Welcome to my webpages !

I am a Lecturer in the School of Mathematics and Statistics at the University of Melbourne.

My research interests are in partial differential equations, geometry and analysis. I am a member of the Analysis group.

In 2012, I obtained my Ph.D. from the University of Cambridge as a student of Mihalis Dafermos. Prior to joining the University of Melbourne in 2018, I held postdoctoral fellowships at the University of Toronto, MSRI, Berkeley, and most recently Sorbonne Université, Paris. Please refer to my curriculum vitae for a detailed academic history.

• AMSI Summer School

  I'm very happy to announce that I will be teaching a course on general relativity at this year's AMSI Summer School in the Mathematical Sciences: General Relativity The summer school will be hosted by ANU, in Canberra, and run for 4 weeks in January 24. Applications close at 11.59pm AEDT, Monday 27 November.

  27 November, 2023
• MATRIX 2023 (Hyperbolic PDEs and Nonlinear Evolution Problems)

  This September I organised another MATRIX program, together with Jesse Gell-Redman, Todd Oliynyk, Andrew Hassell, and Zoe Wyatt. It was held at the Creswick Campus, a couple of hours outside of Melbourne, in rural Victoria, formerly the School of Forestry of the University of Melbourne. I've heard it described "like Oberwolfach in the 70s". The theme of the workshop was Hyperbolic PDEs …

  19 November, 2023
• Highlights in mathematical physics

  I'm giving an informal talk in the series "Highlights in mathematical physics" organised by the mathematical physics group at Melbourne. The talk will be this Friday 2-3pm, room 211 in David Caro Building / Physics: Volker Schlue (05.08.2022): "Mathematical theory of black holes" The other talks this semester are: Anas Rahman (19.08.2022): "Some Applications of Random Matrix Theory: From Quantum Transport to 2D Gravity" Tony …

  3 August, 2022
• Enno Heeren Dirksen

  This summer I spent a few weeks in Paris, and mentioned to a friend on a walk to Père Lachaise that I have a distant relative who may be buried there, and I was quick to dismiss it as a family legend, because a first attempt to find his grave a few years ago had led to nothing. My friend, …

  24 July, 2022
• New position in Melbourne: Lecturer in Analysis

  The School of Mathematics at the University of Melbourne has opened another position in Analysis, and is encouraging applications from any area of partial differential equations, and geometric analysis. A more detailed position description, and information about the application process can be found here: https://careers.pageuppeople.com/422/ci/en/job/900382/lecturer-in-analysis https://www.mathjobs.org/jobs?joblist-534-14133 I have recently joined the School myself, and would be happy to answer any informal inquiries, about …

  9 September, 2019

Recent and upcoming talks

• PDE Colloquium, Universität Münster, April 16, 2024.
• Seminar, Mathematical General Relativity, LJLL, Sorbonne Université, Paris, March 28, 2024.

Past Talks

1. Analysis and PDE seminar, UC Berkeley, US, July 24, 2023.
2. BIRS workshop on Inverse Problems and Nonlinearity, Banff International Research Station, Canada, July 16-21, 2023.

   

3. Conference on Nonlinear Waves and Mathematical Relativity, Tsinghua University, July 10-14, 2023.
4. Pure Mathematics Seminar, Melbourne University, Peter Hall, May 12, 2023.
5. 2023 Australia-NZ Student Conference on Relativity / Cosmology / Astrophysics, May 8-10, 2023.
6. Differential Geometry Seminar and Colloquium, University of Adelaide, March 3, 2022.
7. MUMS Lecture, Melbourne Uni Maths & Stats Society, online talk, October 17, 2020
8. Pacific Rim Conference in Mathematics, Session: PDE: Inviscid Fluid Mechanics and General Relativity, online hosted by UC Berkeley, August 10-11, 2020.



9. ACGRG10, 10th Australasian Conference on General Relativity and Gravitation, Australasian Society for General Relativity and Gravitation, Victoria University of Wellington, New Zealand, December 9-13, 2019
10. AustMS 2019 (Talks in Sessions: Geometric Analysis and PDE; Harmonic and Semi-Classical Analysis), 63rd Annual Meeting of the Australian Mathematical Society, Monash University, Australia, December 3-6, 2019
11. Session D41, DEA 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019



12. Seminar, Institut Mittag-Leffler, Sweden, September 12, 2012
13. Mathematical Physics Seminar, University of Melbourne, Australia, May 14, 2019
14. Pure Mathematics Seminar, University of Melbourne, Australia, May 3, 2019
15. Analysis, PDE, and Geometry Seminar, Monash University, Australia, April 30, 2019
16. Probing the Earth and the Universe with Microlocal Analysis, BIRS, Banff, Canada, April 14 – 19, 2019.
17. Topology and Geometry Seminar, University of Sydney, Australia, March 11, 2019
18. Mathematical General Relativity Seminar, Sorbonne Université, Paris, October 29, 2018.
19. Analysis Seminar, Imperial College, London, October 26, 2018.
20. PDE Seminar, CAMGSD, Técnico, Lisbon, Portugal, September 19, 2018.
21. Conference on Mathematical General Relativity, Institut Henri Poincaré, Paris, May 28-June 1, 2018.
22. Seminar/Forum, University of Melbourne, December 7, 2017.
23. CMC Seminar, Korea Institute for Advanced Study, Seoul, December 4, 2017.
24. Mathematical General Relativity Seminar, Sorbonne Université, Paris, November 20, 2017.
25. General Relativity and Geometric Analysis Reading Seminar, Columbia University, New York, April 21, 2017.
26. Institut Fourier, Grenoble, March 2, 2017.
27. Analysis and PDE Seminar, Stanford University, January 9, 2017.
28. Geometry and Analysis Seminar, Queen Mary University of London, December 6, 2016.
29. Mathematical General Relativity Seminar, Sorbonne Université, Paris, November 21, 2016.
30. Séminaire Laurent Schwartz, Institut des Hautes Études Scientifiques, Paris, October 18, 2016.
31. PDE Seminar, Center for Mathematical Analysis, Geometry, and Dynamical Systems, Técnico, Lisbon, Portugal, June 15, 2016.
32. Analysis and PDE seminar, Imperial College, London, May 13, 2016.
33. Oberseminar Geometrische Analysis, Universität Tübingen, February 12, 2016.
34. Conference in Mathematical General Relativity, Tsinghua Sanya International Mathematics Forum in Sanya, Hainan, China, January 5-9, 2016.



35. Geometric Analysis and Partial Differential Equations Seminar, University of Cambridge, November 2, 2015.
36. Dynamics of Self-Gravitating Matter, Workshop, Institute Henri Poincaré, Paris, October 26-29, 2015.
37. Vienna Relativity Seminar, University of Vienna, (and additional talk in the Geometric Analysis and Physics joint seminar of the University of Vienna and the University of Technology), Vienna, October 22, 2015.
38. Mathematical Aspects of General Relativity, Workshop, Oberwolfach, July 12-18, 2015.
39. Session A1: Mathematical General Relativity, GRG conference: A Centennial Perspective, Penn State, June 8, 2015.
40. International Conference on Black Holes, Fields Institute, Toronto, June 2, 2015.
41. Colloquium, University of Miami, February 2, 2015.
42. Mathematical Problems in General Relativity, Simons Center for Geometry and Physics, January 21, 2015.
43. Analysis and PDEs Seminar, Johns Hopkins University, March 10, 2014.
44. Analysis Seminar, Princeton University, March 3, 2014.
45. Nonlinear Wave Equations and General Relativity Workshop, University of Oxford, January 13-14, 2014.
46. Geometry Seminar, Stanford University, November 11, 2013.
47. MSRI Postdoc Seminar, Evans Hall, University of California, Berkeley, October 25, 2013.
48. MGR Programmatic Seminar, MSRI, Berkeley, October 1, 2013.



49. Miami Waves, FRG workshop, University of Miami, December 18-21, 2012.
50. Analysis Seminar, Princeton University, November 19, 2012.
51. Analysis and Applied Mathematics Seminar, University of Toronto, October 5, 2012.



52. Mathematical Aspects of General Relativity, MFO, Oberwolfach, July 29 – August 4, 2012.
53. Recent Developments in Gravity, NEB 15 Conference, TEI Crete, Chania, June 20-23, 2012.
54. King’s Lunchtime Seminar, King’s College, Cambridge, May 4, 2012.
55. Do we understand gravity?, IOP Gravitational Physics Group, IOP, London, September 16, 2011.
56. Geometric Analysis Seminar, AEI Institute, Potsdam-Golm, April 11, 2011.

Online Talks

• Black holes and the expanding universe (general audience talk aimed at undergraduates), organised by Melbourne Uni Maths & Stats Society and LunchMaths at Monash University, October 17, 2020
• Expanding black hole cosmologies: On the stability problem for Kerr de Sitter space-times, talk in PDE session at the Pacific Rim conference, UC Berkeley, August 2020
• Scattering from infinity for semi-linear wave equations with weak null condition, talk in workshop at Banff International Research Station, April 2019
• Non-existence of time-periodic vacuum spacetimes, talk in workshop at Simons Center for Geometry and Physics, January 2015

Preprints

• (with Hans Lindblad) Scattering for wave equations with sources close to the lightcone and prescribed radiation fields, 50 pages (2023).

  arXiv:2303.10569v4
  Banff International Research Station
  Institute for Advanced Study (Talk by Hans Lindblad)

Publications

1. (with Hans Lindblad) Scattering from infinity for semi linear wave equations satisfying the null condition or the weak null condition, Journal of Hyperbolic Differential Equations 20 (1), 155-218 (2023).

   World Scientific  1711.00822
   Banff International Research Station

2. Decay of the Weyl curvature in expanding black hole cosmologies, Annals of PDE (2022).

   Springer   SharedIt Enhanced PDF   1610.04172
   YouTube, PRCM, UC Berkeley

3. Optical functions in de Sitter, Journal of Mathematical Physics, 62 (8), 082501 (2021)

   AIP   1910.05799

4. (with Grigorios Fournodavlos) On `hard stars’ in general relativity, Annales Henri Poincaré, 20(7), 2135-2172 (2019)

   Springer   1710.00754

5. (with Spyros Alexakis) Non-existence of time-periodic vacuum space-times, J. Differential Geom. Volume 108, Number 1 (2018), 1-62.

   Project Euclid   1504.04592
   Simons Center for Geometry and Physics
   Oberwolfach Report, 3 pages.

6. (with with Spyros Alexakis and Arick Shao) Unique continuation from infinity for linear waves, Advances in Mathematics 286 (2016) 481-544.

   Elsevier   1312.1989
   Banff International Research Station (Talk by Spyros Alexakis)

7. Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies, Communications in Mathematical Physics: Volume 334, Issue 2 (2015), 977–1023.

   Springer   1207.6055v2

8. Decay of linear waves on higher dimensional Schwarzschild black holes, Analysis & PDE 6-3 (2013), 515–600.

   MSP   PDF

Theses

• Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes, Ph.D. Thesis, University of Cambridge (2012), 223 pages.

  DSpace

• The asymptotics of the gravitational field and the memory effect, Dipl.phys. Thesis, ETH Zurich (2008), [expository, after D. Christodoulou, Nonlinear Nature of Gravitation and Gravitational-Wave Experiments, Phys. Rev. Letters (67), no 12, pp. 1486–1489, (1991)], 48 pages.

  PDF

Teaching history

2023 Semester 2

Partial Differential Equations

MAST90133

• Lecture notes

2023 Semester 2

Calculus 2: Advanced

MAST10021

• Lecture notes

2022 Semester 2

Differential Geometry

MAST90143

• Lecture notes

2022 Semester 2

Calculus 2: Advanced

MAST10021

2021 Semester 2

Partial Differential Equations

MAST90133

2021 Semester 2

Calculus 2: Advanced

MAST10021

2020 Semester 2

Calculus 2: Advanced

MAST10021

2019 Semester 1

Measure Theory

MAST90012

• Syllabus (2019)
• Course material (2019)

Lecture Notes

General Relativity (2015)

Students

2025 2022

Louie Bernhardt

PhD

2023 2022

Daniel Traicos

MSc

• Thesis title: “Global Solutions of Semilinear Wave Equations Satisfying the Null Condition”

2022 2021

Sean Wynn

MSc

• Thesis title: “On the non-relativistic limit of crystalline solids in a fully geometric theory of continuum mechanics”

2020 2019

Caelan Turvey

MSc

• Thesis title: “The wave equation on neutron star spacetimes”

2020 2019

Madeleine Johnson

MSc

• Thesis title: “On the notion of global hyperbolicity in Lorentzian geometry”

MSc Theses

If you are a student in the Master’s program of the School of Mathematics and currently looking for a supervisor, then the following topics give you an idea what I have currently in mind:

• Topics in relativistic elasticity. Flavour: Differential Geometry. Literature: Christodoulou & Kaelin (2012)
• Asymptotics for semi-linear wave equations. Flavour: Functional Analysis, PDE. Literature: Luk (2012), Lindblad & Rodnianski (2003), Lindblad & Schlue (2020)
• Waves in de Sitter and Schwarzschild de Sitter. Flavour: General Relativity, PDE. Literature: Dafermos and Rodnianski (2008), Schlue (2012)

Undergraduate Courses

Calculus 2: Advanced (MAST10021)

This course is offered annually for first year students in Semester 2.
I have been teaching it every year since 2020 when it was first introduced.

FAQ about this course:

1. What is the difference between Calculus 2: Advanced and Calculus 2?
   I would say, while the topics covered in this course are broadly the same as in MAST10006 Calculus 2, the material is more “advanced” in the sense that you will see more proofs and concepts, and fewer examples with “ad hoc” solutions. Here we aim to strike a balance between the theoretical underpinnings of Calculus and its practical applications, which also prepares you to take other classes is mathematics & physics.
2. What is the difference between Calculus 2: Advanced and Accelerated Mathematics 2?
   In comparison to Calculus 2 the course Accelerated Mathematics 2 proceeds quicker, but not at a deeper level. The idea of the new courses in the Advanced stream is to offer solid entry courses in mathematics, where the foundations of these subjects are developed with more care, and more emphasis on the underlying proofs is given, both in the presentation of the material, and the skills acquired by the student.
3. How do I enroll?
   Please contact ms-mslc-approvals@ms.unimelb.edu.au for all questions related to enrollment.
4. Is there a book you recommend?
   I like Spivak‘s “Calculus” which gives a beautiful introduction to Calculus at the intended level for this course. In fact, large parts of this course are based on Spivak‘s text. I also recommend Folland‘s “Advanced Calculus”.

Current position

Lecturer in Pure Mathematics (Analysis & Geometry), University of Melbourne

Education

Ph.D. in Pure Mathematics, University of Cambridge, 2012
Diploma in Physics, ETH Zürich, 2008

Curriculum Vitae (PDF)

Academic history

December 2018September 2015

Postdoctoral Researcher

Sorbonne Université

• Funding: ERC consolidator Grant 725589 EPGR, ERC advanced grant 291214 BLOWDISOL, Fondation Sciences Mathématiques de Paris
• Affiliation: Laboratoire Jacques-Louis Lions
• Supervisor: Jérémie Szeftel

July 2017May 2017

Research Associate

Imperial College London

• Supervisor: Gustav Holzegel

July 2014January 2014

Visiting Postdoctoral Research Associate

Princeton University

December 2013August 2013

Postdoctoral Fellow

Mathematical Sciences Research Institute, Berkeley

• Mentor: Hans Lindblad

July 2015August 2012

Postdoctoral Fellow

University of Toronto

• Supervisor: Spyros Alexakis

July 2012October 2008

Ph.D. in Pure Mathematics

University of Cambridge

• Advisor: Mihalis Dafermos
• Thesis title: “Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes”
• Department: Department of Pure Mathematics and Mathematical Statistics
• College: King’s College, Cambridge

April 2008October 2005

Diploma in Physics

ETH Zürich

• Diploma thesis advisor: Demetrios Christodoulou
• Thesis title:“The asymptotics of the gravitational field and the memory effect”