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Jason D. Lotay


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


CALIBRATED GEOMETRY AND GAUGE THEORY

GEOMETRY AND GROUPS

RIEMANNIAN GEOMETRY

PROJECTS


Teaching

I am currently teaching a graduate course on Calibrated Geometry and Gauge Theory in Fall 2022 as Chancellor's Professor at UC Berkeley, California. I am also teaching one 4th year course at Oxford this year: Riemannian Geometry. I have offered 4th year projects and 3rd extended essays at Oxford.

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. I also gave an updated version of the Calibrated Submanifolds course as part of the Summer School: Special Holonomy -- Geometry and Physics at Freiburg on 23-27 September 2019.



Calibrated Geometry and Gauge Theory

During Fall 2022, I am giving a graduate course on Calibrated Geometry and Geometric Flows as part of my role as Chancellor's Professor at UC Berkeley, California. The class is Tuesdays and Thursdays 3:30-5 in Moffitt 106. A description and summary of the course are given below. Preliminary lecture notes will be posted here throughout the course. Recommend reading is Lectures and Surveys on G2-Manifolds and Related Topics. The chapters in the book are also available on arXiv.

Description   Calibrated geometry provides a key tool to study volume-minimizing submanifolds. Gauge theory in higher dimensions potentially gives a way to define new invariants for manifolds with special holonomy. This course provides an overview of these topics and discusses relationships between them. The course will cover fundamental results and techniques in the field and discuss open problems.
  • 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

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

Calibrated Submanifolds

In my LMS-CMI and Freiburg 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.

The original lecture notes and an updated version are available for download. I also provided some problem sheets which are available below.

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 currently offer 2nd/3rd year summer projects, 3rd year extended essays and 4th year/Masters projects at Oxford: some current and past topics are listed below. 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: some past project ideas are listed below.
  • 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 project students: Simi Hellsten* (2nd, Oxford, 2022), Injune Hwang* (2nd, Oxford, 2022), Tom Keany* (2nd, UNIQ+/Lancaster, 2022)

Past project students (* indicates summer project):
2022 Campbell Brawley* (2nd, Oxford, 2022) Lie group and Lie algebra homology and cohomology
Aleksandra-Sasa Bozovic (4th, Oxford, 2022) Morse-Novikov theory
Gabriel Diaz-Aylwin (4th, Oxford, 2022) Towards a mathematical formulation of supersymmetric Yang-Mills theory
Joshua Mann (4th, Oxford, 2022) Combinatorial knot Floer homology
Sergio Serrano (4th, Oxford, 2022) Homological approaches to graph evasiveness
Laura Bradby (3rd, Oxford, 2022) How 'big' is the moduli space of Riemann surfaces?
Samuel Flower (3rd, Oxford, 2022) Surfaces and the classification of 3-manifolds
2021 Dijia Chen* (3rd, UCL, 2021) De Rham cohomology
Yoojin Lee* (3rd, UCL, 2021) Riemannian holonomy
Maksymilian Manko* (2nd, Manchester, 2021) Calibrated geometry and gauge theory on eight-dimensional hyperkähler manifolds
Remy Bohm (4th, Oxford, 2021) Black holes and positive scalar curvature
Michael Bow (4th, Oxford, 2021) Event horizon topology in 4 & 5 dimensional spacetimes
Joseph Miller (4th, Oxford, 2021) The geometry and topology of higher dimensional black hole horizons
Samuel Neil (4th, Oxford, 2021) The geometry and topology of black holes
2020Marek Kurczynski* (3rd, Warwick, 2020) Theorem of three geodesics on S2 using curve-shortening flow
Run Tan (4th, Oxford, 2020) Hamiltonian mechanics in symplectic geometry
2018 Enoch Yiu* (3rd, Oxford, 2018) Cohomology of the moduli space of stable bundles
Yll Buzoku* (3rd, UCL, 2018) Metrics on Milnor's exotic spheres
Laura Wakelin* (3rd, UCL, with Isidoros Strouthos, 2018) Higher homotopy groups
2017 Thomas Foster (4th, UCL, physics, 2017) Symmetric spacetimes in General Relativity
Laura Wakelin* (2nd, UCL, with Jonny Evans, 2017) Hyperbolic 3-manifolds
2016 Brunella Torricelli (4th, ETH, with Jonny Evans, 2016) Lagrangian Floer theory
2015Chris Evans* (2nd, UCL, 2015) An introduction to Riemann surfaces
Jafrin Islam (4th, UCL, 2015) The classification of Riemannian holonomy groups
2014 Rhiannon Graves (4th, UCL, 2014) De Rham cohomology