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Sir Simon Donaldson: Lectures

September 8, 2023
TITLE: Curves in K3 surfaces and fibred calibrated submanifolds

ABSTRACT: We will discuss descriptions of calibrated submanifolds near the “adiabatic limit”. The situation we consider is where most of the ambient manifold is fibred by hyperkahler 4-manifolds and the calibrated submanifolds in question are modelled on fibrations with holomorphic curves as fibers. We will review first the case when the holomorphic curves are 2-spheres, which has been the scene of work by a number of authors (and the subject of previous talks in this Collaboration). We will then go on to consider the case of higher genus, which involves the geometry of moduli spaces of curves in K3 surfaces.

May 24, 2021
TITLE: Asymptotic analysis, moment maps and numerical approximations in Kahler geometry

ABSTRACT: We discuss techniques for obtaining numerical approximations to constant scalar curvature Kahler metrics via embeddings in high-dimensional projective spaces. If L is a positive line bundle over X with curvature a Kahler form \omega the differential geometry of (X,\omega) is reflected in the asymptotics of holomorphic sections of high powers of L and hence in the corresponding projective embeddings of X. Conversely, any metric on the finite dimensional space H^{0}(L^{k}) defines a Kahler metric on X. There is a notion of “balanced” metrics which converge as k\rightarrow \infty to a constant scalar metric, if one exists. These balanced metrics arise as zeros of moment maps and are related to Chow stability in algebraic geometry. There is a simpler theory in the Calabi-Yau case. These ideas lead to iteration procedures for finding numerical approximations. We will illustrate with some examples, particularly in the case of toric manifolds.

Slides of Lecture

January 14, 2021 and January 15, 2021
TITLE: Deformations of singular sets and Nash-Moser theory I, II

ABSTRACT: We consider “multivalued” solutions of certain elliptic PDE, with codimension 2 singular sets. The PDE of primary concern are the Laplace equation on a Riemannian manifold and the nonlinear “maximal submanifold” equation. The multivalued nature is expressed more precisely by saying that the solutions take values in a flat bundle over the complement of the singular set. Solutions of these kinds are relevant in various ways to manifolds of special holonomy, as we will review in the lectures. In particular, in dimension 2, when the singular set is a finite set of points, the square of the derivative of such a multivalued harmonic function is a holomorphic quadratic differential and there are relations to work of Bridgeland and Smith discussed in this meeting.

Slides of Lectures

September 16, 2020
TITLE: Towards enumerative geometry for structures on 4-manifolds

ABSTRACT: There are well-known theories (holomorphic curves, gauge theory on 4-manifolds) in which one defines a virtual fundamental class of solutions to elliptic PDE, which then yields numbers by pairing with suitable cohomology classes. In this talk we discuss the possibilities of extending these ideas to structures on 4-manifolds, namely self-dual conformal and complex structures. The first case leads to compactness questions which are well beyond current understanding but there are interesting questions regarding orientations and coupling to gauge theory. In the second case, for complex structures of “general type” there is a good algebro-geometric KSBA theory of compactified moduli spaces. While the foundational question of the definition of a virtual fundamental class remains open we are able to do some calculations in a model example.

Slides of Lecture

January 6, 2020 and January 7, 2020
TITLE: Gauge theory and special holonomy I,II

ABSTRACT: The bulk of the two lectures will be a review of standard material related to “instanton” solutions of the Yang-Mills equations over manifolds of special holonomy. In the last part we may discuss some more recent and speculative ideas.
1. Differential geometric basics.
2. Formal structures (Floer theory etc.)
3. Rudiments of relevant analysis (monotonicity, small energy estimates)
4. Codimension-4 bubbling and Fueter sections
5 Higher codimension singularities.

Septebmer 12, 2019
TITLE: G2 Geometry and Adiabatic Limits

ABSTRACT: The main focus of the talk will be G2 manifolds with co-associative fibrations and in particular the “adiabatic limit” when the fibers become very small and the structure can be described by solutions of a version of the maximal submanifold equation. Donaldson will discuss progress in the development of this theory and prospects for the future. In particular, he will explain the connection with boundary value problems for G2 structures and descriptions, in part conjectural, of calibrated submanifolds in the adiabatic limit.

Slides of lecture

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

ABSTRACT: We will discuss the deformation theory of adiabatic Kovalev-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 12, 2018
TITLE: An Excursion Through Geometry, Topology and Analysis in 4 Dimensions and Beyond, (Simons Foundation Lecture)

ABSTRACT: The study of differential geometric structures on manifolds has evolved from elementary geometry and calculus to the more complex structures prominent in current research. Work in this area of research has led to significant advances in theoretical physics.

In this lecture, Simon K. Donaldson will explain some basic concepts in modern differential geometry and their historical development. He will discuss connections with analysis of complex variables and illustrate how fundamental existence questions lead to nonlinear partial differential equations. The central section of the talk will discuss four-dimensional K3 surfaces. Donaldson will discuss the special solutions of Einstein’s equation on such surfaces, connected to the quaternions. He will outline some of the analytical techniques used to establish the existence of these. He will also sketch some related questions in higher dimensions which are the scene for much current research activity.

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