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Jason D. Lotay


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


CALIBRATED SUBMANIFOLDS

GEOMETRY AND GROUPS

RIEMANNIAN GEOMETRY

PROJECTS


Teaching

At UCL I taught a fourth year course in Riemannian Geometry. I also taught an ancillary course (Linear Algebra and Calculus), gave an LTCC lecture course for graduate students on Calibrated Geometry and Geometric Flows and taught a second year course in Geometry and Groups. I also offered a lecture course on Calibrated Submanifolds at the LMS-CMI Research School: An Invitation to Geometry and Topology via G2 at Imperial on 7-11 July 2014.



Calibrated Geometry

I gave a graduate course at the LTCC on Calibrated Geometry and Geometric Flows, and my lectures notes are available for download.

In my LMS-CMI graduate course, I gave an introduction to the theory of calibrated submanifolds, with a focus on examples, and described the main techniques involved in their study.

Lecture notes
  • Lecture 1: Minimal submanifolds; introduction to calibrations
  • Minimal submanifolds: definition and examples; first variation of volume; mean curvature vector; minimal graph equation
    Calibrations and calibrated submanifolds; calibrated submanifolds are volume-minimizing; calibrations and holonomy

  • Lecture 2: Complex and special Lagrangian submanifolds
  • Calibrations and complex submanifolds in Kähler manifolds; Wirtinger's inequality
    Special Lagrangian calibration; examples of special Lagrangian submanifolds in Euclidean space and in Calabi--Yau manifolds

  • Lecture 3: Constructing calibrated submanifolds; nonlinear elliptic PDE
  • Methods for constructing calibrated submanifolds
    Nonlinear elliptic PDE: regularity and methods of solution; gluing problems
    Deformations of special Lagrangian submanifolds; special Lagrangian isometric embeddings

  • Lecture 4: Gluing problems; associative and coassociative submanifolds
  • Gluing methods; resolving special Lagrangian self-intersections
    Associative and coassociative calibrations; relationship with complex and special Lagrangian geometry; examples of associative and coassociative submanifolds
    Dirac operator and deformations of associative submanifolds
    Deformations of coassociative submanifolds; coassociative isometric embeddings

  • Lecture 5: Cayley submanifolds; angle theorem; gauge theory
  • Cayley calibration: relationship with other calibrated geometries; examples; deformations of Cayley submanifolds
    The angle theorem; Lawlor necks and Nance calibrations
    Yang--Mills connections; G2 instantons; gauge theory and calibrated geometry (time permitting)
Problem sheets   Problem sheet 1   Problem sheet 2   Problem sheet 3   Problem sheet 4



Geometry and Groups

Geometry attempts to describe and understand the space around us. It is a central activity and main driving force in many branches of mathematics and physics.
In this course we will meet some of the basic examples in geometry, build up fundamental understanding of curvature, and enhance familiarity with groups and group actions outside of pure algebra.
  • 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.


Riemannian Geometry

Differential and Riemannian Geometry provide important tools in modern mathematics, impacting on diverse areas from the pure to the applied.
The first aim of this course is to give a thorough introduction to the theory of manifolds, which are the fundamental objects in Differential Geometry.
The second aim is to describe the basics of Riemannian Geometry, in particular the notion of geodesics and curvature.
Our final objective will be to analyse manifolds with constant curvature, with a focus on the sphere and hyperbolic space.
  • 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.




Projects

I offered fourth year projects at UCL on a number of topics including some listed below. I am also happy to supervise summer undergraduate research projects.
  • 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
Please feel free to contact me if you are interested in pursuing a project with me.

Current students:

Past students: Enoch Yiu (3rd, Oxford), Laura Wakelin (3rd, with Isidoros Strouthos), Thomas Foster (4th, physics), Laura Wakelin (2nd, with Jonny Evans), Chris Evans (2nd), Jafrin Islam (4th), Rhiannon Graves (4th).