

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

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 LMSCMI Research School: An Invitation to Geometry and Topology via G_{2} at Imperial on 711 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 2327 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 3:305 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 G_{2}Manifolds and Related Topics. The chapters in the book are also available on arXiv.
Description Calibrated geometry provides a key tool to study volumeminimizing 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 volumeminimizing; calibrations and holonomy.
 Complex and special Lagrangian submanifolds; the angle theorem
Wirtinger's inequality; complex submanifolds in Kähler manifolds are calibrated.
Special Lagrangian calibration; CalabiYau manifolds; examples. The angle theorem: Lawlor necks and Nance calibrations.
 Calibrated submanifolds and exceptional holonomy
Associative, coassociative and Cayley calibrations; G_{2} 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
YangMills functional and connections. Discussion of gauge theory in low dimensions; instantons and monopoles. HermitianYangMills and stability.
 Gauge theory and exceptional holonomy
Instantons and monopoles on manifolds with special holonomy; instantons minimize YangMills 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. DonaldsonThomas/DonaldsonSegal 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; graduatelevel 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 LMSCMI 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 orbitstabiliser 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, 3transitivity.
 Hyperbolic geometry
Hyperboloid, disc and upperhalf 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; pushforward, Lie bracket, integral curves, flow. Lie derivative of vector fields.
Differential forms; pullback and exterior derivative. Lie derivative of forms and Cartan's formula.
 Orientation and Riemannian metrics
Partitions of unity. Orientability, orientation, volume form, orientationpreserving maps. Existence of Riemannian metrics.
 Riemannian manifolds
Definitions and examples. Isometry and local isometry. Quotient by group action.
Fundamental Theorem of Riemannian Geometry: LeviCivita connection. Christoffel symbols. Covariant derivative and parallel transport.
 Geodesics
Exponential map, Gauss Lemma and geodesics are locally length minimizing. Completeness and HopfRinow Theorem.
 Curvature
Riemann curvature operator and tensor. Sectional, Ricci and scalar curvature.
Spaces with constant curvature; geodesics, isometries and classification.
Theorems of CartanHadamard, BonnetMyers and SyngeWeinstein.
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 Ricciflat 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 CalabiYau 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 
 AleksandraSasa Bozovic (4th, Oxford, 2022)  MorseNovikov theory 
 Gabriel DiazAylwin (4th, Oxford, 2022)  Towards a mathematical formulation of supersymmetric YangMills 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 3manifolds 
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 eightdimensional 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 
2020  Marek Kurczynski* (3rd, Warwick, 2020)  Theorem of three geodesics on S^{2} using curveshortening 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 3manifolds 
2016  Brunella Torricelli (4th, ETH, with Jonny Evans, 2016)  Lagrangian Floer theory 
2015  Chris 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 
