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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; [Schematic of subareas and related areas]

Subfields

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:

Some descriptions of Catastrophe Theory, starting with its creator:

"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

Selected topics at this site


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Last modified 2000/01/24 by Dave Rusin. Mail: