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Jason Lotay: Lectures

June 3, 2019
TITLE: Ancient solutions in Lagrangian mean curvature flow

ABSTRACT: To make progress in the study of Lagrangian mean curvature flow, one
needs to understand singularity formation. Singularities of the flow are
modelled on ancient solutions in complex Euclidean space. I will describe
structural and classification results for ancient solutions which can arise as
singularity models, focusing on the almost calibrated setting.

April 8, 2019
TITLE: Bryant-Salamon metrics and coassociative fibrations

ABSTRACT: The first examples of complete holonomy G2 metrics were constructed by Bryant-Salamon and are thus of central importance in geometry, but also in physics, appearing for example in the work of Atiyah-Witten, Acharya-Witten and Acharya-Gukov. I will describe joint work in progress with Spiro Karigiannis which realises Bryant-Salamon manifolds in dimension 7 as coassociative fibrations. In particular, I will discuss the relationship of this study to gravitational instantons, conical singularities, and to recent work of Donaldson and Joyce-Karigiannis.

September 14, 2018
TITLE: The G2 Laplacian flow

ABSTRACT: The G₂ Laplacian flow was introduced by Bryant as a potential tool for studying the challenging problem of existence of holonomy G₂ metrics. I will introduce the flow, provide a brief survey of the general theory and describe some recent progress.

Slides of lecture

June 6, 2018
TITLE: The G2 Laplacian flow: progress and outlook

ABSTRACT: I will discuss some recent progress in understanding the G2 Laplacian flow, particularly in the presence of symmetries, and describe some open problems and questions.

June 5, 2018
TITLE: The G2 Laplacian flow: introduction and overview

ABSTRACT: The G2 Laplacian flow was introduced by Bryant as a potential tool for studying the challenging problem of existence of holonomy G2 metrics. I will give an introduction to the flow and a brief survey of the general theory.

Video unavailable.

September 10, 2017
TITLE: Invariant coassociative 4-folds via gluing

ABSTRACT: Coassociative 4-folds in R7 with symmetry have been studied by several authors, including Harvey-Lawson and Bryant. Such submanifolds with S1-symmetry locally exist in abundance, but few global examples are known. I will describe joint work with Nicos Kapouleas which produces infinitely many embedded, asymptotically conical, S1-invariant coassociative 4-folds in R7 by a gluing method.

June 8, 2017
TITLE: Calibrated submanifolds of G2 and Spin(7) manifolds with conical singularities

ABSTRACT: I will give a brief survey of known results in the study of compact calibrated submanifolds with conical singularities in exceptional holonomy manifolds. In particular, I will describe their deformation theory, desingularization results and applications, including the construction of examples and potential connections to calibrated fibrations.