Riemannian Spaces with Special Holonomy

MATH 278Z Subject & Catalog Number

Description

This course aims to provide an introduction to the modern field of special holonomy geometry. After discussing what might now be regarded as the “classical” story of holonomy, culminating in Berger’s classification theorem, we will delve into a detailed study of the special and exceptional cases in the classification. The setting for the first third of the course will be Kähler and hyperKähler geometry, with an emphasis on the use of geometric gluing techniques to construct fundamental examples. In the second third of the course, we will discuss the exceptional G2 and Spin(7) geometries. Our focus will again be on non-trivial constructions (for which the aforementioned special geometries will often serve as building blocks) involving clever uses of symmetry and geometric analysis. Such constructions will often live near the boundary of certain degenerations and thus we will find that “spaces”, by which we mean more singular objects than smooth manifolds, play a central role in the theory. In the final third of the course, we will introduce Harvey and Lawson’s notion of a calibrated geometry and study the interplay between special holonomy and special minimal submanifolds known as calibrated submanifolds. The geometric analysis developed in the first two-thirds of the course will serve us in constructing non-trivial examples of compact calibrated submanifolds and if time permits at the end of the course we will discuss related open problems involving non-compactness questions and putative enumerative theories.