The analysis group is broadly interested in the areas partial differential equations, differential geometry and geometric analysis.
Follow the tabs above to find more information about the members of the analysis group, and the PDE Seminar we are running.
Students who are interested in taking courses in analysis, geometry, and PDE, are encouraged to look at the information page for MSc students, and should contact us directly for MSc thesis topics.
Senior Lecturer in Pure Mathematics
Office: Peter Hall 202
Research interests: Microlocal analysis, differential geometry, index theory, spectral asymptotics, singular spaces, scattering theory.
Lecturer in Pure Mathematics
Office: Peter Hall G28
Research interests: Minimal surfaces, isoperimetry, elliptic and parabolic differential equations, geometric measure theory, geometric analysis
Lecturer in Pure Mathematics
Office: Peter Hall 204
Research interests: General relativity, global evolution problems for hyperbolic partial differential equations, geometric analysis.
We are very excited that the MATRIX workshop on Hyperbolic Differential Equations in Geometry and Physics begins today!3 April, 2022
Informal lectures on general relativity
We are offering a few informal lectures on general relativity, Thursdays, Evan Williams Theatre (Peter Hall G03), 4:15-5:15pm, starting this Thursday, March 3. This will not be a formal lecture course at all, instead I am hoping to give a loose introduction to a few research topics in mathematical general relativity, which are accessible to Master's students with interests in analysis, differential geometry, …1 March, 2022
We're launching an Analysis Seminar at the University of Melbourne this year. We are hoping for many talks to happen in person, maybe even en plein air, like the seminar given by our first speaker, Zoe Wyatt. Others will be online, like by our second speaker this Friday, Allen Fang. Either follow these pages, or contact us directly, to be sure …28 February, 2022
Jan Sbierski (University of Edinburgh)
Wednesday, March 30, Peter Hall 213 2:15PM
Title: On holonomy singularities and inextendibility results for Lorentzian manifolds
Given a solution of the Einstein equations a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. This talk will give a basic introduction to the problem of inextendibility of Lorentzian manifolds, beginning with classical methods exploiting a blow-up of curvature to show the inextendibility with a twice continuously differentiable Lorentzian metric and concluding with the presentation of a recent methodology exploiting a blow-up in holonomy to show inextendibility with a locally Lipschitz regular Lorentzian metric.
Brian Krummel (Melbourne University)
Wednesday, March 23, Peter Hall 213 2:15PM
Title: Fine structure of the free boundary for a penalized thin obstacle problem
We consider a two-penalty elliptic boundary obstacle problem, which is motivated by applications to fluid dynamics and thermics. Using monotonicity formulas of Almgren, Weiss, and Monneau, we establish rectifiability of the free boundary and uniqueness of blow-ups at free boundary points. We briefly discuss analogous parabolic problem, which represents a physical system evolving in time. Joint work with Donatella Danielli.
Allen Fang (Sorbonne University)
Friday, March 4, on zoom, 9AM
Title: A new proof for the nonlinear stability of slowly-rotating
Abstract: The stability of black hole spacetimes is a critical question in mathematical relativity. The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter
spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof utilizes spectral methods to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.
Zoe Wyatt (University of Cambridge)
Thursday, February 10, Evan Williams Theatre, 2PM
Title: Stabilising relativistic fluids on slowly expanding cosmological spacetimes
Abstract: On a background Minkowski spacetime, the relativistic Euler equations are known, for a relatively general equation of state, to admit unstable homogeneous solutions with finite-time shock formation. By contrast, such shock formation can be suppressed on background cosmological spacetimes whose spatial slices expand at an accelerated rate. The critical case of linear, i.e. zero-accelerated, spatial expansion, is not as well understood. In this talk, I will present recent work concerning the relativistic Euler and the Einstein-Dust equations for geometries expanding at a linear rate. This is based on joint works with David Fajman, Todd Oliynyk and Max Ofner.
A MSc thesis is a project over 3 semesters typically on a topic close to the current research interests of your supervisor. Talk to us directly to hear what they are!
A list of topics is also maintained on MS Prime
Students who are interested in writing an MSc thesis in the analysis group are encouraged to take the following courses:
- Measure Theory
Measure Theory (MAST90012) is a core course offered every two years in Semester 1.
- Functional Analysis
Functional Analysis (MAST90020) is offered every two years in Semester 1.
- Partial Differential Equations
Partial Differential Equations (MAST90133) is offered every two years in Semester 2.
- Differential Geometry
Differential Geometry (MAST90143) is offered every two years in Semester 2.