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\begincenter{Symplectic Manifolds, Seiberg-Witten Gauge Theory, and Taubes
Minimal Conjecture\\Mathematics 658, Professor Connolly\\
MWF 10:40am-11:30am  Room 326, CCMB }\\
First meeting:  Wednesday, January 15\\
DESCRIPTION OF THE COURSE}\endcenter

(This course will presume no previous knowlege of Gauge Theory. However a
knowlege of vector
bundles, transversality for smooth manifolds, algebraic topology,
Riemannian metrics and
differential forms will be assumed).

An oriented smooth compact 4-manifold $X^4$, is \it irreducible \rm if it
admits no connected sum
decomposition, $X= Y\#Z$, except when $Y$ or $Z$ is homeomorphic to $S^4$.
$X$ is \it
minimal \rm if $X$ admits no decomposition, $X= Y\# \overline {CP^2}$.
Finally $X$ is
symplectic if $X$ comes with a closed real 2-form $\omega$ such that at
every point $x$
$\omega_x$ is nonsingular.

In 1995 , Taubes proved that a simply connected minimal symplectic manifold
must be irreducible
in the above sense.  Then Taubes, Mrowka, and others conjectured that,
conversely, a simply
connected irreducible 4-manifold must be minimal symplectic.  This is the
Minimal Conjecture. But
three months ago, Z.Szabo proved that the Minimal Conjecture was false. He
was aided in this by
the \it Product Formula \rm  of Morgan-Szabo-Taubes.

All of the above work in smooth-4-manifold-topology was proved using ideas
pioneered by
physicists. Beginning with Yang-Mills theory, and culminating, in 1994,
with the work of Seiberg
and Witten, Gauge Theory has become the unrivalled  method for
investigating smooth 4-manifolds.

This course will introduce the student to Seiberg-Witten Gauge Theory. The
heart of this theory
is a $Z$-valued invariant on the set of $Spin^c$-structures on an oriented
4-manifold $X$. This
is the Seiberg-Witten Invariant, $SW:Spin^c(X)\to Z$. It vanishes on
manifolds expressible as
"non-trivial" connected sums. After some applications of this,  the course
will focus on the
Product Formula mentioned above, which gives a means of computing SW in
certain cases. This is
the heart of the course.

	The first major consequence of this formula is the result that a
nonsingular holomorphic curve
in a Kaehler surface, say $C\subset X$, with $C\.C\geq 0 $, must have
minimal genus among all
smooth embedded 2-manifolds in its homology class. The second major
consequence (in combination
with other work) is Szabo's counterexample. We will outline (at least) the
counterexample, show
how the minimal-genus theorem follows from the Product Formula, and then
concentrate on the
proof of the product formula.

\centerline{BIBLIOGRAPHY}
E. Witten: \it Monopoles and Four Manifolds, \rm Math. Research Letters, 1994.

P. Kronheimer and T. Mrowka: \it The Genus of Embedded Surfaces in the
Projective Plane, \rm
Math. Research Letters, 1995.

Morgan, J. : \it Seiberg Witten Invariants and Kahler Manifolds, \rm
Princeton University Press,
1996.

Salamon, D. : \it Spin Geometry and Seiberg-Witten Invariants, \rm Oxford
University Press, to
appear.

Taubes, C.: \it Gromov Invariants and Seiberg-Witten Invariants, \rm Jour.
AMS, 1996.








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