PhD Completion Seminar: Louie Bernhardt

The PhD Completion Seminar of Louie Bernhardt will take place

When: Tuesday, February 10, 2026, 10AM
Where: Peter Hall 162 – Alison Harcourt Seminar Room

followed by tea/coffee in the Staff Tea Room. All welcome!

Title: Stability and scattering problems for expanding cosmologies in general relativity

Abstract: The phenomenon of spacetime expansion is of great importance in general relativity and cosmology. Indeed, our own universe is expanding, and current scientific understanding says the rate of this expansion has changed over time. In this thesis, we carry out a rigorous mathematical study of expanding solutions to the Einstein equations. This study is divided into two separate, but related parts.

In the first part, we investigate the stability of cosmological models in the critical regime of slow expansion. The Friedmann-Lemaitre-Robertson-Walker (FLRW) family of spacetimes, which are spatially homogeneous and isotropic, serve as fundamental models for the universe. Depending on the matter model one prescribes, members of the FLRW family of spacetimes can exhibit many different rates of expansion. We prove that slowly expanding FLRW spacetimes, which are solutions to the Einstein equations coupled to a particular nonlinear scalar field, are nonlinearly stable under small perturbations. We do this for several different matter models, including a relativistic perfect fluid satisfying the Euler equations. In particular, these are the first stability results for the Einstein equations in the regime of decelerated expansion.

In the second part, we study wave equations on expanding black hole spacetimes. Examples of such spacetimes include the Schwarzschild-de Sitter and Kerr-de Sitter families of spacetimes, which respectively model a spherically symmetric black hole in an expanding universe, and a rotating black hole in an expanding universe. We establish a scattering theory for the linear wave equation on Schwarzschild-de Sitter and Kerr-de Sitter spacetimes, with data prescribed at “infinite time”. Then we demonstrate that this scattering theory also holds for the Einstein equations themselves, and construct scattering solutions to the Einstein equations that behave asymptotically like Kerr-de Sitter, from data prescribed at infinity. These scattering solutions possess long-time asymptotics that are consistent with perturbations of Kerr-de Sitter arising from Cauchy data.

Lecturer in geometry and topology

There are currently two positions advertised in the School:

Lecturer in Geometry/Topology

Lecturer in Pure Mathematics

As the name suggests, the first is a search in geometry and topology, broadly construed, and the second is open to all areas in pure mathematics.

We are looking forward to applications which close on December 18 this year.

Please do no hesitate to reach out to us if you have any questions.

Analysis seminar resumes

The Analysis seminar resumes in November.

Serena Dipierro and Enrico Valdinoci are visiting us from the University of Western Australia.

There will be talks on both Thursday, 3 November, and Friday, 4 November. Please check the page Analysis Seminar for the times TBA.

Serena will talk about The strange behaviour of nonlocal minimal surfaces:

Abstract: Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors than the ones minimizing the classical perimeter. Among these peculiar features, an interesting property, which is also in contrast with the pattern produced by the solutions of linear equations, is given by the capacity, and the strong tendency, of adhering at the boundary.

Enrico will talk about Long-range phase coexistence models

Abstract: We will discuss classical and recent results concerning the Allen-Cahn equation and its long-range counterpart, especially in relation to its limit interfaces, which are (possibly nonlocal) minimal surfaces, and to the corresponding rigidity and symmetry properties of flat solutions.

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, or partial differential equations.

Everybody welcome.

-Volker

Analysis Seminar

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 to receive all announcements.