Abstracts for Informal Geometric Analysis Seminar

Monopoles are solutions to the Bogomolnyi equation, which is a PDE for a connection and an Higgs field (a section of an certain bundle) on a 3 dimensional Riemannian manifold. In this talk I plan to introduce these equations. Then I want to tell you some properties of its solutions on R3. Finally, I plan to speak about monopoles on a more general class of noncompact manifolds known as asymptotically conical. My main goal is to explain the geometric meaning of the parameters needed to give coordinates on an open set of the moduli space of monopoles.

We will review some classical problems in Differential Geometry, which lead us to work with manifolds with almost non-negative curvature. In particular, we will explain during the talk why it is natural to wonder weather for these manifolds a topological invariant called Â-genus vanishes (this question was proposed by John Lott in 1997). We will provide a positive answer by investigating sequences of spin manifolds with lower sectional curvature bound, upper diameter bound and the property that the Dirac operator is not invertible.

I will discuss an ongoing project to try to determine which alternating knots are smoothly slice and the related question of which double branched covers of alternating links bound smooth 4-manifolds with the rational homology of the 4-ball. The starting point is Lisca's classification of slice two-bridge knots using and obstruction coming from Donaldson's diagonalisation theorem.

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points.