Analysis Seminar
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.