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 Research
Riemannian geometry, intrinsic metric spaces
Ph. D. Students 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: 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. 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.
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.
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
Stephanie Alexander(PhD 1967) (web page
SBA)
Larry Lipskie(PhD 1975)
Mark Thomas(PhD 1983)
Chien-Hsiung Chen(PhD 1996)
Jeffrey Ho(PhD 1999)
Bishop-O'Neill, 1969
Riemannian geometry, July, 2013
Lie groups, 2013
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