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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, that is, dynamical systems. 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! In the meantime see this summary of some lectures by Singer on the theorem.)

This heading includes the topic of Chaos, well-known in the popular press, but not a particularly large part of mathematics. At best, it provides a paradigm for the phrasing of situations in the applications of mathematics. A quote by Philip Holmes (SIAM Review 37(1), pp. 129, 1995) illustrates this situation well:

  `In spite of all the hype and my enthusiasm for the area, I do not
   believe that chaos theory exists, at least not in the manner of
   quantum theory, or the theory of self-adjoint linear operators. Rather
   we have a loose collection of tools and techniques, many of them from
   the classical theory of differential equations, and a guiding global
   geometrical viewpoint that originated with Poincaré over a hundred
   years ago and that was further developed by G D Birkhoff, D V Anosov
   and S Smale and other mathematicians. I therefore prefer a sober
   description of new tools, rather than grand claims that the problems
   of life, the universe, and everything will shortly be solved.'
(taken from SIAM Review 37(1), pp. 129, 1995). One sometimes hears similar expressions of regret that other topics in this area --- catastrophe theory, dynamical systems, fractal geometry --- have been championed by persons not familiar with the content of the material.

History

Applications and related fields

See also 32-XX, 32CXX, 32FXX, 46-XX, 47HXX, 53CXX; for geometric integration theory, See 49FXX, 49Q15 [Schematic of subareas and related areas]

Subfields

This is among the larger areas within the Math Reviews database. But over half the papers are in subfield 58F (Dynamical systems), one of the largest 3-digit subfields in the database (and containing two(!) of the largest 5-digit areas -- 58F07, Completely integrable systems, and 58F13 Strange attractors; chaos). Apart from 58F, this area would be average in size.

(*) These two sections are to be removed in the year-2000 version of the MSC. A new section 37: Dynamical systems and ergodic theory is being created.

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 electronic Survey articles in Dynamical systems

Some descriptions of Catastrophe Theory, starting with its creator:

Reviews of Chaos and fractal geometry:

"Reviews in Global Analysis 1980-1986", AMS

Chaos bibliography database [Link dead or restricted; May 1999]

Chaos Database

Newsgroups sci.nonlinear, comp.theory.dynamic-sys, comp.theory.cell-automata

There are two mailing lists nonlin_net@complex.nbi.dk (nonlinear systems) and qchaos_net@complex.nbi.dk (quantization/chaos). Here is the information page.

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

Software and tables

DsTool for dynamical systems

Other web sites with this focus

Selected topics at this site


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Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org