September 10, 2017
TITLE: Algebraically special associative submanifolds and special holonomy metrics
ABSTRACT:This talk survey progress in the past year on classifying the algebraically special associative submanifolds in R7, in particular, the ones for which the second fundamental form has nontrivial symmetries, and metrics with special holonomy whose curvature tensors are algebraically special.
September 6, 2016
TITLE: On families of special holonomy metrics defined by algebraic curvature conditions
ABSTRACT: 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.