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Sir Simon Donaldson: Lectures

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

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


September 9, 2016
TITLE: Adiabatic limits of coassocative fibrations