2023 A. Everett Pitcher Lecture Series - Sir Simon K. Donaldson

April 10, 2023 - 7:30pm

Sir Simon K. Donaldson
Simons Center for Geometry and Physics, Stony Brook & Imperial College London

Monday, April 10, 2023
Lecture I (Public Lecture )
Moduli spaces of geometric structures, old and new
Lewis Lab 270
7:30 PM; Lobby Reception at 6:45 PM

Tuesday, April 11, 2023
Lecture II
Multivalued functions, the Torelli Theorem for K3 surfaces and collapsing G_{2} structures
Neville 001
4:30 PM

Thursday, April 13, 2023
Lecture III
Calabi-Yau manifolds and G_{2} manifolds with boundary
Chandler-Ullmann 218
4:30 PM


Lecture 1. "Moduli spaces of geometric structures, old and new”,

There is a distinction in mathematics between objects which are “special” and rigid and those which occur in continuous families. Important examples of the latter arise in the study of spaces endowed with differential geometric structures, such as solutions of Einstein’s equation (in the Riemannian case). Then the parameter spaces are known as “moduli spaces”. We will begin by reviewing the classical example of moduli spaces of Riemann surfaces, or 2-dimensional manifolds with constant curvature. Then we will  mention well-known higher-dimensional examples, such as Calabi-Yau structures. We will introduce the exceptional Lie group G_{2}, related to the octonion number system, and 7-dimensional manifolds with G_{2} structures,  and state some basic properties of their moduli spaces. Nearly all questions about the global structure of these remain open and we will outline some prominent directions of current research in the area.

Lecture 2. "Multivalued functions, the Torelli Theorem for K3 surfaces and collapsing G_{2} structures”

One of the main problems in modern differential geometry is to understand “collapsing” phenomena, where a sequence of Riemannian manifolds converge to a space of lower dimension. In the case of 7-dimensional manifolds with torsion free G_{2} structures there are good reasons to think that an important collapsing mechanism involves coassociative fibrations. (These are analogous to the Special Lagrangian fibrations of Calabi-Yau manifolds which are central to the Strominger-Yau-Zaslow picture of Mirror Symmetry.) In this lecture we will describe some progress towards a theory of collapsing coassociative fibrations.  A key ingredient is the good understanding of moduli spaces of K3 surfaces provided by the Torelli Theorem. Some interesting analysis questions arise, related to the “multivalued” solutions of elliptic equations, with codimension 2 singularities, which have appeared in a number of areas of differential geometry recently.

Lecture 3. "Calabi-Yau manifolds and G_{2} manifolds with boundary”

The deformation theory of Calabi-Yau structures and G_2 structures on closed manifolds is well-understood. In this lecture we will discuss variants of these for manifolds with boundary. In the G_2 case we set up an elliptic boundary value problem. In the case of Calabi-Yau manifolds of complex dimension 3 we focus on those that occur as pseudoconvex domains in {\bf C}^{3}. (This is joint work with F. Lehmann.) The problem then becomes an embedding question, for a 5-manifold equipped with a closed 3-form. This can be viewed as a generalisation of the classical Minkowski problem in affine differential geometry, for a convex surface in {\bf R}^{3}. We describe a generalisation of the affine area to the complex case, and discuss possible analogues of the affine isoperimetric inequality