Celebratio Mathematica

R. S. Palais: “A definition of the exterior derivative in terms of Lie derivatives,” Proc. Am. Math. Soc. 5 : 6 (1954), pp. 902–908. MR 65996 Zbl 0057.13301 article

R. S. Palais: A global formulation of the Lie theory of transformation groups. Ph.D. thesis, Harvard University, 1956. Advised by A. M. Gleason and G. Mackey. Also published as Mem. Am. Math. Soc. 22 (1957). MR 2938771 phdthesis

Andrew Mattei Gleason	Related	Volume
George Whitelaw Mackey	Related

R. S. Palais: “On the differentiability of isometries,” Proc. Am. Math. Soc. 8 : 4 (1957), pp. 805–807. MR 88000 Zbl 0084.37405 article

A. M. Gleason and R. S. Palais: “On a class of transformation groups,” Am. J. Math. 79 : 3 (July 1957), pp. 631–648. MR 0089367 Zbl 0084.03203 article

R. S. Palais: “Imbedding of compact, differentiable transformation groups in orthogonal representations,” J. Math. Mech. 6 : 4 (1957), pp. 673–678. MR 92927 Zbl 0086.02603 article

R. S. Palais: “A global formulation of the Lie theory of transformation groups,” Mem. Amer. Math. Soc. 22 (1957), pp. iii+123. Republication of the author’s 1956 PhD thesis. MR 121424 Zbl 0178.26502 article

R. S. Palais: “Natural operations on differential forms,” Trans. Am. Math. Soc. 92 : 1 (1959), pp. 125–141. MR 116352 Zbl 0092.30802 article

R. S. Palais: “A covering homotopy theorem and the classification of \( G \)-spaces,” Proc. Nat. Acad. Sci. U.S.A. 45 : 6 (June 1959), pp. 857–859. MR 121799 Zbl 0105.16901 article

R. S. Palais: “Extending diffeomorphisms,” Proc. Am. Math. Soc. 11 : 2 (1960), pp. 274–277. MR 117741 Zbl 0095.16502 article

R. S. Palais and T. E. Stewart: “Deformations of compact differentiable transformation groups,” Am. J. Math. 82 : 4 (October 1960), pp. 935–937. MR 120652 Zbl 0106.16401 article

R. S. Palais: “Local triviality of the restriction map for embeddings,” Comment. Math. Helv. 34 : 1 (December 1960), pp. 305–312. MR 123338 Zbl 0207.22501 article

Let \( M \) and \( W \) be differentiable manifolds, \( V \) a compact submanifold of \( W \), and let \( E(W,M) \) denote the space of differentiable embeddings of \( W \) in \( M \) in the \( C^r \) topology where \( 1\leq r \leq \infty \) (see below for precise definitions). In a set of mimeographed notes R. Thom has stated that the restriction map \[ \pi: E(W,M) \to E(V,M) \] defined by \( \pi(f) = f|V \), is a fiber space map in the sense of Serre (i.e. has the covering homotopy property for polyhedra), however the proof indicated shows only that homotopies satisfying certain strong differentiability conditions in the parameter can be lifted. In this paper we will show that not only is the theorem stated by Thom correct but in fact something considerably stronger is true; namely the map \( \pi \) is locally trivial.

W. Ambrose, R. S. Palais, and I. M. Singer: “Sprays,” An. Acad. Brasil. Ci. 32 : 2 (June 1960), pp. 163–178. MR 126234 Zbl 0097.37904 article

Isadore M. Singer	Related	Volume
Warren Arthur Ambrose	Related

R. S. Palais: The classification of \( G \)-spaces. Memoirs of the American Mathematical Society 36. American Mathematical Society (Providence, RI), 1960. MR 177401 Zbl 0119.38403 book

R. S. Palais: “Slices and equivariant embeddings,” Chapter 8, pp. 101–116 in A. Borel, G. Bredon, E. E. Floyd, D. Montgomery, and R. Palais: Seminar on transformation groups (Princeton, NJ, 1958–1959). Edited by A. Borel. Annals of Mathematics Studies 6. 1960. incollection

David Campbell Montgomery	Related
Edwin E. Floyd	Related
Armand Borel	Related
Glen Eugene Bredon	Related

R. S. Palais: “Logarithmically exact differential forms,” Proc. Am. Math. Soc. 12 : 1 (1961), pp. 50–52. MR 123339 Zbl 0196.38903 article

R. S. Palais and T. E. Stewart: “Torus bundles over a torus,” Proc. Am. Math. Soc. 12 : 1 (1961), pp. 26–29. MR 123638 Zbl 0102.38702 article

If a compact Lie group \( P \) acts on a completely regular topological space \( E \) then \( E \) is said to be a principal \( P \)-bundle if whenever the relation \( px = x \) holds for \( p\in P \), \( x\in E \) it follows that \( p = e \), the identity of \( P \). The orbit space \( X = E/P \) is called the base space and the map \[ \pi:E\to X \] carrying \( y \) into its orbit \( P\cdot y \) is called the projection. Suppose now that \( G \) is a Lie group, \( H_i \) a closed subgroup of \( G \) and \( H \) a closed, normal subgroup of \( H_i \) such that \( P = H_i/H \) is compact. Then \( E = G/H \) becomes a principal \( P \)-bundle with base space \( X = G/H_1 \) and projection \[ gH\mapsto gH_i \] under the action \[ p(gH) = gp^{-1}H .\] Such a principal bundle will be called canonical.

The purpose of this note is to show that if \( X \) is a torus of dimension \( n \) and \( P \) a torus of dimension \( m \) then every principal \( P \)-bundle over \( X \) is canonical and further that the group \( G \) lies in an extremely narrow class. Roughly speaking, our method is to try to lift the action of euclidean \( n \)-space up to the total space of the bundle and observe what obstructs this effort.

R. S. Palais: “The cohomology of Lie rings,” pp. 130–137 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. MR 125867 Zbl 0126.03404 incollection

R. S. Palais: “On the existence of slices for actions of non-compact Lie groups,” Ann. Math. (2) 73 : 2 (March 1961), pp. 295–323. MR 126506 Zbl 0103.01802 article

R. S. Palais: “Equivalence of nearby differentiable actions of a compact group,” Bull. Am. Math. Soc. 67 : 4 (1961), pp. 362–364. MR 130321 Zbl 0102.38101 article

R. S. Palais and T. E. Stewart: “The cohomology of differentiable transformation groups,” Am. J. Math. 83 : 4 (1961), pp. 623–644. MR 140613 Zbl 0104.17703 article

If a Lie group \( G \) acts differentiably on a manifold \( \mathscr{M} \) then various spaces of tensor field on \( \mathscr{M} \) become in a natural way modules for the Lie algebra \( \mathscr{G} \) of \( G \) and the cohomology of \( \mathscr{G} \) with these coefficient modules in certain cases carries interesting information about the action of \( G \). In this paper we will discuss this situation, at first in a somewhat more abstract setup, and develop a method for computing these cohomology groups in certain cases. In particular we shall show that if \( G \) is compact and semi-simple then even though these modules are infinite dimensional the conclusions of the First and Second Whitehead Lemmas [1937, 1936] are valid; namely the first and second cohomology groups are trivial. As one consequence we will show that differentiable actions of compact, semi-simple Lie groups admit only trivial infinitesimal deformations, a fact whose global analogue will be found in [Palais and Stewart 1960]. Our second and motivating application of these general cohomology results is to a question initiated by one of the authors in [Stewart 1960]. Namely if a Lie group \( G \) acts differentiably on the base space of a differentiable fiber bundle \( B \) over \( M \) can \( G \) be made to act differentiably on \( B \) so as to be equivariant with respect to the fiber projection and so that each operation of \( G \) on \( B \) is a bundle map. We show here that the answer is yes if \( G \) is compact and simply connected and if the structural group of \( B \) is a solvable Lie group, and moreover that the way of “lifting” the action of \( G \) to \( B \) is essentially unique.

R. S. Palais and R. W. Richardson, Jr.: “Uncountably many inequivalent analytic actions of a compact group on \( R^n \),” Proc. Am. Math. Soc. 14 : 3 (June 1963), pp. 374–377. MR 148796 Zbl 0121.17904 article

R. S. Palais: “Morse theory on Hilbert manifolds,” Topology 2 : 4 (May 1963), pp. 299–340. MR 158410 Zbl 0122.10702 article

R. S. Palais and S. Smale: “A generalized Morse theory,” Bull. Am. Math. Soc. 70 : 1 (1964), pp. 165–172. MR 158411 Zbl 0119.09201 article

R. S. Palais: “On the homotopy type of certain groups of operators,” Topology 3 : 3 (May 1965), pp. 271–279. MR 175130 Zbl 0161.34501 article

R. L. Adler and R. Palais: “Homeomorphic conjugacy of automorphisms on the torus,” Proc. Am. Math. Soc. 16 : 6 (1965), pp. 1222–1225. MR 193181 Zbl 0229.22013 article

Seminar on the Atiyah–Singer index theorem. Edited by R. S. Palais. Annals of Mathematics Studies 57. Princeton University Press, 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.). Russian translation published as Seminar po teoreme At’i–Zingera ob indekse (1970). See also a similarly-titled article in Differential analysis (1964). MR 0198494 Zbl 1103.58013 book

M. F. Atiyah	Related	Volume
Edwin E. Floyd	Related
Robert Thomas Seeley	Related
W. Shih	Related
Robert Martin Solovay	Related
Armand Borel	Related

@book {key0198494m,
	TITLE = {Seminar on the {A}tiyah--{S}inger index
		 theorem},
	EDITOR = {Palais, Richard S.},
	SERIES = {Annals of Mathematics Studies},
	NUMBER = {57},
	PUBLISHER = {Princeton University Press},
	YEAR = {1965},
	PAGES = {x+366},
	NOTE = {With contributions by M. F. Atiyah,
		 A. Borel, E. E. Floyd, R. T. Seeley,
		 W. Shih and R. Solovay. This describes
		 the original proof of the index theorem.
		 (Atiyah and Singer never published their
		 original proof themselves, but only
		 improved versions of it.). Russian translation
		 published as \textit{Seminar po teoreme
		 At'i--Zingera ob indekse} (1970). See
		 also a similarly-titled article in \textit{Differential
		 analysis} (1964). MR:0198494. Zbl:1103.58013.},
	ISSN = {0066-2313},
	ISBN = {9780691080314},
}

R. S. Palais: “Homotopy theory of infinite dimensional manifolds,” Topology 5 : 1 (March 1966), pp. 1–16. MR 189028 Zbl 0138.18302 article

In the past several years there has been considerable interest in the theory of infinite dimensional differentiable manifolds. While most of the developments have quite properly stressed the differentiable structure, it is nevertheless true that the results and techniques are in large part homotopy theoretic in nature. By and large homotopy theoretic results have been brought in on an ad hoc basis in the proper degree of generality appropriate for the application immediately at hand. The result has been a number of overlapping lemmas of greater or lesser generality scattered through the published and unpublished literature. The present paper grew out of the author’s belief that it would serve a useful purpose to collect some of these results and prove them in as general a setting as is presently possible.

R. S. Palais: “Lusternik–Schnirelman theory on Banach manifolds,” Topology 5 : 2 (1966), pp. 115–132. MR 259955 Zbl 0143.35203 article

Several years ago the author, and independently Smale, generalized the Morse theory of critical points to cover certain functions on hilbert manifolds [Palais 1963; Palais and Smale 1964]. Shortly thereafter J. Schwartz showed how the same techniques allowed one also to extend the Lusternik–Schnirelman theory of critical points to functions on hilbert manifolds [1964]. Now while Morse theory, by its nature, is more or less naturally restricted to hilbert manifolds, the natural context for Lusternik–Schnirelman theory is a smooth real valued function on a Banach manifold. In this paper we shall extend Schwartz’s results to cover this more general situation. It is not however simply a desire for generality in an abstract theorem that prompted the present work. The motivation for considering critical point theory in the context of infinite dimensional manifolds comes from the immediate applicability of results to proving existence theorems in the Calculus of Variations. As long as the integrands one considers are of a basically quadratic nature, hilbert manifolds of maps belonging to some Sobolev space \( L_k^2 \) are the natural manifolds in which to formulate the problems. However for more general integrands it becomes necessary to use Banach manifolds, for example manifolds of maps belonging to one of the more general Sobolev spaces \( L_k^p \). Applications in this direction will be found in a forthcoming paper of Browder [1965].

R. S. Palais: “The classification of real division algebras,” Am. Math. Monthly 75 : 4 (April 1968), pp. 366–368. MR 228539 Zbl 0159.04403 article

R. S. Palais: Foundations of global non-linear analysis. Mathematics Lecture Note Series 16. W. A. Benjamin (New York), 1968. MR 248880 Zbl 0164.11102 book

R. S. Palais: “The Morse lemma for Banach spaces,” Bull. Am. Math. Soc. 75 : 5 (1969), pp. 968–971. MR 253378 Zbl 0191.21704 article

R. S. Palais: “When proper maps are closed,” Proc. Am. Math. Soc. 24 : 4 (April 1970), pp. 835–836. MR 254818 Zbl 0189.53202 article

A subset \( F \) of a space \( Y \) is called compactly closed if its intersection with each compact subset of \( Y \) is compact. A Hausdorff space is called a \( k \)-space if every compactly closed set is closed. A map of a space \( X \) to a space \( Y \) is called proper if the inverse image of each compact subset of \( Y \) is a compact subset of \( X \). The following Lemma is a trivial consequence of these definitions.

A subset \( F \) of a space \( Y \) is compactly closed if and only if the inclusion map of the subspace \( F \) into \( Y \) is proper.

It is frequently of interest in applications to know that a given proper map is closed. We provide here a simple proof of a criterion due to G. T. Whyburn [1965].

R. S. Palais: “Critical point theory and the minimax principle,” pp. 185–212 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 15. American Mathematical Society (Providence, RI), 1970. MR 264712 Zbl 0212.28902 incollection

Stephen Smale	Related
S. S. Chern	Related	Volume

R. S. Palais: “\( C^1 \) actions of compact Lie groups on compact manifolds are \( C^1 \)-equivalent to \( C^{\infty} \) actions,” Am. J. Math. 92 : 3 (July 1970), pp. 748–760. MR 268912 Zbl 0203.26203 article

R. S. Palais: “Manifolds of sections of fiber bundles and the calculus of variations,” pp. 195–205 in Nonlinear functional analysis (Chicago, 16–19 April 1968), Part 1. Edited by F. K. Browder. Proceedings of Symposia in Pure Mathematics 18. American Mathematical Society (Providence, RI), 1970. MR 271973 Zbl 0235.58008 incollection

Seminar po teoreme At’i–Zingera ob indekse [Seminar on the Atiyah–Singer index theorem]. Edited by R. S. Palais and A. S. Dynin. Mir (Moscow), 1970. Russian translation of Seminar on the Atiyah–Singer index theorem (1965). MR 0394769 book

R. S. Palais: “Banach manifolds of fiber bundle sections,” pp. 243–249 in Actes du Congrès International des Mathématiciens [Proceedings of the International Congress of Mathematicians] (Nice, France, 1–10 September 1970), Tome 2. Gauthier-Villars (Paris), 1971. MR 448405 Zbl 0326.58008 incollection

R. S. Palais: Notes on the theory of characteristics, 1971. Brandeis University lecture notes. Zbl 0277.58006 misc

R. S. Palais: Equivariant, real algebraic differential topology, I: Smoothness categories and Nash manifolds, 1972. Brandeis University lecture notes. Zbl 0281.57015 misc

R. S. Palais and C. L. Terng: “Natural bundles have finite order,” Topology 16 : 3 (1977), pp. 271–277. MR 467787 Zbl 0359.58004 article

R. S. Palais: “Some analogues of Hartogs’ theorem in an algebraic setting,” Am. J. Math. 100 : 2 (April 1978), pp. 387–405. MR 480509 Zbl 0449.14002 article

Let \( K \) be a field, and let \( f \) be any function from \( K\times K \) into \( K \). For each \( x_0 \in K \) define maps \( f_{x_0} \), and \( f^{x_0} \) from \( K \) to itself by \[ x \mapsto f(x_0,x) \quad\text{and}\quad x \mapsto f(x,x_0) \] respectively, and call \( f \) separately polynomial if all the \( f_{x_0} \) and \( f^{x_0} \) are polynomial functions. Clearly, if \( f \) is a polynomial function, then it is separately polynomial. What additional assumptions, if any, are needed to ensure that a separately polynomial function is polynomial? For what fields \( K \) is it true that every separately polynomial function is polynomial?

More generally, suppose \( X \), \( Y \), and \( Z \) are affine algebraic varieties over \( K \) and \( f \) is a map from \( X\times Y \) to \( Z \) which is separately polynomial. Again we can ask for conditions on \( f \) that will ensure that it is a polynomial map, conditions on \( X \), \( Y \), and \( Z \) that will ensure that all such \( f \) are polynomial, and conditions on \( K \) that ensure \( f \) is polynomial no matter what the varieties \( X \), \( Y \), and \( Z \). We shall find complete answers to some of these questions and partial answers to the rest.

R. S. Palais: “A topological Gauss–Bonnet theorem,” J. Diff. Geom. 13 : 3 (1978), pp. 385–398. MR 551567 Zbl 0436.57006 article

R. S. Palais: “The principle of symmetric criticality,” Comm. Math. Phys. 69 : 1 (October 1979), pp. 19–30. MR 547524 Zbl 0417.58007 article

It is frequently explicitly or implicitly assumed that if a variational principle is invariant under some symmetry group \( \mathbf{G} \), then to test whether a symmetric field configuration \( \phi \) is an extremal, it suffices to check the vanishing of the first variation of the action corresponding to variations \( \phi + \delta\phi \) that are also symmetric. We show by example that this is not valid in complete generality (and in certain cases its meaning may not even be clear), and on the other hand prove some theorems which validate its use under fairly general circumstances (in particular if \( \mathbf{G} \) is a group of Riemannian isometries, or if it is compact, or with some restrictions if it is semi-simple).

R. S. Palais: Real algebraic differential topology, Part 1. Mathematics Lecture Series 10. Publish or Perish (Wilmington, DE), 1981. MR 633063 Zbl 0477.57002 book

R. S. Palais: The geometrization of physics, 1981. Online PDF. Lecture notes from a course held June–July 1981. misc

R. S. Palais, A. Derdzinski, and C.-L. Terng: “Warped products and Einstein manifolds and Hessian PDE,” pp. 44–47 in Proceedings of differential geometry meeting, Münster (Münster, Germany, 1982). 1982. incollection

Andrzej Derdziński	Related
Chuu-Lian Terng	Related

R. S. Palais: “Applications of the symmetric criticality principle in mathematical physics and differential geometry,” pp. 247–301 in Proceedings of the 1981 Shanghai symposium on differential geometry and differential equations (Shanghai and Hefei, China, 20 August–13 September 1981). Edited by C. Gu. Scientific Press (Beijing), 1984. MR 825280 Zbl 0792.53079 incollection

W.-Y. Hsiang, R. S. Palais, and C. L. Terng: “Geometry and topology of isoparametric submanifolds in Euclidean spaces,” Proc. Nat. Acad. Sci. U.S.A. 82 : 15 (August 1985), pp. 4863–4865. MR 799109 Zbl 0573.53033 article

Chuu-Lian Terng	Related
Wu-yi Hsiang	Related

R. S. Palais and C.-L. Terng: “Reduction of variables for minimal submanifolds,” Proc. Am. Math. Soc. 98 : 3 (November 1986), pp. 480–484. MR 857946 Zbl 0627.53050 article

R. S. Palais and C.-L. Terng: “A general theory of canonical forms,” Trans. Am. Math. Soc. 300 : 2 (1987), pp. 771–789. MR 876478 Zbl 0652.57023 article

R. S. Palais and C.-L. Terng: “Geometry of canonical forms,” pp. 133–151 in The legacy of Sonya Kovalevskaya (Cambridge, MA and Amherst, MA, 25–28 October 1985). Contemporary Mathematics 64. American Mathematical Society (Providence, RI), 1987. MR 881460 Zbl 0607.57023 incollection

Chuu-Lian Terng	Related
Sofia Vasilyevna Kovalevskaya	Related

W.-Y. Hsiang, R. S. Palais, and C.-L. Terng: “The topology of isoparametric submanifolds,” J. Diff. Geom. 27 : 3 (1988), pp. 423–460. MR 940113 Zbl 0618.57018 article

It has been known since a famous paper of Bott and Samelson that, using Morse theory, the homology and cohomology of certain homogeneous spaces can be computed algorithmically from Dynkin diagram and multiplicity data. L. Conlon and J. Dadok noted that these spaces are the orbits of the isotropy representations of symmetric spaces. Recently the theory of isoparametric hypersurfaces has been generalized to a theory of isoparametric submanifolds of arbitrary codimension in Euclidean space, and these same orbits turn out to be exactly the homogeneous examples. Even the nonhomogeneous examples have associated to them Weyl groups with Dynkin diagrams marked with multiplicities. We extend and simplify the Bott-Samelson method to compute the homology and cohomology of isoparametric submanifolds from their marked Dynkin diagrams.

Chuu-Lian Terng	Related
Wu-yi Hsiang	Related

R. S. Palais and C.-L. Terng: Critical point theory and submanifold geometry. Lecture Notes in Mathematics 1353. Springer (Berlin), 1988. MR 972503 Zbl 0658.49001 book

M. W. Hirsch and R. S. Palais: “Editors’ remarks: ‘Some basic information on information-based complexity theory’ by B. N. Parlett and ‘Perspectives on information-based complexity’ by J. F. Traub and H. Woźniakowski,” Bull. Am. Math. Soc. (N.S.) 26 : 1 (1992), pp. 1–2. MR 1141473 article

Beresford Neill Parlett	Related
Joseph Frederick Traub	Related
Henryk Woźniakowski	Related
Morris W. Hirsch	Related

incollection R. S. Palais and C.-L. Terng: “The life and mathematics of Shiing Shen Chern,” pp. 18–62 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201340

Chuu-Lian Terng	Related
S. S. Chern	Related	Volume
Shing-Tung Yau	Related

Global analysis in modern mathematics: Proceedings of the symposium in honor of Richard Palais’ sixtieth birthday (Orono, ME, 8–10 August 1991 and Waltham, MA, 12 August 1992). Edited by K. Uhlenbeck. Publish or Perish (Houston, TX), 1993. MR 1278744 Zbl 0920.00058 book

R. Palais: “In appreciation: Delivered after the Chinese banquet at the Lotus Flower,” pp. xxi–xxii in Global analysis in modern mathematics (Orono, ME, 8–10 August 1991 and Waltham, MA, 12 August 1992). Edited by K. K. Uhlenbeck. Publish or Perish (Houston, TX), 1993. Proceedings of a symposium in honor of Richard Palais. MR 1278746 Zbl 0927.01025 incollection

E. Heintze, R. Palais, C.-L. Terng, and G. Thorbergsson: “Hyperpolar actions and \( k \)-flat homogeneous spaces,” J. Reine Angew. Math. 1994 : 454 (1994), pp. 163–179. MR 1288683 Zbl 0804.53074 article

Ernst Heintze	Related
Chuu-Lian Terng	Related
Gudlaugur Thorbergsson	Related

R. S. Palais: “Symmetry, criticality, and condition \( C \),” pp. 77–98 in The Sophus Lie Memorial Conference (Oslo, 17–21 August 1992). Edited by O. A. Laudal and B. Jahren. Scandinavian University Press (Oslo), 1994. MR 1456460 incollection

Bjørn Jahren	Related
Marius Sophus Lie	Related
Olav Arnfinn Laudal	Related

E. Heintze, R. S. Palais, C.-L. Terng, and G. Thorbergsson: “Hyperpolar actions on symmetric spaces,” pp. 214–245 in Geometry, topology, & physics: For Raoul Bott (Cambridge, MA, 23–25 April 1993). Edited by S.-T. Yau. Conference Proceedings and Lecture Notes in Geometry and Topology 4. International Press (Cambridge, MA), 1995. Lectures of a conference in honor of Raoul Bott’s 70th birthday. MR 1358619 Zbl 0871.57035 incollection

Ernst Heintze	Related
Chuu-Lian Terng	Related
Gudlaugur Thorbergsson	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

R. S. Palais: “The symmetries of solitons,” Bull. Am. Math. Soc. (N.S.) 34 : 4 (1997), pp. 339–403. MR 1462745 Zbl 0886.58040 article

In this article we will retrace one of the great mathematical adventures of this century–the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal’s insightful explanation of the surprising results of that experiment (and of a follow-up experiment of their own) in terms of a new concept they called “solitons”. Solitons however raised even more questions than they answered. In particular, the evolution equations that govern solitons were found to be Hamiltonian and have infinitely many conserved quantities, pointing to the existence of many non-obvious symmetries. We will cover next the elegant approach to solitons in terms of the Inverse Scattering Transform and Lax Pairs, and finally explain how those ideas led step-by-step to the discovery that Loop Groups, acting by “Dressing Transformations”, give a conceptually satisfying explanation of the secret soliton symmetries.

R. S. Palais: “The visualization of mathematics: Towards a mathematical exploratorium,” Notices Am. Math. Soc. 46 : 6 (June/July 1999), pp. 647–658. Dedicated to the memory of Alfred Gray. MR 1691563 Zbl 1194.00005 article

B. Palais and R. Palais: “Euler’s fixed point theorem: The axis of a rotation,” J. Fixed Point Theory Appl. 2 : 2 (2007), pp. 215–220. MR 2372984 Zbl 1140.55001 article

R. S. Palais: “A simple proof of the Banach contraction principle,” J. Fixed Point Theory Appl. 2 : 2 (December 2007), pp. 221–223. MR 2372985 Zbl 1140.55301 article

R. S. Palais: “Linear and nonlinear waves and solitons,” pp. 234–239 in The Princeton companion to mathematics. Edited by T. Gowers, J. Barrow-Green, and I. Leader. Princeton University Press, 2008. incollection

William Timothy Gowers	Related
June Elizabeth Barrow-Green	Related
Imre Bennett Leader	Related

R. S. Palais and R. A. Palais: Differential equations, mechanics, and computation. Student Mathematical Library 51. American Mathematical Society (Providence, RI), 2009. MR 2567436 Zbl 1194.34001 book

B. Palais, R. Palais, and S. Rodi: “A disorienting look at Euler’s theorem on the axis of a rotation,” Am. Math. Mon. 116 : 10 (2009), pp. 892–909. MR 2589219 Zbl 1229.15028 article

A rotation in two dimensions (or other even dimensions) does not in general leave any direction fixed, and even in three dimensions it is not immediately obvious that the composition of rotations about distinct axes is equivalent to a rotation about a single axis. However, in [1775], Leonhard Euler published a remarkable result stating that in three dimensions every rotation of a sphere about its center has an axis, and providing a geometric construction for finding it.

In modern terms, we formulate Euler’s result in terms of rotation matrices as follows.

If \( \mathbf{R} \) is a \( 3{\times}3 \) orthogonal matrix (\( \mathbf{R}^T\mathbf{R} = \mathbf{R}\,\mathbf{R}^T = \mathbf{I} \)) and \( \mathbf{R} \) is proper (\( \det\mathbf{R} = +1 \)), then there is a nonzero vector \( \mathbf{v} \) satisfying \( \mathbf{Rv} = \mathbf{v} \).

This important fact has a myriad of applications in pure and applied mathematics, and as a result there are many known proofs. It is so well known that the general concept of a rotation is often confused with rotation about an axis.

In the next section, we offer a slightly different formulation, assuming only orthogonality, but not necessarily orientation preservation. We give an elementary and constructive proof that appears to be new that there is either a fixed vector or else a “reversed” vector, i.e., one satisfying \( \mathbf{Rv} = -\mathbf{v} \). In the spirit of the recent tercentenary of Euler’s birth, following our proof it seems appropriate to survey other proofs of this famous theorem. We begin with Euler’s own proof and provide an English translation from the original Latin. Euler’s construction relies on implicit assumptions of orientation preservation and genericity, and leaves confirmation of his characterization of the fixed axis to the reader. Our current tastes prefer such matters to be spelled out, and we do so in Section 4. There, we again classify general distance preserving transformations, this time using Euler’s spherical geometry in modern dress instead of linear algebra. We note that some constructions present in Euler’s original paper correspond to those appearing in our proof with matrices. In the final section, we survey several other proofs.

Stephen Bellew Rodi, Jr.	Related
Robert Andrew Palais	Related

B. Palais and R. Palais: “Chasles’ fixed point theorem for Euclidean motions,” J. Fixed Point Theory Appl. 12 : 1–2 (December 2012), pp. 27–34. MR 3034850 Zbl 1266.54093 article

R. S. Palais: “A mathematician and an artist: The story of a collaboration,” pp. 1–9 in Mathematics and modern art (Paris, 19–22 July 2010). Edited by C. Bruter. Springer Proceedings in Mathematics 18. Springer (Berlin), 2012. MR 3220154 incollection

Claude Paul Bruter	Related
Luc Bénard	Related

R. S. Palais: “The initial value problem for weakly nonlinear PDE,” J. Fixed Point Theory Appl. 16 : 1–2 (December 2014), pp. 337–349. MR 3346758 Zbl 1320.35017 article

We discuss an extension of the pseudospectral method developed by Wineberg, McGrath, Gabl, Scott, and Southwell [1991] for the numerical integration of the Korteweg–de Vries (KdV) initial value problem. Our generalization of their algorithm can be used to solve initial value problems for a wide class of evolution equations that are “weakly nonlinear” in a sense we will make precise. This class includes in particular the other classical soliton equations, sine-Gordon equation (SGE) and nonlinear Schrödinger equation (NLS). As well as being very simple to implement, this method exhibits remarkable speed and stability, making it ideal for use with visualization tools where it makes it possible to experiment in real time with soliton interactions and to see how a general solution decomposes into solitons. We analyze the structure of the algorithm, discuss some of the reasons behind its robust numerical behavior, and finally describe a fixed point theorem we have found that proves that the pseudospectral stepping algorithm converges.

Richard Sheldon Palais

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