Richard L. Bishop
Professor Emeritus, Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana, Illinois 61801-2975

Office: 329 Illini Hall
Phone: office (217) 244-7339; home (217) 328-6379
FAX: (217) 333-9576
Home address: 3514 N Highcross Rd
Urbana, IL, 61802
e-mail: Richard L. Bishop

General Information
B.S. Case Institute of Technology, 1954
Ph.D. MIT, 1959
Thesis advisor: I. M. Singer
UIUC faculty member since 1959.
Visiting Appointments: UCLA, MIT

Research Riemannian geometry, intrinsic metric spaces

Ph. D. Students
Stephanie Alexander(PhD 1967) (web page SBA)
Larry Lipskie(PhD 1975)
Mark Thomas(PhD 1983)
Chien-Hsiung Chen(PhD 1996)
Jeffrey Ho(PhD 1999)

Recent research paper

In 2004 Professor Stephanie Alexander and I published a paper which gave sufficient conditions for a warped product of metric spaces to have a curvature bound, above or below. Now we have written a paper in which we establish that those sufficient conditions are also necessary. pdf

Notable old publications

In 1964 Richard J. Crittenden and I authored the first book which treated Riemannian geometry in a modern style which has had a lasting presence. It was published under the title "Geometry of Manifolds" in 1964 by Academic Press and reprinted in 2000 by AMS-Chelsea with some corrections. In the last chapter this book contains the original proof of an important new research result by me which is now called the Bishop Volume Theorem. Subsequently this theorem has become a key input to further research, starting with estimates on the growth of the fundamental group of a negatively curved manifold by John Milnor. It was used extensively and very effectively by M. Gromov and some authors have referred (mistakenly) to it as the Bishop-Gromov Volume Theorem. Besides being available for sale from the AMS, it also has a Google eBook version.

In Academic Year 1967-68 Barrett O'Neill and I did an extensive project on manifolds of negative curvature. We published our work under the title "Manifolds of Negative Curvature", 1969, Transactions of the AMS. Up to now (2012) no electronic version has been available, but now it can be downloaded in pdf format:
Bishop-O'Neill, 1969

Recently I have produced lecture notes in pdf form from courses I taught on Riemannian geometry and Lie groups.

The one on Riemannian Geometry uses the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material on differential forms and Lie groups, and the advanced material on Riemannian imbeddings.
Riemannian geometry, July, 2013

The one on Lie groups follows the pattern of Chevalley's book for the basics: it starts with matrix examples, then the basic theory about the Lie group-Lie algebra relation. Then there is a section on topological groups based on Pontryagin's treatment. The material on representation theory ends with the Peter-Weyl theorem. The remaining third is more unusual, covering invariants of group actions, special functions, and actions on differential equations.
Lie groups, 2013

In 1969 I completed a research project concerning the board game of Monopoly. Since the mathematical foundation (probability) was not my specialty, I consulted with a late colleague, Robert B. Ash, to make sure the terminology was correct. There were very few improvements needed, but I asked him to be listed as a joint author. A summary version, "Monopoly as a Markov Process", was published in 1972 in Mathematics Magazine. The main result is a table of expected limit frequencies for all the positions of a token. Subsequently, this same result was obtained by computer simulation; however, my treatment was based on a very accurate Markov process model, from which a further analysis of the rate of convergence could be derived. Moreover, I intended the paper to be primarily educational, illustrating that it was feasible to calculate all the important data for a very complicated Markov process which had an interesting application; this is in great contrast to the toy examples I saw in textbooks. I have no off-prints left, but a more detailed electronic version is available here.