# On families of special holonomy metrics defined by algebraic curvature conditions. Inaugural Conference of the Simons Collaboration "Special Holonomy in Geometry, Analysis, and Physics". Simons Center for Geometry and Physics. September 6, 2016

Invited Talk

There are various methods known now for constructing more-or-less explicit metrics with special holonomy; most of these rely on assumptions of symmetry and/or reduction. Another promising method for constructing special solutions is provided by the strategy of looking for metrics that satisfy algebraic curvature conditions. This method often leads to a study of structure equations that satisfy an overdetermined system of partial differential equations, sometimes involutive sometimes not, and the theory of exterior differential systems is particularly well-suited for analyzing these problems. In this talk, I will describe the ideas and the underlying techniques needed from the theory of exterior differential systems, illustrate the application in the most basic cases, and describe the landscape for the research needed to carry out this program. A similar program is envisioned for finding special calibrated submanifolds of the associated geometries and, if time permits, I will describe some of this work and the initial results.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- September 6, 2016

### Service or Event Name

- Inaugural Conference of the Simons Collaboration "Special Holonomy in Geometry, Analysis, and Physics"

### Host Organization

- Simons Center for Geometry and Physics

### Location or Venue

- Simons Center for Geometry and Physics, Stony Brook, NY