Cluj - Romania, 27 July - 14 August 1998

          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

         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

     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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4.  Institution:

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1. School fee : 50 US Dollars
2. Room (double occupancy) and Meals for 18 days: 200 US Dollars.
   Total: 250 US Dollars (due after admission)


A limited number of grants, covering the transportation  to/from Cluj,
room and board and the school fee,
are available for those participants who, in order to attend the
Summer School, definitely need financial support.

Are you applying for financial support, based on need for
(a) transportation ?

                  Yes                 No

(b) School Fee, Room and Board ?

                  Yes                 No

Depending upon the availability of funds, would you apply for financial
support, independently of need ?

                  Yes                 No


The Application Forms should be sent before April 30, 1998 to the address:

Summer School on Singular Spaces and Monopols
Dipartimento di Matematica "V. Volterra"
Universit\'a degli Studi di Ancona
60131 - Ancona
FAX: 39-71-220.4870

The answer to the applications will be sent out by May 15, 1998.