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The analysis group is broadly interested in the areas partial differential equations, differential geometry and geometric analysis. It is part of the pure mathematics research group.
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.
Jesse Gell-Redman
Senior Lecturer in Pure Mathematics
Office: Peter Hall 202
Research interests: Microlocal analysis, differential geometry, index theory, spectral asymptotics, singular spaces, scattering theory.
Brian Krummel
Lecturer in Pure Mathematics
Office: Peter Hall G28
Research interests: Minimal surfaces, isoperimetry, elliptic and parabolic differential equations, geometric measure theory, geometric analysis
Volker Schlue
Lecturer in Pure Mathematics
Office: Peter Hall 204
Research interests: General relativity, global evolution problems for hyperbolic partial differential equations, geometric analysis.
Postdoctoral researchers
Moritz Doll
Research Fellow
Office: Peter Hall G61
Research interests: Microlocal analysis, spectral and scattering theory.
Graduate researchers
Current PhD students
- Alexius Savvinos
- Jayson Liu
- Louie Bernhardt
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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 …
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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 …
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MATRIX Workshop
We are very excited that the MATRIX workshop on Hyperbolic Differential Equations in Geometry and Physics begins today!
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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, …
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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 …
Richard Melrose (MIT)
Tuesday, October 29, 2024, Peter Hall 162 3:15 PM
Title: Monopoles, moduli spaces and compactification
Abstract: The moduli space of magnetic monopoles of charge k on R^3, for the gauge group SU(2), is a smooth, non-compact, hyperkaehler manifold of dimension 4k. To approach the Hodge theory (Sen conjecture) and other analytic properties of this space the asymptotic behaviour of the metric is described in terms of a compactification, introducing idealized monopole data. I will discuss how this in turn is closely related to asymptotic translations and quantization of configuration spaces. Based in part on joint work with Chris Kottke and Michael Singer.
Louie Bernhardt (Melbourne University)
Tuesday, September 10, 2024, Peter Hall 107 11:00 AM
Title: Linear waves on the expanding region of Schwarzschild de Sitter spacetimes: forward asymptotics and scattering from infinity
Abstract: In this talk I will discuss several new results relating to the linear wave equation on Schwarzschild-de Sitter spacetime. We establish a finite-order asymptotic expansion for solutions in the expanding region. This is accomplished by introducing new higher-order weighted energy estimates that capture the higher-order asymptotics of solutions. We also prove the existence and uniqueness of scattering solutions from data at infinity which possess asymptotics consistent with the forward problem. I will explain how this is achieved via the construction of approximate solutions that capture the desired asymptotics, as well as a new weighted energy estimate suitable for the backward problem. Time permitting, I will discuss how these results generalise to a class of expanding spacetimes which do not necessarily converge back to Schwarzschild-de Sitter at infinity.
Yaozhong Qiu (Imperial College)
Tuesday, September 3, 2024, Peter Hall 107 11:00 AM
Title: Some super-Poincaré, Hardy, and isoperimetric inequalities on Lie groups
Abstract: In the first half of this talk we will introduce functional inequalities and their applications to, for instance, Markov semigroups, spectral theory, and geometric measure theory. In the second half we will show how to prove a super-Poincar\’e and isoperimetric inequality for a type of probability measure defined on a Lie group, passing through some spectral theory and Hardy inequalities along the way.
Andrew Waldron (UC Davis)
Tuesday, August 27, 2024, Peter Hall 107 11:00 AM
Title: Yang—Mills on Conformally Compact Manifolds and Renormalization
Abstract: The Yang—Mills equations are essential to both modern day particle physics and four manifold theory, while conformally compact manifolds have played a crucial role in conformal geometry and string theory. We study the Yang—Mills equations on conformally compact manifolds and in particular the asymptotics of solutions, their renormalized energy functional, the obstruction to smooth solutions and the map from Dirichlet to Neumann
boundary data.
Yann Bernard (Monash)
Friday, May 3, 2024, Peter Hall 107 11:00 AM
Title: Conformally Invariant Problems for 4d Hypersurfaces
Abstract: We discuss a large class of conformally invariant curvature energies for immersed hypersurfaces of dimension 4. The class under study includes various examples that have appeared in the recent literature and which arise from different contexts. We employ Noether’s Theorem to obtain conservation potentials with good analytical dispositions. We show that under natural small-energy hypotheses, critical points satisfy improved energy estimates. Finally, we show how to use Noether’s theorem and the Gauss-Bonnet theorem to construct on hypersurface non-trivial divergence-free and Codazzi symmetric two-tensors of order O(|h|3), where h denotes the second fundamental form.
Jesse Gell-Redman (University of Melbourne)
Friday, December 8, Peter Hall 107 09:30 AM
Title: Scattering theory for nonlinear evolution equations
Abstract: I will discuss a new methodology for proving small data scattering for the nonlinear Schrödinger equation, which avoids the use of Strichartz estimates, and uses instead methods from microlocal analysis. This methodology is flexible and can in principle be applied to massive wave propagation as in the Klein-Gordon or massive Dirac equations. This is joint work with Andrew Hassell and Sean Gomes and with Dean Baskin and Moritz Doll.
Qing Han (Notre Dame University)
Friday, November 17, Peter Hall 107 (Cancelled)
Title: A concise boundary regularity of minimal surfaces in the hyperbolic space
Abstract: The minimal surface equation in the hyperbolic space is given by a quasilinear elliptic equation, which is non-uniformly elliptic and becomes singular on the boundary. In this talk, we discuss recent results of the concise boundary regularity of minimal surfaces in the context of finite regularity, smoothness, and analyticity.
Andras Vasy (Stanford)
Friday, September 15, Peter Hall 107 11:00 AM
Title: The Feynman propagator and self-adjointness
Abstract: In this talk I will discuss the Feynman and anti-Feynman inverses for wave operators on certain Lorentzian (and more generally pseudo-Riemannian) manifolds; these are two inverses which from a microlocal analysis perspective are more natural than the standard causal (advanced/retarded) ones. For instance, for the spectral family of the wave operator, these are the natural inverses when the spectral parameter is non-real. Indeed, I will explain that these connect to the self-adjointness of the wave operator, and the positivity properties that follow.
Brian Krummel (Melbourne University)
Friday, August 18, Peter Hall 107 11:00 AM
Title: Analysis of singularities of area minimizing currents
Abstract: In his monumental work in the early 1980s, Almgren showed that the singular set of an n-dimensional locally area minimizing submanifold T has Hausdorff dimension at most n-2. The main difficulty is that higher codimension area minimizers can admit branch point singularities, i.e. singular points at which one tangent cone is a plane of multiplicity two or greater. Almgren’s lengthy proof showed first that the set of non-branch-point singularities has Hausdorff dimension at most n-2 using an elementary argument based on tangent cone type, and developed a powerful array of ideas to obtain the same dimension bound for the branch separately. In this strategy, the exceeding complexity of the argument stems largely from the lack of an estimate giving decay of T towards a unique tangent plane at a branch point.
We will discuss a new approach to this problem (joint work with Neshan Wickramasekera). In this approach, the set of singularities (of a fixed integer density q) is decomposed not as branch points and non-branch-points, but as a set B of branch points where T decays towards a (unique) plane faster than a fixed exponential rate, and the complementary set S. Using a new intrinsic frequency function for T relative to a plane and a blow-up method of L. Simon and Wickramasekera, we show that T has a unique non-planar tangent cone at Hn-2-a.e. point of S and T is asymptotic to a unique homogeneous harmonic multi-valued function at Hn-2-a.e. point of B. It follows that the singular set of T is in fact countably (n-2)-rectifiable.
Chris Kottke (New College of Florida)
Thursday, June 1, Peter Hall 107 11:00 AM
Title: L2 cohomology of some non-compact moduli spaces
Abstract: Certain non-compact families of moduli spaces arising in geometry come equipped with natural hyperKahler metrics, such as the Hilbert schemes of points
on C2 and the moduli spaces of non-abelian “magnetic monopoles” of a given magnetic charge, among others. Predictions of physics have led to long-standing conjectures concerning these moduli spaces’ L2 cohomology — a geometric invariant, consisting essentially of the dimensions of square-integrable harmonic forms, which is in some sense “in-between” compactly supported and absolute cohomology. The main challenge in proving the conjectures has been to understand how L2 harmonic forms behave in the asymptotic regions of the moduli spaces “near infinity”. I will report on various joint work with F. Rochon and with M. Singer which leads to a proof of the Vafa-Witten Conjecture for Hilbert schemes in all dimensions, and of the Sen Conjecture for monopole moduli spaces in the new case of charge 3.
Hadrian Quan (University of Washington)
Thursday, May 25, Peter Hall 107 11:00 AM (online talk)
Title: Quantum confinement in α-Grushin manifolds and the α-pseudodifferential calculus
Abstract: The $\alpha$-Grushin plane exhibits a sub-Riemannian manifold whose geodesics can pass through singularities of an associated Riemannian metric, but whose Laplace-Beltrami operator can still be self-adjoint. Physically this corresponds to a situation when a classical particle can pass through a singularity, while a quantum particle cannot. In this joint work with Ivan Beschastnyi, we study more general $\alpha$-Grushin Manifolds and, using the tools of geometric microlocal analysis, characterize precisely when operators of this type admit self-adjoint extensions. We achieve by use of a new exotic pseudodifferential calculus, which is closely related to the 0-calculus of Mazzeo first used in study of asymptotically hyperbolic manifolds.
Georgios Mavrogiannis (Rutgers)
Thursday, April 20, Peter Hall 107 11:00 AM (online talk)
Title: Relatively non-degenerate estimates on Kerr de Sitter spacetimes
Abstract: We will start discussing a new method of how to prove exponential decay for the solutions of the wave equation on a Schwarzschild de Sitter black hole spacetime by exploiting a novel “relatively non-degenerate” estimate. This estimate does not degenerate at trapping. The main ingredient in proving this estimate is to commute with a novel vector field that “sees” trapping. Then, we will discuss a natural generalization of the vector field commutation in Schwarzschild de Sitter to the entire subextremal Kerr de Sitter black hole spacetime, by commuting with a pseudodifferential operator. There are more technicalities because of the elaborate nature of trapping. Time permitting we will discuss how to use this black box estimate to prove stability and exponential decay for the solutions of a quasilinear wave equation on Kerr de Sitter.
Paul Minter (Princeton)
Thursday, March 30, Peter Hall 107 11:00 AM (online talk)
Title: The structure and regularity of branched stable minimal hypersurfaces
Abstract: Understanding how smoothly immersed, stable, minimal hypersurfaces can degenerate under uniform volume bounds is a well-known problem within geometric analysis and geometric measure theory. In low dimensions, the work of Schoen-Simon-Yau provides uniform curvature estimates. However, for arbitrary dimensions the problem is still open. A key issue to understand is singular points of higher multiplicity, with a branch point being the main example. A priori, the topological structure about branch points could be very complicated, with, for example, a sequence of “necks” degenerating toward the point; indeed, the branch set could even have positive measure.
In this talk, I will discuss some recent results in this direction. We prove several regularity theorems in this setting, including some uniqueness of tangent cones results, which allow for branch points and make no a priori assumption on the size of the singular set. A key aspect of our argument is being able to prove monotonicity of a frequency function for the linearised problem (i.e blow-ups), despite the blow-ups not satisfying any variational principle a priori (which is the case, for example, in the area-minimising setting and the multiplicity one setting).
Some results are joint with Neshan Wickramasekera (University of Cambridge).
Jacob Shapiro (University of Dayton)
Thursday, March 23, Peter Hall 107 11:00 AM
Title: Semiclassical resolvent estimates in low regularity
Abstract:
One of the central aims of semiclassical and microlocal analysis is to describe how waves scatter or decay by leveraging the so-called classical-quantum correspondence. This principle asserts that there is a deep connection between the functional analytic properties of a Schrödinger or wave-type differential operator, and the dynamics of the underlying Hamiltonian system.
I will survey some recent results concerning energy decay of linear waves in situations where the classical-quantum correspondence breaks down (e.g., the classical dynamics are not well posed). In this situation, the main tool we turn to is a certain semiclassical Carleman estimate, which implies a limiting absorption resolvent estimate for the operator under consideration. Several open problems will be discussed. This talk is based on joint work with Kiril Datchev (Purdue), Jeffrey Galkowski(UC London), and Andres Larrain-Hubach (U Dayton).
Shi-Zhuo Looi (Berkeley)
Thursday, March 16, Peter Hall 107 11:00 AM (online talk)
Title: Asymptotics for odd- and even-dimensional waves
Abstract: In this talk, I will give a survey of recent and upcoming results on various linear, semilinear and quasilinear wave equations on a wide class of dynamical spacetimes in various even and odd spatial dimensions. These results include asymptotics for a wide range of nonlinearities. For many of these results, the spacetimes under consideration have only weak asymptotic flatness conditions and are allowed to be large perturbations of the Minkowski spacetime, provided that an integrated local energy decay estimate holds. We explain the dichotomy between even- and odd-dimensional wave behaviour. Part of this work is joint with Mihai Tohaneanu and Jared Wunsch.
Moritz Doll (Melbourne)
Thursday, November 23, Peter Hall 107 2:15PM
Title: Heat Trace Asymptotics for the Generalized Harmonic Oscillator on Scattering Manifolds
On a scattering manifold, we consider a Schrödinger operator of the form H = -Δ + V(x), where the potential satisfies a growth condition that generalizes quadratic growth for Euclidean space. We follow the approach of Melrose by constructing a suitable space on which the integral kernel of the heat operator is a smooth function and then we use the push-forward theorem to calculate the heat trace asymptotics. This is based on ongoing joint work with Daniel Grieser.
Pierre Portal (ANU)
Thursday, November 10, Peter Hall 107 2:00PM
Title: Hardy spaces for wave equations
I will survey various recent papers that have their origin in my joint work with Andrew Hassell and Jan Rozendaal (building on ground breaking ideas of Hart Smith).
These papers prove well-posedness results for (mostly linear at this stage) wave equations with rough initial data and/or coefficients, by introducing function spaces adapted to the equation, and then deducing results in more classical spaces via an embedding theory. The key idea is that, just as classical function spaces have a Littlewood-Paley description, function spaces adapted to wave equations should be described using the refined Littlewood-Paley decomposition used in the celebrated paper of Seeger, Sogge, and Stein on the Lp boundedness of Fourier Integral Operators. This decomposition is refined in an anisotropic way: it decomposes the momentum side of phase space in a direction dependent manner. In doing so, it captures cancelations arising from destructive interference. These features can then be interpreted as arising from a diffusion phenomenon on phase space, paving the way for the use of parabolic methods that are well suited to rough data and/or coefficients. We will explain these ideas and how they connect to other techniques (including wave packet decompositions and decoupling), and present some recent results, including work of Frey, Hassell, Rozendaal, Schippa, and Yung.
Serena Dipierro (University of Western Australia)
Friday, November 4, Peter Hall 107 1:15PM
Title: The strange behaviour of nonlocal minimal surfaces
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 Valdinoci (University of Western Australia)
Thursday, November 3, Peter Hall 107 2:15PM
Title: Long-range phase coexistence models
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.
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
Kerr-de Sitter
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.
MSc Theses
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
MSc Courses
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.
Past MSc Students
Here is a list of students who have written their MSc theses in the Analysis group:
- Joshua Culbert
- Sean Wynn
- Daniel Traicos
- Edmund Lau
- Lukas Anagnostou
- Yaozhong Qiu
- Caelan Turvey
- Madeleine Johnson
- Alexius Savvinos