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Jason Lotay: Lectures

May 13, 2024
TITLE: G2-instantons, the heterotic G2 system and generalized geometry

ABSTRACT: The heterotic G2 system is a complex system of PDEs arising from heterotic String Theory which intertwines an ambient G2-structure with G2-instantons for this structure. In this talk I will describe recent work, motivated by ideas arising from the physics literature, which re-interprets the heterotic G2 system as a single G2-instanton equation, but on a higher rank bundle. I will also describe how this viewpoint leads to a generalized metric (in the sense of generalized geometry) which is generalized Ricci-flat, and discuss some examples. This is joint work with A.A. da Silva Jr, M. Garcia-Fernandez and H.N. Sa Earp.

Slides of Lecture

September 7, 2023
TITLE: Joyce conjectures for the Lagrangian mean curvature flow of surfaces

ABSTRACT: Building on the seminal work of Thomas-Yau, Joyce formulated an inspirational collection of conjectures concerning the Lagrangian mean curvature flow in Calabi-Yau manifolds which, in particular, relate the flow to notions of stability and the Fukaya category. I will present an overview of recent progress towards these conjectures in the case of Lagrangian surfaces. This is based on joint works with F. Schulze and G. Szekeleyhidi, and also with G. Oliveira.

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)


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.