Seminar über Seiberg-Witten-Theorie im WS 2012/13

Veranstaltungsnr. 51263

Universität Regensburg, Fakultät für Mathematik

Prof. Dr. Bernd Ammann, Dr. Nicolas Ginoux




Zeit und Ort:

S: Mo 16-18 im M103
Tutorium: Di 10-12 im M119 oder nach Vereinbarung

Programm und Link zur G.R.I.P.S.-Seite



Vortragsplan: Namen wegen Datenschutz entfernt

  Datum  
  Thema  
  Sprecher  
  15.10     Spinc structures on manifolds      
  22.10     kein Vortrag     -  
  31.10     The Seiberg-Witten moduli space      
  5.11     Compactness of the moduli space      
  12.11     The parametrized moduli space      
  19.11     Orientability of the moduli space      
  26.11     Structure of the Seiberg-Witten invariants in the case b2+>1      
  3.12     kein Vortrag     -  
  10.12     Structure of the Seiberg-Witten invariants in the case b2+=1     
  17.12     Structure of the Seiberg-Witten invariants in the case b2+=1 [Fortsetzung] / An involution in the theory    
  7.01     The intersection form of a smooth 4-manifold      
  14.01     The intersection form of a smooth 4-manifold [Fortsetzung] / Seiberg-Witten invariants of Kähler surfaces      
  21.01     Seiberg-Witten invariants of Kähler surfaces [Fortsetzung]      
  28.01     Seiberg-Witten invariants of Kähler surfaces [Fortsetzung]      
  29.01     Seiberg-Witten invariants of symplectic manifolds      
  4.02     Seiberg-Witten invariants and rigidity of Einstein metrics      




Module: MSem



Leistungspunkte: 6



Inhaltsangabe / Literatur / empfohlene Vorkenntnisse:

Inhalt: Seiberg-Witten invariants are a very efficient tool for understanding topological and geometrical properties of compact 4-dimensional manifolds. If b2+>1, then these invariants only depend on the smooth structure on the 4-manifold. These invariants yield obstructions to the existence of smooth structures on topological 4-manifolds, they can rule out that certain smooth 4-manifolds carry an Einstein metric, and they yield Mostow rigidity for compact quotients of complex hyperbolic space. On complex surfaces they can be calculated with reasonable effort, and techniques such as gluing formulas allow their calculation on many more spaces. It can also be shown that symplectic 4-manifolds have non-trivial Seiberg-Witten invariants. The goal of the seminar is to learn the definition of these invariants which relies on gauge theoretical methods. We want to learn how to calculate them on complex surfaces, and to study the applications mentioned above. The invariants are also strongly linked to Gromov-Witten invariants, quantum cohomology and Seiberg-Witten-Floer theory, theories that we do not plan to cover in the seminar.

Literatur:
  • S.K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279-315.
  • S.K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45-70.
  • S.K. Donaldson, P.B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford University Press, 1990.
  • T. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, 1997.
  • T. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics 25, American Mathematical Society, 2000.
  • M. Hutchings, C.H. Taubes, An introduction to the Seiberg-Witten equations on symplectic manifolds, Symplectic geometry and topology (Park City, UT, 1997), 103-142, IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, RI, 1999.
  • M. Ishida, C. LeBrun, Spin Manifolds, Einstein Metrics, and Differential Topology, Math. Res. Lett. 9 (2002), no. 2-3, 229-240 (siehe arXiv:math/0107111).
  • D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds, Séminaire Bourbaki, Exp. No. 812, Astérisque 241 (1997), 195-220.
  • P.B. Kronheimer, T.S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797-808.
  • C. LeBrun, Einstein metrics and Mostow rigidity, Math. Res. Lett. 2 (1995), no. 1, 1-8 (siehe arXiv:dg-ga/9411005).
  • C. LeBrun, Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 1, 1-8 (siehe arXiv:0803.3743).
  • J.W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, Princeton University Press, 1996.
  • J.W. Morgan, Seiberg-Witten invariants, in: Nouveaux invariants en géométrie et en topologie, 61-98, Panor. Synthèses 11, Soc. Math. France, 2001.
  • L.I. Nicolaescu, Notes on Seiberg-Witten theory, Graduate Studies in Mathematics 28, American Mathematical Society, 2000.
  • L.I. Nicolaescu, Lectures on the geometry of manifolds, World Scientific Publishing, 1996.
  • D. Salamon, Spin geometry and Seiberg-Witten invariants, Lecture Notes, ETH Zürich, 2000.
  • C.H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809-822.
  • C.H. Taubes, More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), no. 1, 9-13.
  • C.H. Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995), no. 2, 221-238.
  • E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769-796.

    Vorkenntnisse:
  • Differentialgeometrie I und II: Participants should have a solid knowledge in differential geometry, including the most important properties of Dirac operators.
  • Topologie I und II



    Anschlussveranstaltungen: keine



    Zielgruppen: Master



    Prüfungsbestandteile: Vortrag + schriftliche Ausarbeitung + aktive Teilnahme

    Termine und Dauer von Prüfung und erster Wiederholungsprüfung: entfallen

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    Nicolas Ginoux, 30.01.2013