Conformal Geometry, Analysis, and Physics
Abstracts• Maciej Zworski
Title: Outgoing property via Gevrey regularity
Abstract: My small overlap with Robin Graham’s research came fromthe use of scattering matrices in conformal geometry. We also had touse Borel’s lemma to construct certain Poisson operators. My elementary talk will touch upon those concepts in the setting 1D scattering
and will explain a recent approach of Gajic–Warnick to distinguishingoutgoing solutions (hence resonances) using Gevrey regularity. It isbased on joint work with J Galkowski.
• Kengo Hirachi
Title: Normal scale for pseudo-Einstein contact forms and intrinsic
CR normal coordinates
Abstract: In conformal geometry, Robin Graham gave a choice ofgood scale in the conformal class to simplfies the computation involving
the jets of the conformal structure. We will give an analogeous resultin CR geometry based on an earlier work by Jerison-Lee. Our choice ofthe contact form is simpler as we can stay in the class of pseud-Einsteincotact forms, which was not the case for the choice of Jerison-Lee. We
also plan to explain its application to the Szego kernel.
• Yi Wang
Title: Yamabe flow of asymptotically flat metrics
Abstract: In this talk, we will discuss the behavior of the Yamabe
flow on an asymptotically flat (AF) manifold. We will first show thelong-time existence of the Yamabe flow starting from an AF manifold
and discuss the uniform estimates on manifolds with positive Yamabe
constant. This would allow us to prove convergence along the Yamabe
flow on such manifolds. We also prove that the flow will diverge if the
Yamabe constant is nonpositive. It turns out that the behavior of therescaled flow on manifolds with negative Yamabe constant is explicit.
If time permits, we will discuss some further work when the Yamabe
constant is zero. This is joint work with Eric Chen and Gilles Carron.
1• Kostas Skenderis
Title: Ambient metric and the embedding formalism
Abstract: In their 1985 paper on Conformal Invariants Fefferman andGraham developed two constructions that associate a n-dimensional
conformal manifold to a higher dimensional metric. In one of them
the n-dimensional conformal manifold is the conformal boundary of
the n + 1-dimensional (Euclidean) AdS spacetime. This construction is
instrumental in setting up the holographic dictionary in the AdS/CFT
correspondence via the formalism of holographic renormalization. The
second construction associates the n-dimensional conformal manifoldwith a Lorentzian Ricci-flat metric in n + 2 dimensions, the ambient
metric. This second construction have not had a direct physics application to date. In this talk I will argue that the ambient metric may
be used to generalise the embedding formalism for conformal field theories, originally introduced by Dirac in 1936, to curved spacetimes. I
will review how the embedding formalism works in flat space and thenillustrate the generalisation using an example involving a holographic
thermal CFT.
• Jie Qing
Title: Asymptotically hyperbolic Einstein manifolds and conformal
geometry
Abstract: In this talk we will present a survey on our works on asymptotically hyperbolic Einstein manifolds and conformal geometry. We
will start with Fefferman-Graham ambient spaces and the definition of
asymptotically hyperbolic Einstein manifolds. We will then introduce
Mobius coordinate charts and some basic analysis on asymptotically
hyperbolic manifolds.
In the spirit of AdS/CFT correspondence, we will introduce scattering
operators and its relations to renormalized volumes on asymptotically
hyperbolic Einstein manifolds. We will introduce the geometric correspondences: one is on the scattering poles vs the positivity of the
Yamabe constant; the second is on the volume growth vs the Yamabe
constant.
At the end, we will also report some recent work on the uniqueness andcompactness of asymptotically hyperbolic Einstein manifolds. Many of2the reported work are joint work with Alice Chang and Paul Yang.
• Nikolaos Eptaminitakis
Title: Stability for the X-ray Transform on Asymptotically Hyperbolic
Manifolds
Abstract: Given a Riemannian manifold and a function f definedon it, the geodesic X-rayR transform of f is a function on the space ofgeodesics given by γ 7→ γ f ds, where ds is the arc-length measure on
the geodesic γ. Typically the function f is unknown and one is interested in obtaining information about it from its geodesic X-ray transform; for example, one would like to understand whether f is uniquelyand stably determined by the transform or whether an inverse can becomputed. Such problems have been studied extensively in various
settings, starting with J. Radons 1917 seminal paper, where inversion
formulas for the transform were derived on the Euclidean plane. The
topic of the talk will be the stability of the geodesic X-ray transformon simple asymptotically hyperbolic manifolds, a class of non-compact,simply connected, complete manifolds whose behavior near infinity resembles in many ways that of hyperbolic space. The goal is to establishthat small perturbations in the X-ray transform cannot originate fromlarge perturbations of the unknown function f , and find appropriate
spaces to measure such perturbations. In a similar spirit to the work ofStefanov-Uhlmann (2004) on simple compact manifolds with boundary,we study microlocally the stability of the normal operator of the X-ray
transform, which is given by its composition with a backprojection.
We discuss how the normal operator fits within the of the 0-calculusof Mazzeo-Melrose, which is a natural framework for studying pseudodifferential operators on asymptotically hyperbolic manifolds, andconstruct a parametrix, which eventually leads to the desired stability.
• Robert Bryant (Clay Foundation Speaker)
Title: On solitons for the closed G2 -Laplacian flow.
Abstract: I will start by recalling the basic geometry of G2 -structureson 7-manifolds and their relation with special holonony in dimension 7.
Then, I’ll discuss a natural evolution defined for closed G2 -structures,
known as the closed G2 Laplacian flow, that is a natural analog of Ricci
flow for Riemannian metrics. After discussing the short-time existence3and uniqueness question, I will discuss the local ‘generality’ of solitons
for this flow in É. Cartan’s sense, introducing a natural exterior differential system whose integral manifolds correspond to these solitons.
Finally, I will discuss the special case of gradient solitons for this flow,
and the remaining question of their generality.
• Michael Eastwood
Title: Conformal versus CR geometry
Abstract: It is well-known (as seen, for example, in the FeffermanGraham ambient metric construction) that there are close analogies
between conformal differential geometry and hypersurface CR geometry(and the Fefferman circle bundle gives a direct link). What is the CR
analogue of an Einstein metric in a conformal class? I shall explain joint
work with Lenka Zalabova, which gives an answer to this question andto questions like it.
• Qing Han
Title: Asymptotic analysis for extreme Kerr-type harmonic maps
Abstract: Motivated by studies of axially symmetric stationary solutions of the Einstein vacuum equations in general relativity, we study
singular harmonic maps from domains of the 3-dimensional Euclidean
space to the hyperbolic plane, with bounded hyperbolic distance to extreme Kerr harmonic maps. We prove that every such harmonic map
has a unique tangent map at the black hole horizon. The harmonic
map equation restricted to the unit sphere has a singularity at thenorth and south poles. The talk is based on joint work with Marcus
Khuri, Gilbert Weinstein, and Jingang Xiong.
• Yueh-Ju Lin
Title: Conformally variational Riemannian invariants and polydifferential operators
Abstract: Conformally variational Riemannian invariants (CVIs) arehomogeneous scalar invariants which appear as the conformal gradient of a Riemannian functional. In this talk, I will present several
stability and rigidity results for CVIs that generalize similar known
results for the scalar curvature and Q-curvature. I will talk about constructing a formally self-adjoint conformally covariant polydifferential
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operator associated with a given CVI. In addition, I discuss a complete
classification of tangential bi-differential operators in terms of ambient Laplacian. This result is a curved analogue of conformally covariant bi-differential operators on spheres classified by Ovsienko-Redouand Clerc. As time permits, I will also present a family of sharp,
fully nonlinear Sobolev inequalities involving the Paneitz operator andOvsienko-Redou operator. This talk is based on several joint works
with Jeffrey Case and Wei Yuan.
• Daniel Fox
Title: Minimal algebraic hypersurfaces in spheres from quasicomposition algebras
Abstract: A cubic form on a metrized vector space determines a commutative algebra for which the given metric is invariant. Hsiang asked
for the classification of irreducible homogeneous polynomials whose regular level sets are codimension one minimal cones. For cubic forms thisis equivalent to the classification of certain commutative algebras called
Hsiang algebras by Tkachev. The regular level sets of the cubic formsof the Hsiang algebras studied by Vladimir Tkachev give examples ofalgebraic minimal hypersurfaces in spheres that are not homogeneous.
He showed these come in an infinite family constructed from Clifford
systems and finitely many exceptional cases. The talk will describe a
tripling construction that associates with an n-dimensional metrized
algebra with involution a 3n-dimensional metrized commutative algebra and will explain that the triple algebra is a Hsiang algebra if andonly if the original algebra is a quasicomposition algebra. The quasicomposition algebras include algebras in dimensions 1, 2, 3, 4, 6, 7, 8:
Hurwitz algebras, commutative para-Hurwitz algebras, cross productalgebras, six-dimensional color algebras, and probably nothing more.
This yields a unified construction of certain of the Hsiang algebras andraises a number of further questions. This is joint work in progresswith Vladimir Tkachev.
• Pawel Nurowski
Title: Poincare-Einstein approach to Penrose’s Conformal Cyckic Cosmology
Abstract: I will discuss two consecutive eons in Penroses conformal
5cyclic cosmology, and study how the matter content of the past eon determines the matter content of the present eon. I will use the PoincareEinstein type of expansion to determine the metric in the past eon, andby the reciprocity hypothesis will produce the metric in the present eon.
Then, using the Einstein equations in the present eon, i will show what
has happened to the matter after its transition through the Big Bang.
I will show that a single spherical wave from the previous eon, in the
new eon, splits into three portions of radiation: the two spherical waves(one which is a damped continuation from the previous eon, the otherwhich is focusing in the new eon as it encountered a mirror at the Big
Bang surface), and a scattered radiation described by the statisticalphysics.
• Richard Melrose
Title: Boundary weights and Hodge cohomology
Abstract: I will discuss some aspects of the spaces of pseudodifferential operators intended to allow the identification of the Hodge cohomology of various complete Riemann metrics on the interiors of compact
manifolds with corners. The boundary weights of the title refer to theappearance of incomplete metrics in the model problems which arise in
the construction of a generalized inverse of the Hodge-Dirac operator.
• Rafe Mazzeo
Title: The Hitchin equations and ALG spaces
Abstract: There is a twelve-parameter family of Hitchin equations
on the four-punctured sphere, the parameters representing different‘boundary conditions at the punctures. For each allowable choice ofparameters, the corresponding moduli space of solutions carries thenatural structure of a four-dimensional hyperKaehler manifold of ALG
type, and with D4 topology. There is a twelve dimensional family ofsuch hyperKaehler metrics. In work with Fredrickson, Swoboda andWeiss, we show that the map from Hitchin parameters to ALG metrics
with this topology is surjective.
• Azahara Dela Torre
Title: The fractional Yamabe problem with singularities: from Conformal Geometry to the study of non-local semilinear elliptic PDEs.
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Abstract: The so called Yamabe problem in Conformal Geometry
consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem
becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean
space, allowing the presence of positive-dimensional singularities canbe transformed into reducing the non-linearity to a Sobolev-subcriticalpower. When, instead of scalar, we work with non-local curvature weneed to deal with the pertinent non-local semilinear elliptic PDE.
In this talk, we will focus on the study of the solutions of these non-local(or fractional) semilinear elliptic PDEs which represent metrics which
are singular along positive-dimensional singularities. In collaboration
with Ao, Chan, Fontelos, Gonzlez and Wei, we covered the constructionof solutions which are singular along (zero and positive-dimensional)smooth submanifolds. This was done through the development of new
methods coming from conformal geometry and Scattering theory forthe study of non-local ODEs. Due to the limitations of the techniques
we used, the particular case of “maximal dimension for the singularity
was not covered. In a recent work, in collaboration with H. Chan, we
covered the construction for this specific dimension using asymptotic
analysis and the stability and semi-linearity of the equation. Moreover we developed new methods to study the singular behavior of thesolutions.
• Claude LeBrun
Title: Kodaira Dimension and the Yamabe Problem, Revisted
Abstract: Dimension four provides a surprisingly idiosyncratic setting
for the interplay between scalar curvature and differential topology.
This peculiarity becomes especially pronounced when discussing the
Yamabe invariant or sigma constant of a smooth compact manifold;and Seiberg-Witten theory makes this especially apparent for those
4-manifolds that arise as compact complex surfaces. For compact complex surfaces of Kaehler type, I showed in the late 1990s that the signof the Yamabe invariant is always determined by the Kodaira dimension, and calculated the Yamabe invariant exactly in all cases where
it is non-positive. In this talk, I will describe recent joint work withMichael Albanese that generalizes these results to all complex surfaces
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of non-Kaehler type, except those of class VII. However, I will also
explain why excluding this exceptional class is absolutely essential forthese purposes.
• Andras Vasy
Title: Waves on black hole backgrounds and the stability problem
Abstract: I will discuss the behavior of waves on black hole backgrounds as well as its relationship to, and implications for, the blackhole stability problem.
This talk is based on joint work with Peter Hintz, Dietrich Haefner andOliver Lindblad Petersen.
• Lionel Mason
Title: Holomorphic discs and self-dual Einstein metrics from conformalboundary data
Abstract: A global twistor construction for conformally self-dual split
signature metrics on S 2 ×S 2 was developed 15 years ago by the speaker
and Claude LeBrun. This encodes the conformal metric into the location of a finite deformation of the real slice in CP3 ; the construction
entails determining a family of holomorphic discs in CP3 whose boundaries lie on the deformed real slice. This talk adapts the construction
to resolve this version of a problem studied by Graham & Lee 30 years
ago to construct global Einstein metrics from conformal boundary data.
This problem can be resolved straightforwardly in the case of vanishingcosmological constant, but remains conjectural when nonzero.
• Andreas Karch
Title: Submanifolds in conformal Geometry and their Physics Applications
Abstract: Holography, the equivalence between a conformal field theory and gravity in a higher dimensional bulk, provides a very concreteconnection between ideas from conformal geometry and pressing questions in modern physics. In this talk I’ll review the crucial role submanifolds have played in this context and present some recently developed
applications in the context of the studies of quantum information inboth gravity and field theory.
8• Jeffrey Case
Title: Some Q-curvature operators in CR geometry
Abstract: In conformal geometry, Q-curvature operators are differential operators on differential forms which have a “Q-like” transformation law when restricted to closed forms. In CR geometry, the firstexample of a Q-curvature operator is the P 0 -operator. In this talk,I introduce some Q-curvature operators on ∂ b -closed and on d-closed
differential forms, and discuss their application to the Q-curvature ofclosed strictly pseudoconvex CR manifolds in dimension five.
• Andreas Cap
Title: Tractors and the mass of asymptotically hyperbolic manifolds
Abstract: This talk reports on my joint work arXiv:2108.01373 withRod Gover that is motivated by the concept of mass for asymptotically hyperbolic metrics. We work in the setting of conformally compact metrics on arbitrary manifolds with boundary (at infinity) and aclass of metrics that are asymptotically hyperbolic in a weak sense andasymptotic to each other to appropriate (higher) order. This makes
sure that all the metrics in the class induce the same conformal infinity on the boundary. We then associate to two metrics in the class a
two-parameter family of ””relative masses””, that are top degree forms
on the boundary with values in the standard tractor bundle of theconformal infinity.
The construction is manifestly independent of coordinates and it is easily seen to be equivariant with respect to a natural family of diffeomorphisms. Much more subtle considerations shows that a one-parametersubfamily satisfies an additional invariance condition with respect todiffeomorphisms asymptotic to the identity. Assuming that the givenclass contains hyperbolic metrics, this allows us to associate an invariant to a single metric in the class. In the case of hyperbolic space
and the class of metrics determined by the Poincaré metric, the forms
we construct can be integrated to parallel sections of the tractor bundle and this recovers that mass of asymptotically hyperbolic metrics
introduced by Wang and Chrusciel-Herzlich.

9• Edward Witten
Title: Volumes and Ensembles in 3d Gravity
Abstract: There has been a longtime puzzle concerning the role ofconnected manifolds with disconnected boundaries in the duality between quantum gravity on a D-manifold X and an ordinary quantum
theory on the boundary of X. Some of these studies have suggested thatthe boundary theory in this duality is really an ensemble average. Herewe will explore this question in the case of D=3 using facts from classical hyperbolic geometry. The upshot is to argue that carefully selected
observables are not affected by “wormholes,” consistent with the ideathat the boundary theory is a well-defined quantum theory. (Based on
https://arxiv.org/pdf/2202.01372.pdf with J.-M. Schlenker.)• Rod Gover
Title: Conformal submanifolds, distinguished submanifolds, and integrability
Abstract: For conformal geometries of Riemannian signature, we provide a comprehensive treatment of the basic local theory of embedded
submanifolds of arbitrary codimension. We provide three distinct, but
equivalent, fundamental invariants of submanifolds. For any one ofthese normal objects its derivative by the pullback of the conformal
tractor connection recovers the data of the tractor second fundamental
form. The vanishing of this, so the normal object being parallel, determines a notion of distinguished submanifold. For the case of curves
this exactly characterises conformal circles, while for hypersurfaces itis the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that thiscoincides with a property of ambient conformal circles remaining inthe submanifold, a property that we term weakly conformally circular.
If this holds and the conformal circles coincide with the submanifold
distinguished curves then the submanifolds is said to be strongly conformally circular; this holds if and only also a second tractor invariant
vanishes, this invariant is a Fialkow type tensor invariant. We provethat the property of being distinguished is also captured by a typeof moving incidence relation. This second characterisation is used toprovide a very general theory and construction of quantities that arenecessarily conserved along the submanifold. The formalism thus leads
10to a conformal submanifold first integral theory that generalises theideas available for curves. We prove that any normal solution to an
equation from the class of first BGG equations can yield such a conserved quantity, and we show that it is easy to provide explicit formulae
for these. We also show that for normal solutions of conformal KillingYano equations, a certain zero locus of the solution is necessarily adistinguished submanifold.
This is joint work with Daniel Snell• Andrew Waldron
Title: Quantization and Contact Structures
Abstract: A contact form on an odd dimensional manifold determinesa generic classical mechanics. This can quantized by generalizing Fedosov’s approach to symplectic structures. This gives quantum mechanics, generally covariant with respect to choices of time, generalized
position and momentum. Just as many structures in general relativity(GR) are effectively studied by considering causal structures–i.e. conformal classes of metrics–it is also useful to consider conformal classes ofcontact forms aka contact structures. In the case of GR, key machinery
for the study of conformal manifolds is tractor calculus. This theory
is a parabolic geometry, and thus extends nicely to contact structures
and, as will be explained, their quantization. Key notions are parallel
sections with respect to distinguished connections; just as these provide the link between conformal and Einstein structures in GR, here
they relate contact structures to quantum mechanics. The general theory will be illustrated for the example of S 3 with its standard contact
structure.
• Eric Bahauad
Title: Analytic semigroups, bounded geometry and geometric flows
Abstract: Semigroup methods provide an elegant abstract frameworkfor studying parabolic evolution equations. In this talk I will describe
joint work with Guenther, Isenberg and Mazzeo providing a criterion
for a geometric elliptic operator to generate an analytic semigroup on
weighted Hlder spaces on manifolds with bounded geometry. I will thenoutline a few applications to geometric flows.

11• Travis Willse
Title: Chains of path geometries on surfaces
Abstract: A path geometry on a surface Σ can be viewed as (1) akind of nondegenerate 2-parameter family of unparameterized curvesin Σ, or, slightly more generally, (2) a pair of line distributions on
a 3-manifold spanning a contact distribution, or (3) an equivalenceclass of second-order o.d.e.s modulo point transformations. Path geometries are a classic, low-dimensional example of parabolic geometries: They are formally analogous to 3-dimensional CR structures,and they share with CR-structures a notion of chains, a preferred classof curves in P(T Σ). In this project, joint with Gil Bor (C.I.M.A.T.),
we investigate the geometry of chains of path geometries. We give a
simple but apparently new characterization of projective path geometries (those whose paths are the geodesics of some projective structure)in terms of chains, we describe the chains of four geometrically natural homogeneous path geometries, and we raise some open questions.
https://arxiv.org/pdf/2201.09141.pdf• Jack Lee
Title: Some Remarks on the 5-Dimensional CR Embedding Problem
Abstract: CR manifolds are the abstract models of real hypersurfaces
in Cn . But abstract CR manifolds cannot always be realized as such
hypersurfaces, even locally. Louis Boutet de Monvel proved in 1975
that every compact strictly convex CR manifold of dimension at least
5 can be embedded locally as a hypersurface, and globally in higher
codimension. On the other hand, generic 3-dimensional CR structures
cannot be locally embedded.
The noncompact case is more subtle. Masatake Kuranishi proved in1982 that every strictly pseudoconvex CR manifold of dimension atleast 9 is locally embeddable, and shortly thereafter Takao Akahori
was able to push the dimension down to 7. That left the 5-dimensional
noncompact case open, and it remains open to this day.
I’ll review the history of the problem, explain some of the reasons why
it is harder than the other dimensions, and suggest some strategies forgoing forward.

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