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Sir Simon Donaldson: Lectures

June 7, 2019
TITLE: Deformations of branched harmonic functions and Kovalev-Lefschetz fibrations

ABSTRACT: We will discuss the deformation theory of adiabatic Kovale-Lefschetz fibrations, assuming some background from the previous lecture. We will review some of the relevant elliptic theory for functions with the appropriate singularities along a link and explain how Nash-Moser theory can be applied to a related simplified, linear, problem. In the last part of the lecture we will outline an approach to the nonlinear case, using an iteration scheme.

June 5, 2019
TITLE: Adiabatic associative and co-associatives

ABSTRACT: The notion of an “adiabatic Kovalev-Lefschetz fibration” is a proposal to model the behaviour of G2 manifolds with co-associative fibrations, which are expected to exist in many examples. The first part of the lecture will begin reviewing the set-up, involving a PDE for a section of a flat bundle over the complement of a link in the 3-sphere. In the second part of the lecture we will discuss descriptions of calibrated submanifolds in these fibrations. We will focus on associative sub manifolds diffeomorphic to S2× S1 corresponding to closed orbits of a gradient vector field in the complement of the link. We will explain how to go from these orbits to both formal power series and genuine solutions of the associative equation. We will then discuss a number of variants of the idea, which will involve more difficult analysis.

September 11, 2018
TITLE: G2-structures on manifolds with boundary and positive mean curvature.

ABSTRACT: This is a continuation of one of my talks at the June meeting, but I will begin by reviewing background to make the lecture self-contained. We will discuss the multilinear algebra of 3-forms in 6 and 7 dimensions and explain that there is an intrinsic notion of a 3-form with “postive mean curvature” on a 6-manifold. When the 3-form is the restriction of a G_{2}-structure to the boundary of a 7-manifold this notion is consistent with the standard definition in Riemannian geometry. Combining this observation with well-established comparison theorems in Riemannian geometry we derive geometric inequalities, such as a volume bound, for torsion-free G_{2}-structures with positive mean curvature boundary data and discuss the possible extension to closed structures.

June 8, 2018
TITLE: Adiabatic limits, multi-valued harmonic functions and the Nash-Moser-Zehnder theory

ABSTRACT: This lecture is a report on work in progress. We will begin by reviewing an adiabatic limit of the G2 equations, for K3-fibred manifolds. This involves data comprising a link in the 3-sphere, a flat bundle over the complement of the link and a section of this bundle which locally parametrizes a maximal submanifold, with branching over the link. Then we will turn attention to a simpler model problem involving “branched”, or multi-valued, harmonic functions. These also have some connection with work of Taubes, Takahashi, Walpuski , Haydys and Doan. We will explain how, given a suitable analytical set-up, an extension by Zehnder of the Nash-Moser theory can be applied to study the deformation theory of these branched functions.

June 8, 2018
TITLE: G2 manifolds with boundary

ABSTRACT: We discuss the problem of finding a torsion-free G2 structure on a 7-manifold with boundary where the restriction of the 3-form to the boundary is prescribed. We will begin by reviewing the mulitilinear algebra of 3-forms in 6 and 7 dimensions and the connection with almost-complex structures. The main point of the lecture will be to explain how to set up the problem as a nonlinear elliptic boundary value problem. This leads, in a standard way, to a Kuranishi model for the deformation theory and we discuss the obstruction space that arises. We will also explain that there is an intrinsic notion of “mean-convex” boundary data, and in this case one can derive various explicit geometric bounds on solutions.

September 14, 2017
TITLE: Collapsing co-associative fibrations

ABSTRACT: We will begin by explaining how “maximal” sub manifolds in spaces of indefinite signature arise as formal collapsing (or adiabatic) limits of G_2 manifolds with co-associative fibrations. Then we will discuss some analytical problems which arise in developing this idea, mostly having to do with the critical sets where the fibres become singular and the maximal submanifolds have branch points. In one direction we will discuss the deformation theory of these sets and in another we outline the relevance of recent constructions (by Yang Li and others) of new Calabi-Yau metrics on \mathbb{C}^3.  
Slides of lecture

June 9, 2017
TITLE: 3+4 dimensional reductions of G2 holonomy—collapsing and boundary value problems

ABSTRACT: The theme of the talk will be various simplifications of the equations defining G_2 holonomy structures, involving spaces that are fibred by either 3-dimensional or 4-dimensional manifolds. A central feature  in these situations is an equation for “positive triples” of 2-forms over 4-manifolds,  which is a generalisation of the hyperkahler condition. In one direction we will explain how this leads to a conjectural adiabatic limit of the G_2 equations for manifolds with co-associative “Kovalev-Lefschetz”  fibrations,  describing Riemannian collapsing to a 3-dimensional limit.  In another direction we will discuss boundary value problems for positive triples and an extension of the Gibbons-Hawking construction, which leads to a formulation in terms of the  Monge-Ampere equation on domains in  \mathbb{R}^{3}.

September 9, 2016
TITLE: Adiabatic limits of coassocative fibrations

September 7, 2016
TITLE: Introduction to formal aspects of gauge and submanifold theory

September 6, 2016
TITLE: Introduction to global questions around special holonomy