## Columbia Mathematics Department Colloquium

*Genus-minimizing surfaces in three and four dimensions*

by

# Jacob Rasmussen

## Cambridge University and the Simons Center for Geometry and Physics

Abstract:

Given a manifold M and a homology class x in H_{2}(M), what is the minimal genus of an embedded surface representing x? I'll discuss what we know about this question in dimensions 3 and 4, and describe applications to topics such as knots with lens space surgeries and the construction of an exotic smooth R^{4}.