The school will
discuss two main topics: global analysis on spaces
with singularities and the Seiberg  Witten equations and their
applications. These topics constitute two very active directions of
the
current research, relevant both for mathematics and physics. Beyond
the
independent interest for studying each of these subjects, separately,
there
is a deep reason for studying them together. Indeed, the present Seiberg

Witten theory is discussed exclusively on differentiable four dimensional
manifolds. In spite of this, there are some basic questions arising
in the
Donaldson  Seiberg  Witten theory which could not be addressed within
the
category of differentiable manifolds: their topological invariance
as well
as finding combinatorial definitions or constructions.
Previous fundamental work
due to Kirby and Siebenmann, Sullivan,
Donaldson, Donaldson and Sullivan show that the universe of four dimensional
manifolds is extremely complicated, in which the topological and
differentiable structures are extreme ends of a stratified world evolving
through quasi conformal, Lipschitz, and combinatorial structures. There
is a
foundational conjecture due to Sullivan, which states that an appropriate
Dirac package used in the Seiberg  Witten theory does not exist for
a quasi
conformal or Lipschitz gauge unless the gauge contains a smooth structure.
The courses presented at this school will introduce the students into
these
two topics, offering them the possibility, at the same time, to attack
foundational problems of the Donaldson  Seiberg  Witten theory.
The course given by
M. Hilsum will present results due to Teleman 
Donaldson  Sullivan and Hilsum, which extend the Atiyah  Singer index
theory from the category of smooth manifolds to the category of Lipschitz
and quasi conformal manifolds. In these spaces the singularities are
densely
distributed.
Although these theories
require the weakest amount known of
analytical structure, they allow one to discuss index phenomena only
on
manifolds. The index theory can be extended in another direction, to
stratified spaces, where the manifold requirement is dropped and a
smooth
structure on the strata is required in change. This theory, due primarily
to
Cheeger, on the analytical side, and to Goresky  MacPherson, on the
geometrical side, will be presented in J.P. Brasselet and A. Legrand's
course. The stratified spaces are relevant in the study of algebraic
singularities.
The unifying theory for all these contexts
is offered by the non
commutative differential geometry. The course given by N. Teleman will
present some basic techniques of non commutative geometry needed in
the
study of singular spaces, with a special attention to the problem of
combinatorial Pontrjagin classes.
The course given by Y. Yomdin
will discuss quantitative problems in
the theory of singularities, with applications, shedding a different
prospective in this study.
The course given by A. Teleman
will present the Seifert  Witten
theory with applications.
The school addresses primarily
to the new Ph.D. holders and Ph.D.
candidates in mathematics and physics. The school will consist of lectures
as well as seminars and workshops. The courses will be held at the
Mathematics
Department of the Cluj University. Cluj is the capital of Transylvania.
The housing will be provided in the Cluj University complex.
J. P. Brasselet (University of Marseille, France),
A. Legrand (University of Toulouse, France):
Global Analysis on Pseudomanifolds.
N. Teleman (University of Ancona, Italia) :
Topics on Non Commutative Geometry.
M. Hilsum ( Institut Henry Poincar\'e, Paris):
Index Theory on Lipschitz and Quasi conformal Manifolds.
Y. Yomdin ( Weizman Institute, Israel):
Quantitative Theory of Singularities.
A. Teleman ( Universitatea din Bucuresti, Romania and Universit\"at
Z\"urich):
Seiberg  Witten Equations and Applications.
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Summer School on Singular Spaces and Monopols
Dipartimento di Matematica "V. Volterra"
Universit\'a degli Studi di Ancona
60131  Ancona
ITALIA
FAX: 3971220.4870
Email: TELEMAN@ANVAX1.UNIAN.IT
The answer to the applications will be sent out by May 15, 1998.