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Jason Lotay: Lectures

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March 16, 2023
TITLE: Geometric flows and special holonomy: past, present and future

ABSTRACT: Two geometric flows have been of particular interest to this Simons Collaboration on Special Holonomy: Lagrangian mean curvature flow and G2-Laplacian flow. I will give a brief overview of progress made on these two topics by the Collaboration, describe current research directions and discuss challenges for the future.

September 13, 2022
TITLE: Neck pinches in Lagrangian mean curvature flow

ABSTRACT: For a Lagrangian mean curvature flow which could converge to a special Lagrangian, the simplest singularities are modelled on special Lagrangians called Lawlor necks and are called neck pinches by Joyce. I will describe recent joint work with F. Schulze and G. Szekeleyhidi on Lagrangian surfaces, which shows that we can recognize a neck pinch using only information about the tangent flow at the singular time, that we can flow through the singularity and that there is a relation between the singularity formation and notions of stability.

September 10, 2021
TITLE: Some remarks on contact Calabi-Yau 7-manifolds

ABSTRACT: In geometry and physics it has proved useful to relate G2 and Calabi–Yau geometry via circle bundles. Contact Calabi–Yau 7-manifolds are, in the simplest cases, non-trivial circle bundles over Calabi–Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures. They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull–Strominger system), which is a coupled set of equations arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi–Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.

Slides of Lecture

October 26, 2020
TITLE: Special holonomy and branes: observations and themes

ABSTRACT: In this talk, I will first make some remarks about various geometric objects motivated by the study of physics and branes. I will then describe some relevant research themes at the interface of physics and geometry, primarily from a mathematical perspective.
Slides of Lecture

September 18, 2020
TITLE: Deformed G2-instantons

ABSTRACT: Deformed G2-instantons arise as “mirrors” to certain calibrated cycles in G2 geometry, providing an analogue to deformed Hermitian-Yang-Mills connections, and are critical points of a Chern-Simons-type functional. I will describe an elementary construction of the first non-trivial examples of deformed G2-instantons, and their relation to 3-Sasakian geometry, nearly parallel G2-structures, isometric G2-structures, obstructions in deformation theory, the topology of the moduli space, and the Chern-Simons-type functional.

Slides of Lecture

September 14, 2020
TITLE: Singularities in the G2-Laplacian flow (a contribution to Discussion II)

Slides

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.

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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.