Geometry and Topology of Manifolds with Exceptional Holonomy

An important theme in differential geometry is the study of Riemannian manifolds with special holonomy. The
possible "prime" holonomy groups were classified by Berger, and include two exceptional cases: 7-manifolds with
holonomy G_2 and 8-manifolds with holonomy Spin(7). That these are realised at all was first proved by Bryant in
1985, and the existence of closed manifolds with exceptional holonomy was proved by Joyce in 1995. Since then
there has been increasing progress on understanding exceptional holonomy manifolds.
In the context of closed manifolds with exceptional holonomy, the G_2 side has in recent years seen more progress
in understanding; for example, the diffeomorphism types, and the calibrated geometry and gauge theory of examples.
The project will study similar problems for Spin(7)-manifolds, in particular from Joyce's weighted projective space
construction.
Initially, during the first 12 months, the focus will be on topological questions: for example, computing invariants,
investigating applicable classification results and the relation to rational homotopy theory. This will build familiarity
with the area and with the particular constructions, while still potentially leading to some publishable results in the
short term. A primary source for this work is Joyce's 'Compact Manifolds with Special Holonomy', while the
topological background material will include Hatcher's 'Algebraic Topology', and Milnor and Stasheff's 'Characteristic
Classes'.
More analytical problems concerning calibrated geometry (e.g. searching for fibrations by calibrated submanifolds)
and gauge theory will be considered in the longer term. In order to progress to these problems, the necessary
background material, such as understanding of Sobolev spaces and other functional analysis tools, as well as objects
including K3 surfaces and Cayley submanifolds, will be developed through TCC courses and independent reading.