Jason D. Lotay
			Professor of Pure Mathematics
			University of Oxford
			Fellow at Balliol College

• Introduction to calibrations
Minimal submanifolds: definition and examples; first variation of volume; minimal graphs. Calibrations and calibrated submanifolds; calibrated submanifolds are volume-minimizing; calibrations and holonomy.


• Complex and special Lagrangian submanifolds; the angle theorem
Wirtinger's inequality; complex submanifolds in Kähler manifolds are calibrated. Special Lagrangian calibration; Calabi-Yau manifolds; examples. The angle theorem: Lawlor necks and Nance calibrations.


• Calibrated submanifolds and exceptional holonomy
Associative, coassociative and Cayley calibrations; G₂ and Spin(7) manifolds; examples; relations to complex and special Lagrangian geometry.


• Constructing calibrated submanifolds and moduli problems
Construction methods via reductions to ODEs. Deformation theory of calibrated submanifolds; links to elliptic PDE and spin geometry. Gluing methods for nonlinear PDE to construct compact examples.


• Introduction to gauge theory in higher dimensions
Yang-Mills functional and connections. Discussion of gauge theory in low dimensions; instantons and monopoles. Hermitian-Yang-Mills and stability.


• Gauge theory and exceptional holonomy
Instantons and monopoles on manifolds with special holonomy; instantons minimize Yang-Mills functional.


• Constructing solutions to gauge theoretic equations and moduli problems
Construction methods via reductions to ODEs. Deformation theory of instantons; links to elliptic PDE and spin geometry.


• Links between calibrated geometry and gauge theory
Calibrated submanifolds and limits of instantons and monopoles; Fueter sections. Examples of instantons on compact manifolds via gluing. Donaldson-Thomas/Donaldson-Segal program; conjectured links to enumerative invariants and Floer theory. Mirror symmetry and new gauge theories from calibrated geometry (time permitting).


• Open problems
Discussion of key problems in the field; graduate-level problems.

Calibrated Geometry and Geometric Flows

Calibrated Submanifolds

• Platonic solids
Symmetry groups: using the orbit-stabiliser theorem to count symmetries and identifying symmetry groups by their actions. Classification.


• Isometries of Euclidean space
Galilean group and orthogonal group. Every rotation is a composition of reflections.
Rotations in 3D: every rotation has an axis, quaternionic picture of rotations. Rotations in 4D: quaternion action.


• Spherical geometry
Geodesics, spherical triangles, spherical trigonometry. Area controls angle surplus.


• Möbius transformations
The Riemann sphere. Stereographic projection. Conformality and preservation of straight lines and circles, 3-transitivity.


• Hyperbolic geometry
Hyperboloid, disc and upper-half plane models. Geodesics, distances and hyperbolic triangles. Failure of parallel postulate. Area controls angle deficit.
Isometries of hyperbolic space as PSL(2,R). Parabolic, elliptic, hyperbolic elements and their fixed points.

• Manifolds
Definitions and examples. Smooth map, diffeomorphism and local diffeomorphism. Quotient by group action.
Tangent vectors and tangent space. Differential of smooth map. The tangent bundle. Vector bundles; sections and trivial bundles.


• Vector fields and differential forms
Vector fields; push-forward, Lie bracket, integral curves, flow. Lie derivative of vector fields.
Differential forms; pull-back and exterior derivative. Lie derivative of forms and Cartan's formula.


• Orientation and Riemannian metrics
Partitions of unity. Orientability, orientation, volume form, orientation-preserving maps. Existence of Riemannian metrics.


• Riemannian manifolds
Definitions and examples. Isometry and local isometry. Quotient by group action.
Fundamental Theorem of Riemannian Geometry: Levi-Civita connection. Christoffel symbols. Covariant derivative and parallel transport.


• Geodesics
Exponential map, Gauss Lemma and geodesics are locally length minimizing. Completeness and Hopf-Rinow Theorem.


• Curvature
Riemann curvature operator and tensor. Sectional, Ricci and scalar curvature.
Spaces with constant curvature; geodesics, isometries and classification.
Theorems of Cartan--Hadamard, Bonnet--Myers and Synge--Weinstein.

• De Rham cohomology
  One of the greatest challenges in geometry is: how do we know when two spaces are different? An important way to distinguish spaces is using invariants. Given any manifold, one can define a collection of vector spaces using the differential forms on the manifold called the de Rham cohomology. De Rham cohomology is an invariant of the manifold which is in fact dual to singular homology, and classes in de Rham cohomology have canonical representatives which have "least energy" known as harmonic forms (in the case of functions they are just the solutions to Laplace's equation). De Rham cohomology is a fundamental tool in differential topology which has many applications throughout geometry and topology.

Prerequisites: Multivariable analysis


• Holonomy
  In Riemannian geometry, so on curved spaces, parallel transport gives a map between the tangent spaces at the start and end point of a curve. In flat space parallel transport is just translation, but in other Riemannian manifolds it can be far more interesting. If your curve happens to be a loop, parallel transport around the loop gives you an isometry of the initial tangent space, and by taking different loops based at the same point you can form a group using the parallel transport maps. This group is called the holonomy group and is an invariant of the Riemannian manifold. For flat space the holonomy group is trivial but for the sphere it is the special orthogonal group. The classification of holonomy groups is very surprising, with connections to the quaternions and octonions as well as Ricci-flat and Einstein metrics, and inspires hot topics in current research.

Prerequisites: Basic differential geometry


• Calibrated geometry
  Minimal surfaces have formed a fundamental part of mathematics for more than 250 years, with important contributions from key figures in mathematics such as Euler, Lagrange, Gauss and Weierstrass, and continue to play a major role in current reseach. The minimal surface equation is a second order partial differential equation, so is very difficult to solve and analyse in general. In 1982, Harvey and Lawson introduced the notion of calibrated submanifolds, which are minimal but are defined by a first order equation. Calibrated geometry includes the classical subject of complex geometry in Kaehler manifolds, but also relates to current research in Calabi--Yau manifolds and manifolds with exceptional holonomy, Lagrangian mean curvature flow, gauge theory, and theoretical physics.

Prerequisites: Multivariable analysis and basic differential geometry