Conference rusure::math

1. Homotopy theory is a topic in algebraic topology
4. Experimental mathematics refers to the use of computers to calculate one or more special cases of some mathematical speculation.  Instead of proving theorems, one uses statistics and inductive inference.
7. Chaos theory focuses on functions with irregular dynamic behavior.
   Synergetics is not used as the name of any field of mathematics, to my knowledge.  It is the study of how several systems can produce results greater than just the union of their operations.
   Large scale order refers to the characteristic of some graphical presentations of models, where microscopic views are irregular but macroscopic views are more or less regular.

Hope this helps.

PLEASE use a little common sense in laying out replies. I don't know of 
many people who have 256-character-wide screens :-) - here's .-1 in a more 
readable layout.

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1. Homotopy theory is a topic in algebraic topology
4. Experimental mathematics refers to the use of computers to calculate one
or more special cases of some mathematical speculation.  Instead of proving
theorems, one uses statistics and inductive inference. 
7. Chaos theory focuses on functions with irregular dynamic behavior.   
Synergetics is not used as the name of any field of mathematics, to my
knowledge.  It is the study of how several systems can produce results
greater than just the union of their operations.    Large scale order
refers to the characteristic of some graphical presentations of models,
where microscopic views are irregular but macroscopic views are more or
less regular. 

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CFD is Computational Fluid Dynamics, or how to soak up CPU cycles in an 
attempt to solve fluid flow problems. (The president of Apple recently
told about how Apple uses Crays to help analyse the flow and cooling of
plastics used in making workstation boxes.) 

    I don't have a clue how these diverse topics come together as in the
    base note, but here are some pointers.

1. homotopy theory

    As someone mentioned, this is in algebraic topology; it has to do
    with the equivalence classes of "paths" which can be deformed into
    each other in a manifold, and generalizations.  For example, on a
    torus there are two simple closed loops that can't be deformed into
    each other, and any closed curve whatsoever can be made up of chains
    of these two - they generate the fundamental group for the torus.

    You'd want to look at a book on algebraic topology, such as Hilton and
    Wylie, "Homotopy Theory". The theory was invented by Poincare.

2. Lie algebras

    A Lie algebra is a linear vector space equipped with a non commutative,
    non associative, law of "multiplication", which satisfies the
    so-called "Jacobi identity".  The tangent space at the origin of
    a Lie group is a Lie algebra under the Lie bracket.  An even simpler
    example is the vector cross product.  Any introductory book on Lie
    groups will discuss this - there are books on Lie algebras themselves
    but it's best to start in the context of Lie groups where they came
    from because the Lie algebra books love to go off the deep end with
    what seems to be unmotivated abstract algebra.

3. Stein spaces

    I don't know the precise definition, but basically this has to do
    with a complex manifold (a topological space with a local structure
    like an n dimensional complex vector space) that satisfies some
    sort of cohomology relation on the the differential forms on that
    manifold.  I know that Griffiths and Harris discuss this in their
    book on algegbraic geometry, but don't have a more precise definition
    handy.

4. Experimental mathematics

   This could be anything from noodling around with Mandelbrot sets
   on your PC to doing quantum chromodynamics calculations on a cray.
   I'd know it if I saw it at any rate...

5. CFD ( I really don't know what CFD stands for)

   I'm pretty sure this is indeed computational fluid dynamics.  It's a really
   hard problem because even a little perturbation somewhere in a fluid
   can profoundly influence the long-term behaviour.  The SIAM is a good
   place to look for more info and they have a number of recent books
   in their catalog.

6. completely integrable systems

    Well, my brain is realy a sieve, but I have read something on this
    at some point.  In noninear dynamical systems one can sometimes
    make changes of variables in order to express the long term behaviour
    of the system in a kind of "closed form".  It's discussed in books
    on dynamics like Abraham and Marsden "Foundations of Mechanics" or
    in books by V.I. Arnold.

    Hope this limited info helps - the DEC library does have some
    of the info you will need.  I can try and look thru my stuff at home
    and see if some more precise definitions are possible.

    - Jim

Many thanks for all of you who are taking time to explain some of these 
subjects.

Let me extract the part of the paragraph from the preface to the book which
impelled my curioisity:

"Further, the kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is used 
(non-trivially) in regional and theoretical economics; algebraic geometry 
interacts with physics; the Minkowsky lemma, coding theory and the structure of
water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras
are relevant to filtering; and prediction and electrical engineering can use
Stein spaces. And in addition to this there are such new emerging subdisciplines
as "experimental mathematics", "CFD", "completely integrable systems", "chaos,
synergetics and large-scale order", which are almost impossible to fit into
the existing classification schemes. They draw upon widely different sections of
mathematics. ..."

Swamy

    re : .2
    
    That's funny, because when I was working for Cray, we used to have 
    Mac systems for engeneering purposes %-)
    
    Christophe