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\begin{document}

\begin{center}
{\Large{\underline{MATH 658, Spring 1999 - Seiberg-Witten theory}}}
\end{center}
\begin{center}
Liviu I. Nicolaescu
\end{center}



 This course is the sequel to Math 657 currently running.   What is Seiberg-Witten theory?

Before  giving you a general idea I want to say I find this
subject very exciting because  it has something to offer to
anyone, and it does that in an unconventional way.

This very young theory is about  some nonlinear elliptic equations
and their applications to  low dimensional differential topology.
These equations  are formulated in  geometrically sensitive terms
(such as spinors and connections)  and because of this, the
geometry has a considerable impact on their analytic features. The
topological relevance of these equations comes into play when we
start counting in a suitable manner their solutions. The resulting
integer turns out to be a very fine invariant of a 4-manifold.

The fall semester covers the general   theory: what are these
equations, what are their properties how to count their solutions
and why the above count is a topological invariant.

In spring I want to put all these facts to work. I  want to show
how to {\em effectively} count solutions and then  discuss the
topological and geometric implications of these computations. This
count is easier if the background geometry is richer.  That is why
I will focus on algebraic surfaces and,   towards the end, on
symplectic 4-manifolds.  With a relatively modest technical
luggage one can prove   results  inaccessible before November
1994, the official birthdate of this theory.  I plan to  describe
Kronheimer and Mrowka's proof of the  Thom conjecture, to give a
proof of Donaldson's theorem on negative definite {\em smooth}
4-manifolds and to  discuss Taubes'  symplectic  nonvanishing
theorem.

Math 657 is a pre-requisite for the Spring sequel, but  if  you
have  a basic knowledge of elliptic equations and  Dirac operators
you will have little trouble catching-up.  Even if you do not plan
to become an expert  in this subject, the ideas and techniques we
will discuss in class have a wide applicability, and may prove to
be relevant in your research as well.

\begin{center}
{\bf Helpful reference}
\end{center}

\noindent J.W. Morgan: {\sl The Seiberg-Witten equations and
applications to the topology of smooth 4-manifolds}, Princeton
University Press, 1996.







\end{document}
