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Fakultät für Mathematik Universität Regensburg
Seminar on Gauge Theory
SoSe 2018

Bernd Ammann, Raphael Zentner

Type of course (Veranstaltungsart)

The subjects of the seminar are Donaldson's instanton theory and Seiberg-Witten theory. Both theories associate invariants to closed 4-manifolds. These invariants yield obstructions against positive scalar curvature metrics, even in situations when index theoretical methods fail. For example, in dimension 4 there exist simply-connected closed non-spin manifolds without metrics of positive scalar curvature, which is amazing, as there are no such manifolds in any other dimension. Donaldson's instanton theory uses moduli spaces of connections with self-dual curvature. Such connections yield solutions to the Yang-Mills equations, which can be considered as a Riemannian analogue of the vacuum Maxwell equations on 3+1-dimensional spacetime. An infinite-dimensional group, the "gauge group" acts on the space of solutions, but the quotient of the space of solutions by the gauge group is a finite-dimensional manifold, which provides these invariants. Seiberg-Witten theory is structurally similar, but uses spinors instead of connections. The seminar will provide the foundations of these two gauge theories, and thus it will provide the knowledge to understand the main topics in summer school and a conference of the CRC Higher Invariants in July 2018. There also examples homeomorphic manifolds

  • Freed, Uhlenbeck, Instantons and Four-manifolds, MSRI Publications, Springer-Verlag
  • J. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds

Recommended previous knowledge
Profound knowledge about manifolds and riemannian metrics

Monday 16-18


Course homepage
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

  • Organisational meeting/distribution of topics: Monday, February 5th, 11:30 M119 (Büo B.
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Regelungen bei Studienbeginn vor WS 2015 / 16
  • Benotet:
    • O. g. Studienleistung und Bestehen der o. g. Prüfungsleistung
  • Unbenotet:
    • O. g. Studienleistung und Bestehen der o. g. Prüfungsleistung
BSem, MV, MSem

Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn
vor WS 15/16