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Robert Bryant: Lectures

September 9, 2022
TITLE: On Solitons for the closed G2-Laplacian Flow

Slides of Lecture

September 18, 2020
TITLE: The geometry of pseudoholomorphic curves in the 6-sphere

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. If time permits, I will discuss some of what I know about the Gromov compactification of certain moduli spaces of pseudoholomorphic curves in the 6-sphere.

Slides of Lecture

September 12, 2019
TITLE: An Overview of the Progress and Goals of the Special Holonomy Simons Collaboration

ABSTRACT: This talk will serve as an introduction to the meeting, including background on the area of special holonomy and an overview of the fundamental existence results, progress made by our collaboration (and others) and what we see as the major goals and challenges in current research in special holonomy.
Slides of lecture

June 8, 2018
TITLE: On solitons for the closed G2-Laplacian flow

ABSTRACT: After discussing the necessary background about techniques from exterior differential systems, I will present some results about the local structure of solitons for the Laplacian flow on closed G_2-structures. In particular, their local generality will be discussed, along with other aspects, as time permits.

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.

Slides of lecture

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.

Slides of lecture