Geometry Seminar

Voorjaarssemester 1999

Reading Seminar for students, Mondays, 16.00-17.30, room WI 401, begin: 18th January

The plan of this seminar is to read together through the book Lectures on Seiberg-Witten Invariants by J.D. Moore (on reserve in the reading room of our library). Each student who wants to receive credit for this seminar (7 study points) has to give one of the fourteen 90 minute talks. Graduate students and staff are very welcome to attend and to participate actively by giving talks not distributed among students. As background knowledge a first course on manifolds should suffice. Vector bundles and connections will be reviewed in the first talk. A knowledge of homology theory, as developed in my course Introduction to Geometric and Algebraic Topology will be useful and at certain points essential, but I hope that this and all other necessary background material can be picked up during the seminar.

During the 1980's, Simon Donaldson used methods originating from mathematical physics (Yang-Mills equations) to prove remarkable theorems about the topology of 4-manifolds (winning him a Fields medal in 1986) of the following type:

There exist topological 4-manifolds without differentiable structure.
There exist differential 4-manifolds which are homeomorphic but not diffeomorphic.

In 1994 a new set of equations was proposed by Seiberg and Witten, which give many of the results of Donaldson theory, as well as many new results. The principal idea is that the solution spaces to these equations give invariants that depend not only on the topology of the manifold (like the fundamental group, say), but also on its differential structure. This seminar intends to develop Seiberg-Witten theory in sufficient depth to understand the differential geometric aspects of the theorems listed above. Here are the titles of the individual talks with chapter references and additional literature where appropriate.


Date	Title	Chapter	see also	Speaker
18.01.	Vector bundles and connections	1.2, 1.3	[MS]	Koos van Winden
25.01	Curvature of connections, characteristic classes	1.4, 1.5	[H]	Hansjörg Geiges
01.02	The universal bundle	1.6	[MS]	Gerben Dirksen
08.02	No seminar (Dies Natalis)
15.02	Classification of connections	1.7		Otto van Koert
22.02	Hodge theory	1.8	[W]	Willem van Briemen
01.03	Spin-, Spin^c- and almost complex structures	2.1-2.3		Bernard Meulenbroek
08.03	Clifford algebras	2.4	[LM]	Otto van Koert
15.03	The Spin-connection	2.5	[LM]	Bob Planqué
22.03	The Dirac-operator	2.6	[LM]	Erdal Emsiz
12.04	The Atiyah-Singer index theorem	2.7	[LM]	Erik van Erp
19.04	The Seiberg-Witten equations and compactness of the moduli space	3.1-3.3		Raoul Pietersz
26.04	Transversality	3.4		Christoph Hummel
03.05	Topology of 4-manifolds	3.5		Hansjörg Geiges
10.05	Seiberg-Witten invariants and Dirac operator on Kähler surfaces	3.6, 3.7		Martin Lübke

Additional literature: (Also on reserve in the reading room of our library)


[H]	D. Husemoller:	Fibre bundles, 3^rded., Springer, 1994.
[LS]	H. Lawson, M. Michelson:	Spin Geometry, Princeton Univ. Press, 1989.
[MS]	J. Milnor, J. Stasheff:	Characteristic Classes, Princeton Univ. Press, 1974.
[W]	F. Warner:	Foundations of differentiable manifolds and Lie groups, Springer, 1983.

For further information and assistance in preparing your talks you may contact
, or .

H.G.

, February 23, 1999.