## Richard S. Palais

Adjunct Professor, Mathematics

School of Physical Sciences

School of Physical Sciences

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

University of California, Irvine

Dept. of Math. RH 410H

UC Irvine

Mail Code: 3875

Irvine, CA 92697

Dept. of Math. RH 410H

UC Irvine

Mail Code: 3875

Irvine, CA 92697

**Research Interests**

Mathematical Visualization

**Websites**

**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

**Grants**

The National Science Foundation DUE Award #0514781

**Professional Societies**

AMS, MAA, ACM, AAAS

**Other Experience**

Professor of Mathematics

Brandeis Univ. 1963—2003

Brandeis Univ. 1963—2003

**Last updated**

06/13/2014