IN-PERSON MATHEMATICAL PHYSICS SEMINAR
RUTGERS UNIVERSITY
HILL 705
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Paul Feehan – Rutgers University
Date/Time/Location
Thursday, November 14th, 2:00 pm; Hill Center 705
Gauge theory and the Bogomolov-Miyaoka-Yau inequality
The Bogomolov-Miyaoka-Yau (BMY) inequality is a topological property of most compact complex surfaces. It was proved by Miyaoka (1977) using methods of algebraic geometry and by Yau (1977/78) as a consequence of his solution to the Calabi Conjectures, obtained by solving the fully nonlinear, second-order complex Monge-Ampere equation. It is a well-known conjecture that the BMY inequality should hold, far more generally, for smooth 4-manifolds with non-zero Seiberg-Witten invariants, which includes symplectic 4-manifolds (by a result of Taubes). For the past 5 years, we have worked with Leness to prove this conjecture by solving the anti-self-dual Yang-Mills (ASD) equation on a complex rank-2 vector bundle (with topological constraints) over such 4-manifolds. To solve the ASD equation, we use a version of Morse theory (valid on singular spaces) on the moduli space of solutions to the non-Abelian monopole equations. In this talk, I shall describe our progress towards a resolution of the conjecture.