Newsgroup sci.math.research 6625

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Some of the more classical & mathematical ones:
 * The first name that always comes to my mind on 
   that subject is Hermann Weyl. I think his books 
   on "Symmetry", "Group Theory & Quantum Mechanics",
   "Space, Time, Matter" should still be insprirational 
   for beginners. After all he was one of the pioneers
   looking for applications of group theory in nature.
 * As a student I remeber having used lecture notes by
   Res Jost (another guy from Zurich), which impressed
   me because of their mathematical clarity. Especially
   a clean way of describing many particle systems, their 
   gauge and permutation symmetries, and how those mesh
   when considering the representation theory. 
   I don't know if they have been published, but I would
   guess a lot has been taken from Wolgang Pauli's lectures
   (yet another guy from Zurich), which I think are published.
   In both lectures one should find also a variety of more
   concrete applications to special experiments. 
   Personally I'd also look  into the really old stuff by Wigner.
 * A few days ago I looked again in Thirring's book on 
   quantum mechanics in his mathematical physics series.
   (this guy now is from Vienna, which is not far from Zurich)
   One thing from there, I'd probably couldn't resist teaching 
   in a course on group theory and QM is the derivation of the
   spectrum & degeneracies of the hydrogen atom, using only 
   the representation theory of SO(3), and no diff. eqs.
   (The trick is that with the Runge-Lentz-vector you really
     have an SO(4) symmetry). In this and the next book also
   many particle systems are nicely introduced and applications 
   discussed.
 * As a sophemore undergrad. I once took a course/seminar on
   "GT & QM" organized by mathematicians, which was basically
   following the book by Simms, Springer Lecture Notes in Math,
   (Vol. 52)  It is quite limited in its scope but it's a rather
   nice derivation of the general formalism fo relativistic
   quantum mechanics from first physical principles (causality,
   states & observables, ...) in a really clean way (e.g., via
   that Mackey-stuff with induced & small representations.) It
   also deals with SU(3), can't remember whether it was color or
   flavor though. 
    The only thing that stuck in my mind from Hamermesh's book
   is a rather lengthy explanation of what a projective representation
   is, but I think  Simms' book not only makes more sense in the 
   mathematical presentation, there you also have a much clearer
   physical reasoning, where the projective things come from.
 * Unfortunately I don't know the literature in solid state. Of
   course the standard stuff on cristallographic groups should be 
   explained in a lot of books.( Not very riveting; unfortunately
   we are not in hyperbolic 3-space, where that would include, e.g.,
   all of the platonic solid groups.) There is of course the more
   recent subject of quasi-cristals, which can often be decribed
   as "projections" of higher dimensional crystals, and related to
   that the stuff with tilings that don't have naive translation 
   invariance. There must a lots of articles out there, but I don't
   know much about serious literature.
     The thing on group theory and solid states physics that really
   fascinated me as a student (and probably still would) are 
   classifications of defects (vortices etc.) in crystals and fluids,
   given the topology of the "inner configuration space" of the
   particles. The latter is usually topological group, G, or more
   generally a homogeneous space, G/N, and the singularities are
   classfied in terms of the fundamental groups of G/N. I remeber 
   having a lot of fun talking in a solid state student seminar
   about what happens, when, e.g., superfluid helium changes phases 
   and that configuration space changes. I'm not sure if I can 
   retrieve the relevant literature, but I could go deep-digging
   (it was 9 years ago) if you can't get it from anyone else.
I guess you're right that in view of the wealth and beauty of
the subject, most monographs are rather pale and not even 
pedagogically brilliant. There should be a Hermann Weyl IIIrd
or so out there, who can put all what is scattered across the 
literature into one book or series about the subject of goups
in nature. 
  My selection obviouosly reflects the facts that I'm mathematically
inclined, have conservative tastes, and did my PhD in Zurich.
I hope there is still one or the other thing, from which you can draw 
ideas for your course.
Thomas

[ Article crossposted from soc.culture.ukrainian ]
[ Author was Pan Tofli (pyz@panix.com) ]
[ Posted on 10 Jan 1997 12:24:05 -0500 ]
Greetings,
We've set up a new section at Infomeister-Ukrainian for
Conferences/Announcements.  The first one which we have listed is:
"Symmetry in Nonlinear Mathematical Physics, Second International
Conference"
 -- July 7-13, 1997, Kyiv, Ukraina 
For more information on this conference please use one of the following
web coordinates:
http://www.osc.edu/ukraine.html#CONF
                or
http://www.osc.edu/ukraina.html#CONF (for Ukrainian KOI8-speaking web
browsers)
Similar types of notices will gladly be accomodated.  Much thanks to Prof.
Roman Andrushkiw for providing this one.
Max Pyziur
pyz@panix.com

Let M be a Banach manifold.
Obs: A topological space X is said to be regular when every point has a 
fundamental system of closed neighborhoods, or equivalently, a point p 
and a closed set F such that p is not in F always have disjoint 
neighborhoods.
Obviously if M is finite dimensional then M is regular, since M is 
locally compact. If M is infinite dimensional the M is never locally 
compact and so comes the question of whether M is regular (to be precise: 
a Banach manifold M is a set M together with an atlas of charts taking 
values in open sets of Banach spaces, such that the overlappings are 
C^\infty - no topological assumptions made, like in Lang's Differential 
Manifolds).
Of course, if M is not Hausdorff then M can not be regular too, so let's 
supose that M is Hausdorff. Most natural examples of Banach manifolds 
(like functions spaces) are all regular (at least the examples I know).
Regularity is very very important: You need it to build partitions of 
unit (I have never seen a book about Banach manifolds taking regularity 
for assumption. They usually make a mistake somewhere when the related 
problems appear). Let phi:U->U' be a chart in M (where U is open in M and 
U' is open in some Banach space E).
Suppose you have a function f:U' -> R which is C^\infty and whose support 
(i.e., the closure in U' (not in E) of {x:f(x)<>0}) is also closed in E. 
Then we can define g:M -> R by g(x)=f(phi(x)) for x in U and g(x)=0 for x 
not in U. In order to conclude that g is C^\infty you need to know that 
the inverse image of supp f by phi is closed in M (you know only that 
it's closed in U!).
In finite dimension it's reasonable to suppose that supp f is compact and 
so there is no problem. In infinite dimension the only way to solve the 
problem is to suppose that M is regular (with this assumption it's easy 
to solve the problem by shrinking phi a little).
Another problem: Let M be a Riemannian Hilbert manifold. Define the 
distance d(x,y) (where x,y in M) by the infimum of the lengths of 
piecewise C^\infty arcs conecting x and y. 
We want to prove that (M,d) is a metric space. The problem is to prove 
the d(x,y)=0 imply x=y. If you pay good attention you will see that you 
need M to be regular for that, or else the usual proofs won't work (what 
makes sense, since a metric space is always regular).
Well, the whole thing is: I don't know any example of a non-regular 
Banach manifold (neither natural examples nor artificial ones). But I'm 
pretty sure that they exist. Any thoughts?
Daniel Victor Tausk - tausk@ime.usp.