R. Ogilvie and M. U. Schmidt,
Quaternionic Analysis of Conformal Maps and the Willmore Functional,
Book manuscript (2025), x+229 pp. Link

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

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

University of Mannheim

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 harmonically immersed tori in the 3-sphere.

Let me explain. A circle (the edge of a disk) is a 1-sphere, a sphere (the surface of a ball) is a 2-sphere; likewise the 3-sphere is the boundary of a 4-dimensional ball. Just as we can make a 2-sphere by gluing together two hemispheres along the equator, we can model a 3-sphere as two balls glued together on their outsides. "Torus" is a fancy word for donut-shapped. To understand harmonically immersed, imagine a piece of rubber stretched over a frame. If we poke or pull it, when we let go, it will spring back to least stretched position. The same happens with soap films stretched on wire loops. We can also say that these shapes are minimizing the total tension. The mathematical term for this is harmonically immersed.

So to rephrase: In my PhD thesis, I investigated relaxed donuts inside a nightmare craft project. Why would anybody be interested in that? Well, mathematically it's very interesting because all such surfaces can be described in a very nice way. Each surface can be described by a handful of numbers. But not every set of numbers is valid. To visualize this, we interpret the set of numbers as coordinates. To make the image on the right, I put a dot if the numbers described a harmonic torus. Obviously I can't draw infinitely many dots, so this is an approximation. I also connected some of the dots together to make the overall shape easier to see. What features can you notice?