one, be harder to represent complex objects and stuff, which some extensions may be use- you could get some savings by removing strings like "title" which XTMF repeats a lot, but at the cost of it no longer being self-describing, making extensions harder and making debugging more annoying- I am not convinced that metadata size is a significant issue
        gollark: I mean, "XTMF with CBOR/msgpack and compression" was being considered as a hypothetical "XTMF2", but I'd definitely want something, well, self-describing.
        gollark: Also also, why a binary format?
        gollark: Also, XTMF can do runtime update, you just need to allocate, say, 4KB at the start of the tape, and write metadata to that. The offsets might be fiddly, though.

        References
        • Thurston, William P. (1982). "Three dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. 6 (3): 357–382. doi:10.1090/S0273-0979-1982-15003-0.
        • Brin, Matthew G.; Jones, Gareth A.; Singerman, David (2013). "Commentary on Robert Riley's article "A personal account of the discovery of hyperbolic structures on some knot complements"". Expositiones Mathematicae. 31 (2): 99–103. arXiv:1301.4599. MR 3057119.
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