# The geometry of pseudoholomorphic curves in the 6-sphere. Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems. Duke University. September 18, 2020

Invited Talk

ABSTRACT: The 6-sphere has a well-known G_2-invariant almost complex structure and its pseudoholomorphic curves have interesting properties: They are minimal (though not homologically minimizing) and the cone on such a curve is an associative 3-fold whose local singular structure is well-understood. I will briefly survey some of the basic known results on the geometry of such pseudoholomorphic curves, in particular exploring their similarities and differences with holomorphic curves in \mathbb{CP}^3. Using a connection with the G_2(\mathbb{C})-invariant holomorphic 2-plane field on the complex 5-quadric discovered by Cartan, I will explain how one can construct pseudoholomorphic curves with some prescribed singularities. I will discuss some of what I know about the Gromov compactification of certain moduli spaces of pseudoholomorphic curves in the 6-sphere.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- September 18, 2020

### Service or Event Name

- Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems

### Host Organization

- Duke University

### Location or Venue

- virtual