PACIFIC NORTHWEST GEOMETRY SEMINAR

                             1995 Fall Meeting

                           University of Oregon
                              Eugene, Oregon

                 Saturday and Sunday, October 14-15, 1995

=========================================================================== ===
                                 SCHEDULE:

Saturday, October 14
--------------------

10:00-10:45     S.-S. Chern
                "Riemannian Geometry as a special case of Finsler Geometry"

10:45-11:00     Tea

11:00-12:00     David Bao (University of Houston)      
                "Some Global Issues in Finsler Geometry"

12:00-14:00     Lunch

14:00-15:00     Susan Tolman (MIT)  
                "Examples of non-Kaehler Hamiltonian torus actions"

15:30-19:00     Picnic

Sunday, October 15
------------------

09:30-10:30     Richard Hamilton (UC San Diego)
                "Remarks on the Ricci flow on 3 manifolds"

11:00-12:00     Juha Pohjanpelto (Oregon State University)
                "Invariant variational bicomplexes -- new applications
                of the Gelfand-Fuks cohomology of vector fields"

All lectures will be in Wilamette Hall Room 110; tea will be in the atrium
outside.

The picnic on Saturday evening will be held at Mt. Pisgah Arboretum, a
great area for casual hiking.  The picnic will be open to all seminar
participants and their guests.  Maps will be provided.

=========================================================================== ===
                           FOR MORE INFORMATION

For more information about this program, contact one of the organizers at
U of O:

Boris Botvinnik <botv...@poincare.uoregon.edu>
Peter Gilkey    <gil...@math.uoregon.edu>
                (503-346-4717)
Jim Isenberg    <j...@newton.uoregon.edu>
John Leahy      <le...@math.uoregon.edu>

For general information about the PNGS, check out the PNGS World-Wide Web
site:

  http://www.math.washington.edu/~lee/PNGS/

It contains information about upcoming PNGS meetings, including a copy of
this announcement, as well as general information about the PNGS and a
historical record of all PNGS meetings and speakers.

Hello,

If you have a 2D surface (I myself am interested in surfaces in R^3)
you can *locally* map it to the plane with a conformal map (and so to any
other surface). Is it known under which conditions a surface can be mapped
*globally* to the sphere S^2? Is there any method for constructing such
conformal maps?

--

Position available at CERFACS

We are looking for someone at post-doc level with experience, interest,
and background (with publications) in both mathematics and computer
science.
The position is open in the framework of an ESPRIT contract in connection
with European reseach centres and industrial partners. For this project
we want someone with experience both in parallel computing and
optimization techniques. Basically, we have to identify the most
appropriate methods for solving optimization problems (arising in
Multi Body Simulation) suitable for a implementation on a parallel
platform. The type of target computer has to be selected according to the
parallel features of the selected numerical techniques, it may be either
shared memory multi-CPU workstation or network of computers. Once a
technique and a platform have been selected, the implementation will
have to be performed.  Some issues related to parallel computing on a
network of computers will also be addressed (fault-tolerance, load
balancing, ...).
The research to be developed in the framework of this contract is clearly
industrially oriented.

  The position is in the Parallel Algorithm Project at CERFACS,
where the main research activities are :
      -  Large sparse matrix calculations:
               -  direct linear solvers
               -  iterative linear solvers, preconditioning
               -  eigensolvers
      -  Domain decomposition
      -  Large scale nonlinear systems and optimization
      -  Reliability of numerical software
      -  Heterogeneous computing

 For detailed information about CERFACS
   URL address : http://www.cerfacs.fr/algor

 For more information about Toulouse, the city where CERFACS is located
   URL address : http://www.cict.fr/toulouse/EBienvenue.html

Theorem. Let V be a locally convex topological vector space and let {E_n}
be an increasing sequence of finite dimensional subspaces of V whose union
is dense in V. Given O open in V, let O_n denote the intesection of O with
E_n, and let O_oo denote the union of the O_n with the inductive limit
topology. If V is  metrizeable, or more generally if O is paracompact,
then the inclusion of O_oo into O is a homotopy equivalence.

   This can be applied to answer questions like that of Asimov. If M is a
smooth infinite dimensional submanifold in V, then in reasonable cases
(e.g., in the Hilbert space case) we can replace M by a tubular neighborhood
O that has M as a deformation retract (so M --> O is a homotopy equivalence)
and then apply the above theorem to O.

  For typical applications see my paper "On the homotopy type of certain groups
of operators", Topology, 3, (1965), in which I show various infinite dimensional
linear groups have the homotopy type of the inductive limits of their finite
dimensional counterparts.

SLEIGN2 is a package for computing eigenvalues, eigenfunctions, and
approximating the continuous spectrum of regular and singular Sturm-
-Liouville problems. It was developed by P.B. Bailey, W.N. Everitt,
B. Garbow and A. Zettl.

  The Fortran code comes with a user friendly interface, including
"help", and associated files and papers. These are available
through a Web browser at URL http://www.math.niu.edu/~zettl/SL2/ or
by anonymous FTP to ftp.math.niu.edu, in /pub/papers/Zettl/Sleign2.

Hello proba&statist world,

Does anybody knows the answer to the following question:

Let $\pi_x$ be a Poisson distributed r.v. with the parameter $x>0$. There are
explicit
expressions for its n-th moment, e.g.
\[
E \pi_x^n = \sum_{k=1}^n \frac{(-1)^k}{k!} [\sum_{m=1}^k (-1)^m {k\choose m}
m^n] x^k
\]

My question: how that staff grows when $x$ is fixed and $n \to \infty$?

In other words: what is the asymptotic expansion of the series:
\[ e^{-x} \sum{k=1}^\infty \frac{k^n x^k}{k!} \]

=========================================================================
                          Serguei ZOUEV

This new policy -- the SBIR "Fast Track" -- was approved for implementation
by Under Secretary of Defense (Acquisition & Technology) Dr. Paul Kaminski in
early June.  Its purpose is to significantly increase DoD's success in
converting SBIR research into affordable, high-performance products which
serve military and commercial customers.

People who have been trying and failing to get old editions of This
Week's Finds and other stuff by ftp may be pleased to hear that ftp is
working again here.  See the end of this issue for how to do it.  Also,
in a while I'll be getting a www homepage, and This Week's Finds will
become available in that format.  It may take a long time before I get
all the issues nicely cross-linked, but I hope eventually to do so.

This time I'll finish up talking about "ADE classifications" for a
while, although there is certainly more to say.  Recall where we were:
the following diagrams correspond to the simple Lie algebras, and they
also define certain lattices, the "root lattices" of those Lie algebras:

A_n, which has n dots like this:

o---o---o---o

B_n, which has n dots, where n > 1:

          4
o---o---o->-o

C_n, which has n dots, where n > 2:

          4
o---o---o-<-o

D_n, which has n dots, where n > 3:

              o
             /
o---o---o---o
             \
              o

E_6, E_7, and E_8:

      o               o                   o
      |               |                   |
o--o--o--o--o   o--o--o--o--o--o    o--o--o--o--o--o---o

F_4:                 G_2:        

      4                6          
o---o->-o---o        o-<-o        

The dots in one of these "Dynkin diagrams" correspond to certain set of basis
vectors, or "roots", of the lattice.  The lines, with their decorative
numbers and arrows, give enough information to recover the lattice from
the diagram.  In particular, two dots that are not connected by a line
correspond to roots that are at a 90 degree angle from each other, while
two dots connected by an unnumbered line correspond to roots that are at
a 60 degree angle from each other.  Numbered lines mean the angle
between roots is something else, and the arrows point from the longer to
the shorter root in this case, as partially explained in "week63".

However, we will now concentrate on the cases A, D, and E, where all the
roots are 90 or 60 degrees from each other, and they are all the same
length --- usually taken to be length 2.  These are the "simply laced"
Dynkin diagrams.  I want to explain what's so special about them!  But
first, I should describe the corresponding lattices more explicitly, to
make it clear how simple they really are.  

The following information, and more, can be found in Chapter 4 of:

1) Sphere Packings, Lattices and Groups, J. H. Conway and N. J. A.
Sloane, second edition, Grundlehren der mathematischen Wissenschaften
290, Springer-Verlag, 1993.  ISBN 0-387-97912-3.

which I described in more detail in "week20".  

So, what are A, D, and E like?  

A:  We can describe the lattice A_n as the set of all (n+1)-tuples
of integers (x_1,...,x_{n+1}) such that

                     x_1 + ... + x_{n+1} = 0.

It's a fun exercise to show that A_2 is a 2-dimensional hexagonal
lattice, the sort of lattice you use to pack pennies as densely as
possible.  Similarly, A_3 gives a standard way of packing grapefruit,
which is in fact the densest lattice packing of spheres in 3 dimensions.
(Hua has claimed to have shown it's the densest packing, lattice or not,
but this remains controversial.)

D: We can describe D_n as the set of all n-tuples of integers
(x_1,...,x_n) such that

                     x_1 + ... + x_n is even.

Or, if you like, you can imagine taking an n-dimensional checkerboard,
coloring the cubes alternately red and black, and taking the center of
each red cube.  In four dimensions, D_4 gives a denser packing of
spheres than A_4; in fact, it gives the densest lattice packing
possible.  Moreover, D_5 gives the densest lattice packing of in
dimension 5.  However, in dimensions 6, 7, and 8, the E_n lattices are
the best!

E: We can describe E_8 as the set of 8-tuples (x_1,...,x_8) such
that the x_i are either all integers or all half-integers --- a
half-integer being an integer plus 1/2 --- and

                     x_1 + ... + x_8 is even.

Each point has 240 closest neighbors.  Alternatively, as described in
"week20", we can describe E_8 in a slick way in terms of the
quaternions.  Another neat way to think of E_8 is in terms of the
octonions!  Not too surprising, perhaps, since the octonions and E_8 are
both 8-dimensional, but it's still sorta neat.  For the details, check
out

2) Geoffrey Dixon, Octonion X-product and E8 lattices, preprint
available as hep-th/9411063.

Briefly, this goes as follows.  In "week59" we described a
multiplication table for the "seven dwarves" --- a basis of the
imaginary octonions --- but there are lots of other multiplication
tables that would also give an algebra isomorphic to the octonions.
Given any unit octonion a, we can define an "octonion X-product" as
follows:

b X c = (b a)(a* c)

where a* is the conjugate of a (as defined in "week59") and the product
on the right-hand side is the usual octonion product, parenthesized
because it ain't associative.  For exactly 480 choices of the unit
octonion a, the X-product gives us a new multiplication table for the
seven dwarves, such that we get an algebra isomorphic to the octonions
again!  240 of these choices have all rational coordinates (in terms of
the seven dwarves), and these are precisely the 240 closest neighbors of
the origin in a copy of the E_8 lattice!  The other 240 have all irrational
coordinates, and these are the closest neighbors to the origin of a
*different* copy of the E_8 lattice.  (Here we've rescaled the E_8
lattice so the nearest neighbors have distance 1 from the origin,
instead of sqrt(2) as above.)

Once we have E_8 in hand, we can get its little pals E_7 and E_6 as
follows.  To get E_7, just take all the vectors in E_8 that are
perpendicular to some closest neighbor of the origin.  To get E_6, find
a copy of the lattice A_2 in E_8 (there are lots) and then take all
the vectors in E_8 perpendicular to everything in that copy of A_2.

So, now that we have a nodding acquaintance with A, D, and E, let me
describe some of the many places they show up.  First, what's so
great about these lattices, apart from the fact that they're the
root lattices of simple Lie algebras, with a special "simply-laced"
property?   I don't think I really understand the answer to this in
a deep way, but I know various things to say!  

First, Wit's theorem says that the A, D, and E lattices and their direct
sums are the only integral lattices having a basis consisting of vectors
v with ||v||^2 = 2.  Here a lattice is "integral" if the dot product of
any two vectors in it is an integer.  In fact, any integral lattice
having a basis consisting of vectors with ||v||^2 equal to 1 or 2 is a
direct sum of copies of A, D, and E lattices and the integers, thought
of as a 1-dimensional lattice.

This makes ADE classifications pop up in various places in math and
physics.  For example, there is a cool relationship between the ADE
diagrams and the symmetry groups of the Platonic solids, called the
McKay correspondence.  Briefly, this is what you do to get it.  First,
learn about SO(3) and SU(2) from "week61" or somewhere.  Then, take the
symmetry group of a Platonic solid, or more generally any finite
subgroup G of SO(3).  Since SO(3) has SU(2) as a double cover, you can
use get a double cover of G, say G~, sitting inside SU(2).  For example,
if G was the symmetry group of the icosahedron, G~ would be the icosians
(see "week24").  Since G~ is finite, it has finitely many irreducible
representations.  One of these will be 2-dimensional, coming from the
spin-1/2 representation of SU(2).  Draw a dot for each of the
irreducible representations, but draw the dot for this 2-dimensional
representation with red ink.  Now, when you tensor this 2d rep with any
other irreducible rep R, you get a direct sum of irreducible reps; draw
one line for the dot from R to each other dot for each time that other
irreducible rep appears in this direct sum.  What do you get?  Well, you
get an "affine Dynkin diagram" of type A, D, or E, which is like the
usual Dynkin diagram but with an extra dot thrown in.  And you get all
of them this way!

In fact, playing around with this stuff some more, you can get the
affine Dynkin diagrams of the other simple Lie algebras.  There is a lot
more to this... you should probably look at:

3) John McKay, Graphs, singularities and finite groups, in Proc. Symp. Pure
Math. vol 37, Amer. Math. Soc. (1980), pages 183- and 265-.

John McKay, Representations and Coxeter Graphs, in "The Geometric Vein"
Coxeter Festschrift (1982), Springer-Verlag, Berlin, pages 549-.

4) Pavel Etinghof and Michael Khovanov, Representations of tensor
categories and Dynkin diagrams, preprint available as hep-th/9408078.

McKay does lots of other mindblowing group theory.  He's clearly in tune
with the symmetries of the universe... and occaisionally he deigns to
post to the net!  A beautiful way of thinking about the McKay
correspondence in terms of category theory appears in the paper by
Etinghof and Khovanov; what we are really doing, it turns out, is
classifying the representations of the tensor category of unitary
representations of SU(2).  This tensor category is generated by one
object, the spin-1/2 representation, meaning that every other
representation sits inside some tensor power of that one.  This way of
thinking of it is important in

5) Jurg Froehlich and Thomas Kerler, Quantum Groups, Quantum Categories,
and Quantum Field Theory, Springer Lecture Notes in Mathematics 1542,
Springer-Verlag, Berlin, 1991.