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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 58: Global analysis, analysis on manifolds

Introduction

Global analysis, or analysis on manifolds, studies the global nature of differential equations on manifolds. In addition to local tools from ordinary differential equation theory, global techniques include the use of topological spaces of mappings. In this heading also we find general papers on manifold theory, including infinite-dimensional manifolds and manifolds with singularities (hence catastrophe theory), as well as optimization problems (thus overlapping the Calculus of Variations.

(The real introduction to this area will have to summarize the Atiyah-Singer Index Theorem!)

History

Applications and related fields

For dynamical systems, ergodic theory, and chaos see 37: Dynamical Systems

For fractals see 28: Measure Theory.

For geometric integration theory, See 49FXX, 49Q15

See also 32-XX, 32CXX, 32FXX, 46-XX, 47HXX, 53CXX;

Subfields

• 58A: General theory of differentiable manifolds
• 58B: Infinite-dimensional manifolds
• 58C: Calculus on manifolds; nonlinear operators, see also 47HXX
• 58D: Spaces and manifolds of mappings (including nonlinear versions of 46EXX)
• 58E: Variational problems in infinite-dimensional spaces
• 58H: Pseudogroups, differentiable groupoids and general structures on manifolds
• 58J: Partial differential equations on manifolds [See also 35-XX] [new in 2000]
• 58K: Theory of singularities and catastrophe theory [See also 37-XX] [new in 2000]
• 58Z05: Applications to physics

Until 2000 there were two (large) sections 58F and 58G which have since been moved to 37: Dynamical systems and ergodic theory.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Some descriptions of the traditional areas of global analysis:

• Smale, S.: "What is global analysis?", Amer. Math. Monthly 76 1969 4--9. MR38#5248
• Morse, Marston: "What is analysis in the large?", Amer. Math. Monthly 49, (1942). 358--364. MR3,292a
• Ambrosetti, Antonio; Prodi, Giovanni: "A primer of nonlinear analysis", Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, Cambridge, 1993. 171 pp. ISBN 0-521-37390-5 MR94f:58016
• Ewald, Günter: "Probleme der geometrischen Analysis", (German: Problems of geometric analysis), Bibliographisches Institut, Mannheim, 1982. 156 pp. ISBN 3-411-01633-7 MR84g:58001
• Alexander, J. C.: "A primer on connectivity", Fixed point theory (Sherbrooke, Que., 1980), pp. 455--483, Lecture Notes in Math., 886; Springer, Berlin-New York, 1981. MR83e:58013

Some descriptions of Catastrophe Theory, starting with its creator:

• Thom, René: "What is catastrophe theory about?" Synergetics (Proc. Internat. Workshop, Garmisch-Partenkirchen, 1977), pp. 26--32. Springer, Berlin, 1977. MR58#18536
• Stewart, Ian: "Consumer guide to catastrophe theory", Southeast Asian Bull. Math. 2 (1978), no. 1, 13--16. MR81a:58017
• Poston, Tim; Stewart, Ian: "Catastrophe theory and its applications", Surveys and Reference Works in Mathematics, No. 2. Fearon-Pitman Publishers, Inc., Belmont, Calif., 1978. 491 pp. ISBN 0-273-01029-8 MR58#18535

"Reviews in Global Analysis 1980-1986", AMS

E-text: "Invariance Theory, the heat equation, and the Atiyah-Singer index theorem", by Gilkey.

Software and tables

Other web sites with this focus

• Nonlinear sites (Good place to start; the following have not yet been examined too closely).
• Sci.nonlinear FAQ
• UK Nonlinear Dynamics Groups
• Nonlinear Dynamics and Topological Time Series Analysis Archive
• Complexity Online (Complex systems)
• Research in Applied Nonlinear Mathematics
• Here are the AMS and Goettingen resource pages for area 58.

Selected topics at this site

• The Poincaré Eternal Return theorem.
• Can you hear the shape of a drum? (no) Citations, URLs, and a summary.