Title: The geometry of null-infinity
Speaker: Jörg Frauendiener
Abstract: The concept of null-infinity was formulated by R. Penrose in 1962 in order to give a geometric description of asymptotically flat space-times. Since then, this idea has been fundamental in many applications of general relativity, not the least being the rigorous definitions of gravitational radiation and global quantities, such as energy-momentum and angular momentum. In this talk, I will present a different view on the (degenerate) geometric structure of null-infinity, which draws attention to the idea of a cut-system. The interplay between a cut-system and conformal invariance is explored, leading to an invariant definition of a Minkowski space-time "at infinity". Some consequences of this structure will be discussed.
Title: Memory of Robinson-Trautman waves
Speaker: Glenn Barnich
Abstract: The (non-linear) memory effect for Robinson-Trautman waves is explicitly worked out. In a first step, we construct the combined frame rotation and coordinate transformation in which Robinson-Trautman waves are manifestly locally asymptotically flat at future null infinity. This allows us to apply well-established results on how to derive the memory effect in this context. In a second step, we construct a suitably improved generalized mass aspect that provides a local Lyapunov function for the flow in the sense that it is manifestly positive. News-free solutions are studied in detail and shown to coincide with the vacuum sector of Euclidean Liouville theory. They can be obtained from a Schwarzschild black hole by a rescaling and applying a boost inside the BMS4 group at future null infinity.
Title: The Geometry of Black Hole Interiors: Singularities and Cosmic Censorship Conjecture
| Speaker: | Maxime Van de Moortel, Rutgers University |
| Abstract: |
| The nature of spacetime singularities inside black holes is a fundamental problem in General Relativity, central to Penrose’s Strong Cosmic Censorship Conjecture. While significant progress has been achieved on the stability of black hole exteriors, the interior dynamics and the structure of the resulting singularities remain comparatively unexplored. In this talk, I will present a resolution of the singularity structure inside dynamical black holes in spherical symmetry. We prove that contrary to classical expectations, the singularity has a hybrid geometric character: a weak null component where the metric extends continuously, and a strong spacelike component where the metric is inextendible. I will discuss why this configuration is conjectured to be generic and provides a new direction within the Strong Cosmic Censorship program. |
Title: Local structure theory of Einstein manifolds with boundary
Speaker: Lan-Hsuan Huang
Abstract: We discuss results on the structure of compact Einstein manifolds in terms of the conformal boundary metric and the mean curvature. In three dimensions, we confirm M. Anderson's conjecture by showing that, generically, Einstein metrics are locally parametrized by such boundary data. We also obtain analogous results in higher dimensions for Einstein manifolds with a nonpositive constant. This is joint work with Zhongshan An.
Title: Poincaré-Einstein manifolds: conformal structure meets metric geometry
Speaker: Ruobing Zhang
Abstract: A Poincaré-Einstein manifold is a complete non-compact Einstein manifold with negative scalar curvature which can be conformally deformed to a compact manifold with boundary, called the conformal boundary or conformal infinity. Naturally, such a space is associated with a conformal structure on the conformal infinity. A fundamental theme in studying these geometric objects is to relate the Riemannian geometric data of the Einstein metric to the conformal geometric data at infinity which is also called the AdS/CFT correspondence in theoretical physics.
Title: The AdS/CFT correspondence and conformal geometry
Speaker: Kostas Skenderis
Abstract: The talk will start with an introduction/overview of Conformal Field Theory (CFT), the AdS/CFT correspondence and the applications of conformal geometry methods to them. The emphasis of the talk will be on open questions and conjectures motivated by physical arguments that may be amenable to rigorous mathematical analysis.
Title: The Killing connection and a Calabi operator for locally symmetric spaces
Speaker: Thomas Leistner
Abstract: The prolongation of the Killing equation for vector fields gives rise to a connection, the Killing connection. Its parallel sections can be identified with Killing vector fields, i.e. with the kernel of the Killing operator, and hence it provides a useful tool when studying symmetries. On the other hand, the Killing connection can also be used when analysing the range of the Killing operator. For spaces of constant sectional curvature, Calabi found a second order linear differential operator that provides exact local integrability conditions for the range of the Killing operator. We generalise this result by providing such a second order operator for most Riemannian and Lorentzian locally symmetric spaces, and identify those for which exactness fails. As our approach uses the Killing connection, we are lead to analyse the range of a connection on a vector bundle in general. This is joint work with Federico Costanza, Mike Eastwood, and Benjamin McMillan.
Title: CR structures in Lorentzian conformal geometry: old and new results
Speaker: Arman Taghavi-Chabert
Abstract: Cauchy-Riemann three-manifolds are known to underlie the geometry of certain classes of Einstein metrics on Lorentzian four-manifolds, notably thanks to the work of mathematical relativists from the Oxford and Warsaw schools. In this talk, I will provide a conceptual approach to this subject by relating these Einstein Lorentzian four-manifolds to Fefferman's well-known canonical conformal structure associated to a strictly pseudo-convex CR three-manifold. In this way, conformal infinity naturally arises from the existence of a so-called almost Einstein scale and can be identified as cross-sections of Fefferman's bundle.
Title: Petrov's Other Classification
Speaker: Ian Anderson
Abstract: The Russian mathematical physicist A. Z. Petrov is certainly best known for his algebraic classification of the Weyl tensors for a 4-dimensional spacetime metric. It is perhaps less well-known that Petrov also gave a lengthy classification of the possible (infinitesimal) isometry groups for a 4-dimensional spacetime. In this talk I will focus on two aspects of this latter classification. Firstly, I will discuss the case where the isometry group of the spacetime metric acts simply transitively --- that is to say, the spacetime is a 4-dimensional Lie group. Here Petrov's classification is incomplete. In joint work with Charles Torre, we obtained a complete solution to the equivalence problem for spacetime groups. I will summarize our results and discuss applications to general relativity and conformal geometry. The role of various tractor connections will be highlighted. Secondly, I will briefly discuss a special class of spacetimes with cohomogeneity-1 isometry groups which also appear in Petrov. In the Riemannian setting, isometry group actions always admit a slice -- that is, a cross-section on which the isotropy subgroups are all the same. The Petrov examples do not admit a slice --- but nevertheless can be systematically classified by newly developed Lie-algebraic methods.
Title: A Boundary DeTurck Trick
Speaker: Stephen McKeown
Abstract: It is well known that the DeTurck trick puts the Einstein equations in elliptic gauge; but Anderson has shown that several natural boundary value problems do not satisfy the Lopatinskii-Shapiro conditions. We show that one may apply a DeTurck trick to the boundary value equations so that, under favorable conditions, the CMC umbilic boundary value problem becomes elliptic.
Title: Conformally Kähler–Einstein metrics on toric surfaces
Speaker: Rosa Sena-Dias
Abstract: Chen–LeBrun–Weber developed a strategy for finding conformally Kähler–Einstein metrics, which they use to show the existence of such a metric on $\mathbbCP^2\sharp 2\overline{\mathbbCP^2}$. In this talk, we will explain how this strategy led us to both new and old examples of conformally Kähler–Einstein metrics either in the noncompact setting or with cone angles. We discuss degenerating families of such metrics. We adopt the same perspective to present classification results for conformally Kähler Ricci-flat metrics on complements of divisors, building on a recent construction due to Biquard and Gauduchon. This is based on joint work with Gocalo Oliveira.
| Title: | Diffeomorphisms of 4-manifolds, complex surface singularities, and gauge theory |
| Speaker: | Juan Muñoz-Echániz, Stony Brook University |
|
Title: Einstein—Maxwell gravitational instantons
Speaker: Maciej Dunajski
Abstract: I will describe a construction (joint work with Bernardo Araneda) of new asymptotically flat Einstein—Maxwell gravitational instantons which generalise the Chen—Teo Ricci—flat metric and thus provide a counter example to the Euclidean Einstein-Maxwell Black Hole Uniqueness Conjecture. These solutions result from a combination of SU(∞) Toda techniques with the hidden symmetries of the anti—self-dual Yang—Mills equations with SL(3, C) gauge group. I will also review other examples of Einstein—Maxwell instantons (joint work with Sean Hartnoll).
Title: Integrable deformations and stability of the Ricci flow
Speaker: Alex Waldron
Abstract: I will introduce a joint paper with Max Stolarski in which we take a new approach to the stability theorems for Ricci flow on Ricci-flat ALE spaces due to Deruelle-Kroncke and Kroncke-Petersen.
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Title: Asymptotically Euclidean Solutions of the Constraint Equations with Prescribed Asymptotics
Speaker: James Wheeler
Abstract: I will discuss recent work on the construction of asymptotically flat vacuum initial data sets in General Relativity via the conformal method. My collaborators (Lydia Bieri, David Garfinkle, Jim Isenberg, and David Maxwell) and I have demonstrated that certain asymptotic structures may be prescribed a priori through the method's seed data, including the ADM momentum components, the leading- and next-to-leading-order decay rates, and anisotropy in the metric's mass term, yielding a recipe to construct initial data sets with desired asymptotics. As an application, we discuss a simple numerical example, with stronger asymptotics than have been presented in previous work, of an initial data set whose evolution does not exhibit the conjectured antipodal symmetry between future and past null infinity.
Title: Ricci flow and integral curvature pinching
Speaker: Eric Chen
Abstract: Early applications of the Ricci flow by Hamilton and others characterized Riemannian manifolds with certain pointwise curvature pinching conditions as spherical space forms. In some cases, curvature pinching in averaged, integral senses can extend such results on topological restrictions. I will describe some works on critical, scale-invariant integral curvature pinching and smoothing obtained using the Ricci flow and consequences of Perelman's W-entropy, joint with Guofang Wei and Rugang Ye.
Title: Gravitational instantons and harmonic map
Speaker: Song Sun
Abstract: Gravitational instantons are by definition 4-dimensional complete Ricci-flat metrics with finite curvature energy. Previous studies mainly focused on gravitational instantons with special geometries, known as hyperkahler or conformally Kahler metrics. These special cases have been essentially classified in recent years. I will present joint work with Mingyang Li on the construction of an infinite family of new gravitational instantons, using axisymmetric harmonic maps. These do not have special geometries and the construction is non-perturbative. This work is motivated by questions related to black hole uniqueness questions in general relativity. Time permitting, we will also discuss some open problems.
Anna Jove Campabadal, University of Barcelona
TBA
TBA
Title: Local and global invariants of Poincare-Einstein manifolds and their conformal boundaries
Speaker: Jeffrey Case
Abstract: In 1985, Fefferman and Graham introduced (formal) one-to-one correspondences between conformal manifolds, Poincare-Einstein (PE) manifolds, and ambient spaces. This provides a powerful way to construct local conformal invariants on the boundary and global invariants of the interior PE metric. One illuminating example comes from the renormalized volume: for even-dimensional PE manifolds, it is a global invariant of PE manifolds that can be computed via a Gauss-Bonnet-type formula , while for odd-dimensional PE manifolds it comes with an "anomaly" whose integral is the total Q-curvature. In this talk I will describe a broader connection between conformal invariants and conformal anomalies based on Albin's renormalized curvature integrals, as well as a new technique for computing global integral invariants of Einstein manifolds, including the Gauss-Bonnet-type formula for the renormalized volume. Moreover, I will show that in even dimensions eight and larger, there are scalar conformal invariants and Weyl anomalies that integrate to zero; indeed, in high dimensions there are drastically fewer integrated conformal invariants (resp. integrated anomalies) than there are local conformal invariants. This talk is primarily expository, though the last part is based on recent joint work with Ayush Khaitan, Yueh-Ju Lin, Aaron Tyrrell, and Wei Yuan.
Filip Zivanovic, SCGP
TBA