## Richard S. Palais

Adjunct Professor, Mathematics

School of Physical Sciences

Ph.D., Harvard Univ., 1956, Mathematics

Phone: (949) 226-1188

Email: palais@uci.edu

University of California, Irvine

Dept. of Math. RH 410H

UC Irvine

Mail Code: 3875

Irvine, CA 92697

School of Physical Sciences

Ph.D., Harvard Univ., 1956, Mathematics

Phone: (949) 226-1188

Email: palais@uci.edu

University of California, Irvine

Dept. of Math. RH 410H

UC Irvine

Mail Code: 3875

Irvine, CA 92697

**Research Interests**

Mathematical Visualization

**URLs**

**Academic Distinctions**

1963-64 NSF Senior Postdoctoral Fellow & Member, Princeton Institute for Advanced Study

1965-67 Sloan Foundation Research Fellow

1965-82 Editor, The Journal of Differential Geometry

1966-68 Chairman, Dept. of Mathematics, Brandeis University

1968-69 Member, Princeton Institute for Advanced Study

1965-82 Editor, The Journal of Differential Geometry

1966-69 Editor, Transactions of the American Mathematical Society

1968-72 Member, NSF Advisory Panel for Mathematics

1970 Invited Speaker, International Congress of Mathematicians, Nice

1971-81 Trustee, American Mathematical Society

1974 Convenor of the Panel on Global Analysis and Analysis on Manifolds, Helsinki International Congress of Mathematicians

1974-75 Member, Princeton Institute for Advanced Study

1976-78 Professor, Univ. of Calif., SantaCruz (on leave from Brandeis)

1978-80 AMS Representative to CBMS

1979-83 Member, U.S. Committee for Mathematics

1979 Founding Chairman of TUG (TEX Users Group)

1980 Elected Fellow of AAAS

1981 Invited Speaker, US-China Symposium on Differential Geometry and Differential Equations, Shanghai-Anhui

1981-82 Gastprofessor, Universitat Bonn Member, Max-Planck-Institut fur Mathematik

1984-85 Member, Mathematical Sciences Research Institute, Berkeley

1987 Visiting Professor, Nankai Institute of Mathematics, Tianjin, China

1987 Chairman, AMS Committee on Communications

1988-89 Chairman, AMS Committee on the Electronic Exchange of Information

1990-1995 Managing Editor, Bulletin of the AMS, Research Expository Papers

1996-1997 Chair of the AAAS Section A (Mathematics)

2006 Awarded First Prize for Illustration, with partner Luc Bénard, in NSF/Science Magazine 2006 International Science and Engineering Visualization Challenge

2007- Editor for Surveys and Research Expository Papers, The Journal for Fixed Point Theory and its

Applications

2009 Awarded First Prize for Illustration, with partner Luc Bénard, in NSF/Science Magazine 2006 International Science and Engineering Visualization Challenge

1965-67 Sloan Foundation Research Fellow

1965-82 Editor, The Journal of Differential Geometry

1966-68 Chairman, Dept. of Mathematics, Brandeis University

1968-69 Member, Princeton Institute for Advanced Study

1965-82 Editor, The Journal of Differential Geometry

1966-69 Editor, Transactions of the American Mathematical Society

1968-72 Member, NSF Advisory Panel for Mathematics

1970 Invited Speaker, International Congress of Mathematicians, Nice

1971-81 Trustee, American Mathematical Society

1974 Convenor of the Panel on Global Analysis and Analysis on Manifolds, Helsinki International Congress of Mathematicians

1974-75 Member, Princeton Institute for Advanced Study

1976-78 Professor, Univ. of Calif., SantaCruz (on leave from Brandeis)

1978-80 AMS Representative to CBMS

1979-83 Member, U.S. Committee for Mathematics

1979 Founding Chairman of TUG (TEX Users Group)

1980 Elected Fellow of AAAS

1981 Invited Speaker, US-China Symposium on Differential Geometry and Differential Equations, Shanghai-Anhui

1981-82 Gastprofessor, Universitat Bonn Member, Max-Planck-Institut fur Mathematik

1984-85 Member, Mathematical Sciences Research Institute, Berkeley

1987 Visiting Professor, Nankai Institute of Mathematics, Tianjin, China

1987 Chairman, AMS Committee on Communications

1988-89 Chairman, AMS Committee on the Electronic Exchange of Information

1990-1995 Managing Editor, Bulletin of the AMS, Research Expository Papers

1996-1997 Chair of the AAAS Section A (Mathematics)

2006 Awarded First Prize for Illustration, with partner Luc Bénard, in NSF/Science Magazine 2006 International Science and Engineering Visualization Challenge

2007- Editor for Surveys and Research Expository Papers, The Journal for Fixed Point Theory and its

Applications

2009 Awarded First Prize for Illustration, with partner Luc Bénard, in NSF/Science Magazine 2006 International Science and Engineering Visualization Challenge

**Research Abstract**

In recent years I have become interested in mathematical visualization, and one of my major ongoing projects is the development and continued improvement of a program called 3D-XplorMath. This is a tool for aiding in the visualization of a wide variety of mathematical objects and processes. Based on what I have learned from my experience in writing this program, I wrote an essay called "The Visualization of Mathematics: Towards a Mathematical Exploratorium" that appeared in the June/July 1999 issue of the Notices of the American Mathematical Society. With the help of Xah Lee, and Luc Benard I have created a Gallery of visualizations, mostly produced using 3D-XplorMath. Luc Benard and I won first prize in the NSF/Science Magazine Scientific Visualization Challenge in 2006 and again in 2009. Our 2006 entry appeared on the cover of the Sept. 22,2006 issue of Scince Magazine.

I have also recently been thinking about integrable, one-dimensional wave equations---an area that is usually referred to as soliton mathematics. I wrote an expository article for the Bulletin of the AMS (October 1997 issue) called The Symmetries of Solitons and I was invited to write an article on wave equations for the Princeton Companion to Mathematics

I have also recently been thinking about integrable, one-dimensional wave equations---an area that is usually referred to as soliton mathematics. I wrote an expository article for the Bulletin of the AMS (October 1997 issue) called The Symmetries of Solitons and I was invited to write an article on wave equations for the Princeton Companion to Mathematics

Publications

A definition of the exterior derivative in terms of Lie derivatives, Proc. Amer. Math. Soc. 5 (1954) 902-908.

**
**

On the differentiability of isometries, Proc. Amer. Math. Soc. 8 (1957) 805-808.

**
**

A Global Formulation of the Lie Theory of Transformation Groups, Memoirs of the AMS #22(1957 ).

**
**

On a class of transformation groups, (with A.M. Gleason), Amer. J. Math. 79 (1957),631-648.

**
**

Imbedding of compact, differentiable,transformation groups in orthogonal representations, J. Math. and Mech. 6 (1957), 674-678.

**
**

A covering homotopy theory and the classification of G-spaces, Proc. Nat. Acad. Sci.U.S.A. 45 (1959), 857-859.

**
**

Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959),125-141.

**
**

Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 274-277.

**
**

Slices and equivariant embeddings, (a chapter from Seminar on Transformation Groups, Ann. of Math. Study #6) 1960

**
**

Sprays, (with W. Ambrose and I. M. Singer), An. Acad. Brasil Ci. 32 (1960) 163-178.

**
**

The Classification of G-spaces, Memoirs of the Amer. Math. Soc. 36 (1960).

**
**

On the cohomology of Lie rings, Proc. of Symposium on Differential Geometry, Tucson, Feb. 1960.

**
**

On the local triviality of the restriction map for embeddings, Comm. Math. Helv. 34 (1960), 306-312.

**
**

Deformations of compact differentiable transformation groups, (with T. E. Stewart) Amer. J. Math. 82 (1960) 935-937.

**
**

Logarithmically exact differential forms, Proc. Amer. Math. Soc. 12 (1961), 50-52.

**
**

Torus bundles over tori, (with T. E. Stewart), Proc. Amer. Math. Soc. 12 (1961), 26-40.

**
**

On the existence of slices for actions of non-compact Lie groups, Annals of Math. 73 (1961), 295-323.

**
**

The cohomology of differentiable transformation groups, (with T. E. Stewart) Amer. J. Math. 83 (1961), 623-644.

**
**

Equivalence of nearly differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362-364.

**
**

Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340.

**
**

Uncountably many inequivalent actions of a compact group on R^n, (with R. Richardson), Proc. Amer. Math. Soc. 14 (1963), 374-377.

**
**

A generalized Morse theory, (with S. Smale), Research Announcement, Bull. Amer. Math. Soc. 70 (1964), 165-172.

**
**

Seminar on the Atiyah-Singer Index Theorem, Annals of Math. Study No. 4 (1964).

**
**

On the homotopy type of certain groups of operators, Topology 3 (1965) 271-279.

**
**

Homeomorphic conjugacy of automorphisms on the Torus, (with L. Adler), Proc. Amer. Math. Soc. 16 (1965) 1222-1225.

**
**

Homotopy theory of infinite dimensional manifolds, Topology 5 (1966) 1-16.

**
**

Lusternik-Schnirelman theory on Banach Manifolds Topology 5 (1966) 115-132.

**
**

Foundations of Global Non-linear Analysis, Benjamin and Co. New York, 1968.

**
**

The classification of real division algebras, Amer. Math. Monthly, 75, (1968 ) 366-368

**
**

Manifolds of sections of fiber bundles and the calculus of variations, Proc.Symp.on Non-linear Analysis, vol. XVIII, Amer. Math. Soc.

**
**

Critical point theory and the minimax principle, Proc. Symp. on Global Analysis, vol. XV, Amer. Math. Soc. 185-212.

**
**

The Morse Lemma for Banach spaces, Bull. Amer. Math. Soc. 75 (1969) 968-971.

**
**

When proper maps are closed, Proc. Amer. Math. Soc. 24 (1969) 835

**
**

C^1 action of compact Lie groups on compact manifolds are C^1 equivalent to C-infinity actions, Amer. Jour. Math. 88 (1970) 748-760.

**
**

Banach manifolds of fiber bundle sections, Proc. International Cong. Math. Nice, 1970, vol. 2, 243-249, 1971.

**
**

Natural bundles have finite order, (with C. L. Terng), Topology, 16 (1977) 271-277.

**
**

A topological Gauss-Bonnet theorem, Jour. of Diff. Geometry, 13 (1978) 385-398.

**
**

An analogue of Hartogs' theorem in an algebraic setting, Amer. Jour. Math. 100 (1978), 387-406.

**
**

The Principal of Symmetric Criticality, Comm. Math. Physics, 69 (1979) 19-30.

**
**

The Geometrization of Physics, Tsing-Hua University Press, Hsinchu, Taiwan (1981).

**
**

Real Algebraic Differential Geometry, Math. Lecture series #10, Publish or Perish Inc. 1981.

**
**

Warped product Einstein manifolds and Hessian PDE, (with A. Derdzinski and C.-L. Terng), preliminary report in Proc. of 1982 Munster Conference on Differential Geometry.

**
**

Applications of the symmetric criticality principal in mathematical physics and differential geometry, Proc. Symp., Shanghai/China 1981, 247-301 (1984).

**
**

Geometry and topology of isoparametric submanifolds, (with W. Y. Hsiang and C. L. Terng), (Research announcement), Proc. Nat. Acad. Sci. USA, 82(1985) 4863-4865.

**
**

Reduction of variables for minimal submanifolds, (with C.L. Terng), Proc. Amer. Math. Soc., 98 (1986) 480-484.

**
**

A general canonical form theorem, (with C.L. Terng), Transactions Amer. Math. Soc., 300 (1987) 771-789.

**
**

Geometry of canonical forms, (with C.L. Terng), (Proceedings of the Kovaleskaya Symposium) Contemporary Mathematics, 64 (1987) 133-151.

**
**

Topology of isoparametric submanifolds (with W.Y. Hsiang and C.L. Terng), Jour. of Diff. Geometry, 27 (1988) 423-460.

**
**

Critical Point Theory and Submanifold Geometry (with C.-L.Terng), Lecture Notes in Mathematics #1353,(280 pages) Springer-Verlag, NY (1988)

**
**

The Life and Mathematics of Shiing-Shen Chern, in "Chern---A Great Geometer of the Twentieth Century", edited by S.T. Yau, International Press, Hong Kong, 1992.

**
**

Hyperpolar actions and k-flat homogeneous spaces, (with E. Heintze, G. Thorbergsson, and C.-L. Terng) J. reine angew. Math, 454 (1994) 163--179

**
**

Symmetry, Criticality, and Condition C, In "Proceedings of the Sophus Lie Memorial Conference", edited by O. A. Laudal and B. Jahren, Scandinavian University Press (1994) 77-98

**
**

Hyperpolar actions on symmetric spaces, (with E. Heintze, G. Thorbergsson, and C.-L. Terng) In "Geometry, Topology, and Physics For Raoul Bott", edited by S.T. Yau, International Press (1995) 214-245

**
**

The Symmetries of Solitons,Bull. Amer. Math. Soc. 34 (1997), 339-403.

**
**

The Visualization of Mathematics: Towards a Mathematical Exploratorium, Notices Amer. Math. Soc., 46, No.6 (June-July 1999)

**
**

A Simple Proof of the Banach Contraction Principle, The Journal for Fixed Point Theory and its Applications, 2 (2007) 221--223

**
**

Euler's Fixed Point Theorem: The Axis of a Rotation (with Bob Palais), The Journal for Fixed Point Theory and its Applications, 2 (2007) 215--220

**
**

Linear and Nonlinear Waves and Solitons, in The Princeton Companion to Mathematics, T. Gower Ed., Princeton Univ. Press 2008, 234-239

**
**

An Introduction to Ordinary Differential Equations and Classical Mechanics, (with Robert A. Palais) AMS STML-51, ISBN-10: 0-8218-2138-5 (2009)

**
**

Euler's Theorem on the Axis of a Rotation: Proofs Old and New (with Bob Palais and Stephen Rodi), Amer. Math. Monthly, 116, (2009 ) 892-909

**Grant**

The National Science Foundation DUE Award #0514781

**Professional Society**

AMS, MAA, ACM, AAAS

**Other Experience**

Professor of Mathematics

Brandeis Univ. 1963—2003

Brandeis Univ. 1963—2003

**Link to this profile**

**Last updated**

06/13/2014