From: d012560c@aol.comNoBasura (D012560c)
Subject: Re: Lecture Notes on the Internet
Date: 18 Jun 2000 16:51:18 GMT
Newsgroups: sci.math
Summary: [missing]

>I've recently found many excellent sets of lecture notes on mathematical
>topics on the internet, and it has occurred to me that for every one I
>know about there are probably a hundred that I don't. I would be
>interested in receiving any recommendations for lecture notes or
>expository articles at the upper undergraduate/ lower graduate level. If
>you have written any or just run across some that are interesting,
>please post the URL. Unfortunately I can't read postscript but can read
>PDF

To view PS files, download for free GhostScript. It can also view PDF files.
http://www.ghostscript.com/

For notes on measure theory, go to www.probability.net. 

For notes on category theory and combinatorics, go to
http://www.brics.aau.dk/LS/Ref/BRICS-LS-Ref/BRICS-LS-Ref.html
You will find there: 
LS-95-1  Jaap van Oosten. Basic Category Theory. January 1995. vi+75 pp. 
and 
LS-95-4 Dany Breslauer and Devdatt P. Dubhashi. Combinatorics for Computer
Scientists. August 1995. viii+184 pp. 

Also try the links from http://www.geometry.net/
==============================================================================

From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: Query about Category Theory.
Date: 24 Jun 2000 01:47:36 GMT
Newsgroups: sci.math
Summary: [missing]

This article forwarded from the CT mailing list.
-------------------------------------------------

Date: Wed, 21 Jun 2000 15:23:33 +0100
From: Ronnie Brown <r.brown@bangor.ac.uk>
Subject: [Fwd: categories: A question on: <newsgroup sci.math>]

I have a paper with Loday on calculating the third homotopy group of a
suspension of a K(G,1)

51. (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'',
{\em Topology} 26
(1987) 311-334. 

in terms of a new tensor product of non abelian groups. It would not be a
sensible task to rewrite the paper without category theory (considered as a
unifying principle, as a mode for efficient calculation in certain
algebraic structures,  and as a supplier of new algebraic structures). 

To go further back (and higher, of course) Grothendieck's extension of the
Riemann-Roch Theorem uses category theory explicitly. In the 1950's people
were trying to give algebraic proofs of this theorem, when AG came up with
an algebraic proof of a vast generalisation. Then there is the categorical
background to the proof of the Weil conjectures ....

But the question is misplaced - in the early part of the 20th century, I
expect some would ask if set theory was really necessary for a confirmed
problem solver! The history of maths shows that maths greatest contribution
to science, culture and technology has been in terms of expressive power,
to give a language for intuitions which enables exact description,
calculation, deduction. (Exam question: discuss the last statement, with an
emphasis on particular examples!) It also allows for the *formulation* of
new problems, which perhaps cause old interests to lapse as people perceive
there are more exciting things to do. It is a narrow view to regard
`important maths' as necessarily that which solves well known problems, and
so leave evaluation as akin to a sports league table: how old is the
problem? who has worked on it? etc, etc. The progress of maths is much more
complicated and interesting than that! It won't help the public image of
maths if it is seen that mathematicians believe the most important aspect
of their subject is a (to the public) bizarre set of problems which seem to
interest no one else. 

However, it is important that these questions be asked, together with
questions on modes for evaluating `good maths'. As AG remarked in a letter,
maths was held back for centuries for lack of the `trivial' concept of
zero! For more discussion, look at 

http://www.bangor.ac.uk/ma/CPM/cdbooklet/knots-m.html
                                         knots2.html

So let the debate be broadened and continued!

Ronnie Brown

(will someone please forward this to the newsgroup?)

PS I am trying to trace a combinatorics problem solved in the 1950s using
categories of paths, and which was at the time held up as the sort category
theory could not do! Maybe my memory is failing! But the question put on
the newsgroup is an old war horse! 
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