M. Hamilton and D. Kotschick: Mathematical Gauge Theory II

• Place and time: Tue 10-12 and Thu 14-16 in A 027
• Exercise class: Wed 12-14 (tentative!) in A 027
• Summary: This course is about the applications of gauge theory to the geometry and topology of four-dimensional manifolds. It continues the introduction to mathematical gauge theory taught by one of us (MH) last spring. Students who know the basics of the geometry of (principal) fiber bundles can attend this course, even if they did not attend last semster.
  We shall start with a discussion of spin^c structures and Dirac operators. We will give a geometric introduction to smooth four-manifolds, discussing examples, the intersection form, perhaps some homotopy theory of four-manifolds, and embedded surfaces. We then develop the basics of Seiberg-Witten gauge theory on four-manifolds, and we apply this theory to the study of both topological and geometric properties of four-manifolds. The latter are related to the existence of complex and symplectic structures, and of special Riemannian metrics.
  Time permitting, we shall also cover some topics closer to the physics of gauge theories.
• Intended audience: Master and PhD students of mathematics and/or physics.
• Prerequisites: Some knowledge of differential geometry and topology.
• References:
  S. K. Donaldson and P. B. Kronheimer: The Geometry of Four-Manifolds. Oxford University Press 1990.
  J. W. Morgan: The Seiberg-Witten equations and applications to the topology of smooth four-manifolds. Mathematical Notes, 44. Princeton University Press, Princeton, NJ, 1996.
  R. E. Gompf and A. I. Stipsicz: 4-Manifolds and Kirby Calculus, American Math. Soc. 1999.
• Exams: The course is worth 9 ECTS points. There will be an exam in February 2017.