R. Ogilvie,
Whitham deformations of nodal spectral curves of the sinh-Gordon equation,
Preprint (2026), 28 pp. Link

N. Hasse, R. Ogilvie and M. U. Schmidt,
A new invariant for doubly periodic solutions of the sinh-Gordon equation,
Preprint (2026), 25 pp. Link

R. Ogilvie,
Weakly conformal maps into \(\mathbb{R}^3\) and \(\mathbb{R}^4\),
2025 MATRIX Annals (2026), 19 pp. Link

R. Ogilvie and M. U. Schmidt,
Quaternionic Analysis of Conformal Maps and the Willmore Functional,
Under review (2025), x+229 pp. Link
This long work, co-authored with Prof Martin Schmidt, is the culmination of my postdoc project. Many of the 18 chapters contain ideas and results of sufficient significance and substance to warrant their own papers, but due to their tight interdependence they will be published all together.

E. Carberry and R. Ogilvie,
The space of equivariant harmonic tori in the 3-sphere,
J. Geom. Phys. 157 (2020), 103808, 22 pp. Link

E. Carberry and R. Ogilvie,
Whitham deformations and the space of harmonic tori in \(\mathbb{S}^3\),
J. Lond. Math. Soc. (2) 99 (2019), no. 3, 945-964. Link

R. Ogilvie,
Deformations of Harmonic Tori in \(\mathbb{S}^3\),
Doctoral dissertation, University of Sydney (2017). Link

Arxiv

Constrained Willmore surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^4\),
Conference “Surface Theory: Geometry, Analysis, and Experiments”, TSIMF, Sanya, China (2026). Slides

Kodaira and Weierstrass representations of (weakly) conformal maps,
Mini-course, conference “Integrable Geometry: Smooth and Discrete”, MATRIX Institute, Australia (2025). Slides

Existence of constrained Willmore surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^4\),
Invited talks, University of Sydney and University of New South Wales, Australia; Beijing Institute of Mathematical Sciences and Applications (BIMSA), China (2025).

Willmore tori and their spectral curves,
Invited talk, University of Sydney, Australia (2020).

Harmonic tori in \(\mathbb{S}^3\),
Poster, conference “Differential Geometry and its Applications”, Czech Republic (2019).

Moduli space of harmonic tori in \(\mathbb{S}^3\),
Invited talks, Leibniz University Hannover, Germany; University College Cork, Ireland; University of Bath and University of Leicester, UK; Australian Mathematical Society Conference (2017). Slides

Deformations of harmonic maps,
AMSI Winter School on Algebra, Geometry and Physics, University of Queensland, Australia (2015). Slides

Deformations of harmonic maps,
Conference “The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles”, Institute for Mathematical Sciences, Singapore (2014).

Harmonic maps and spectral curves,
Invited talks, Universities of Mannheim and Tübingen, Germany (2014).

Harmonic maps in \(\mathbb{S}^3\),
Australian Mathematical Society Conference (2013). Slides

Geometric measure theory,
AMSI Summer School (2011).

Riemannian Geometry Lecture Notes
Dynamical Systems Exercises
Intro PDE Lecture Notes and Exercises

University of Mannheim

2025/26

• MAA301 Analysis I (Exercise Lecture)
• MAA302 Analysis II (Exercise Lecture)
• MAA514 Analysis III (Lecture)

2024/25

• MAA510 Introduction to PDEs (Lecture and Tutorials)
• MAA408 Dynamical Systems and Stability (and variants)

2023/24

• MAA510 Introduction to PDEs (Lecture and Tutorials)
• MAA524 Riemannian Geometry (Lecture and Tutorials)

2022/23

• MAA514 Analysis III
• MAA510 Introduction to PDEs
• MAA504 Elliptic PDEs
• MAA501 Riemann surfaces (Reading Course)

2021/22

• MAA510 Introduction to PDEs
• MAA504 Elliptic PDEs

2020/21

• MAT301 Analysis I (Exercise Lecture)
• MAT302 Analysis II (Exercise Lecture)
• MAA514 Analysis III
• MAA510 Introduction to PDEs

2019/20

• MAA504 Elliptic PDEs
• MAA501 Riemann surfaces

2019

• University of Sydney
  • MATH1005 Statistics (Summer School)
  • MATH1011 Applications of Calculus (x3)
  • MATH1021 Calculus of One Variable (Summer School)
  • MATH1111 Introduction to Calculus (x2)
• University of Notre Dame, Australia
  • MATH5502 Statistics (Course Instructor)
• Australian Catholic University
  • MATH107 Foundations of Mathematics (x2)
  • SCIT101 Science for Technologists (Lecturer for maths component)

2018

• University of Sydney
  • MATH1004 Discrete Mathematics
  • MATH1064 Discrete Mathematics for Computing
  • MATH023 Multivariable Calculus and Modelling (x2)
  • MATH1921 Calculus of One Variable (Advanced)
  • Mathematics 2U Bridging Course
• Australian Catholic University
  • MATH107 Foundations of Mathematics (x2)
  • MATH104 Functions and Calculus (x2)

2017

• University of Sydney
  • MATH1001 Differential Calculus (Lecturer)
  • MATH1002 Linear Algebra (x2)
  • MATH1003 Integral Calculus and Modelling (x2)
  • MATH1004 Discrete Mathematics (Lecturer)
  • MATH1005 Statistics (x2)
• Australian Catholic University
  • MATH107 Foundations of Mathematics (x2)
  • MATH104 Functions and Calculus (x2)

University of Sydney

2016

• MATH1001 Differential Calculus (x3)
• MATH1002 Linear Algebra (x2)
• MATH1003 Integral Calculus and Modelling
• MATH1004 Discrete Mathematics (x3)
• MATH3968 Differential Geometry (Advanced)
• Mathematics 2U Bridging Course

2015

• PHAR1812 Basic Pharmaceutical Sciences (Lecturer for maths component)
• MATH1013 Mathematical Modelling (x3)
• MATH1014 Linear Algebra (x2)

2014

• MATH1011 Applications of Calculus (Lecturer)
• MATH1003 Integral Calculus and Modelling (x2)
• MATH3968 Differential Geometry (Advanced)

2013

• MATH1011 Applications of Calculus (Lecturer)
• MATH1903 Integral Calculus and Modelling (Advanced, x2)
• PHAR1822 Physical Pharmaceutics and Formulations A (x3)
• Mathematics 2U Bridging Course

2012

• MATH1013 Mathematical Modelling (x3)
• MATH1901 Differential Calculus (Advanced, x2)
• MATH1902 Linear Algebra (Advanced, x2)

2011

• MATH1002 Linear Algebra (x2)
• MATH1014 Linear Algebra (x2)

Unless otherwise noted, these are the tutorials I taught.

The image to the right is one component of the moduli space of harmonic tori in the 3-sphere.

The 3-sphere \(\mathbb{S}^3\) is the set of unit vectors in \(\mathbb{R}^4\), just as the ordinary sphere is the set of unit vectors in \(\mathbb{R}^3\). A map \(F\) from a torus into \(\mathbb{S}^3\) is harmonic if it is a critical point of the Dirichlet energy \(\tfrac12\int |dF|^2\). For surfaces it is closely related to minimal surfaces, which are surfaces that have the least amount of area for their given edges, i.e. the shapes soap films take.

In general, solutions of a nonlinear PDE like the harmonic map equation are hard to describe. This one, however, is an integrable system, and every harmonic torus is determined by algebraic data: a hyperelliptic curve, called the spectral curve, together with a few extra parameters. So a torus that is a solution of a PDE becomes a point in a finite-dimensional space of algebraic data. Not every choice of data closes up into a torus, though. The periodicity conditions cut out a subset, and this subset is the moduli space.

In my thesis, and in later work with Emma Carberry, I showed that for spectral curves of each genus \(g\), this moduli space is a smooth two-dimensional manifold away from a small exceptional set. For low genus, \(g = 0, 1\), we were able to parametrize it explicitly. To make the picture, I computed many points in one component and joined some of them up to show the shape. Its edges are the interesting part: there the spectral curve degenerates, and the family can pass through into tori of a different genus. What features can you notice?