Mathematics Colloquia and Seminars

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Let L be an oriented link in S^3, thought of as the boundary at infinity of 4-dimensional hyperbolic space H^4. I will describe a conjecture which says that the Alexander polynomial of L counts the complete connected oriented minimal surfaces in H^4 which are asymptotic to L at infinity. The Alexander polynomial is simple to compute from a plane diagram of the link, meanwhile finding minimal surfaces is typically very hard. If true, the conjecture would give a simple combinatorial existence theorem for minimal surfaces filling L. I will explain what is already known about this conjecture and what (I hope!) remains to be done. A key role in the story is played by the twistor space Z of H^4. This is a symplectic 6-manifold whose J-holomorphic curves are in 1-1 correspondence with minimal surfaces. The Alexander polynomial conjecture can be thought of as a statement about the Gromov-Witten invariants of Z. Whilst Gromov-Witten invariants give a guide as to how to proceed, the whole story has to be rewritten because of the way the geometry of Z becomes singular at infinity. In particular, the theory of minimal surfaces in H^4 plays a crucial role in compactness arguments.