Pierre-Alexandre Arlove
Title: Time functions and monotone paths on orderable space of Legendrians.
Abstract: I will first remind that the notions of causality in Lorentzian geometry and orderability in contact
geometry both come from a closed proper cone structure. I will then show that any orderable space of
Legendrians admits time functions and that a Legendrian isotopy in this space is monotone if and only if
it is non-negative. This talk is based on joint works with Simon Allais.
Peter Cameron
Title: The extension problem for Lorentzian manifolds in low regularity
Abstract: I will be interested in the problem of extending a manifold with smooth Lorentzian metric to a larger
manifold where the extended metric is required only to be continuous. I will discuss examples where there is
one "natural" extension and will ask whether one could extend the topological and/or differentiable structure
differently while still obtaining a manifold with continuous metric. This work is motivated by classical general
relativity, where it is conjectured that black holes occurring in nature contain "weak null singularities'' in their
interiors. These correspond to boundaries across which the spacetime (described by a manifold with smooth
Lorentzian metric) cannot be extended with sufficient regularity for even a weak notion of the Einstein equations
to hold. However this singularity is "weak" in the sense that one can extend as a manifold with continuous metric.
Dylan Cant
Title: Floer cohomology for positive orbits of contact vector fields
Abstract: This talk is concerned with the question: "does a contact vector field X on the ideal boundary Y
of a Liouville manifold W have a closed orbit which is positively transverse to the contact distribution?"
I will describe recent joint work with I. Uljarević which discusses a Floer cohomology approach to this question.
As application, we provide a positive answer to the above question when a certain invariant is non-zero.
The invariant depends only on the domain where X is positively transverse, and should be thought of
as a "selective symplectic cohomology." Other aspects of the theory will be discussed, including its
applications to contact non-squeezing questions.
Carla Cederbaum
Title: On ruled hypersurfaces in static spacetimes and ensuing rigidity results
Ruled hypersurfaces have a long tradition in mathematics and engineering; in general relativity, timelike hypersurfaces
ruled by null geodesics – called “photon surfaces” -- turn out to be relevant for optical phenomena such as trapping of light.
After introducing the relevant definitions and some important examples of photon surfaces, we will discuss both their
abundance in static, spherically symmetric spacetimes and their scarcity in general static, (electro-)vacuum,
asymptotically flat spacetimes. The scarcity results can be phrased as rigidity or uniqueness theorems for the
Schwarzschild and Reissner—Nordström spacetimes (in four and higher spacetime dimensions). We will sketch
several different proofs of these rigidity results. Physically speaking, this asserts that static, (electro-)vacuum,
photon surfaces have no hair.
Time permitting, we will give a brief outlook on the situation in stationary spacetimes such as the Kerr spacetime.
The abundance result we will present is joint work with Galloway and with Jahns and Vičánek Martínez.
The rigidity results we will present are based on joint works with with Borghini and Cogo, with Cogo and Fehrenbach,
with Cogo, Leandro, and Paulo dos Santos, with Galloway, and with Jahns and Vičánek Martínez.
Georgios Dimitroglou Rizell
Title: Non-shrinking of Legendrians in prequantization bundles and non-squeezing of open subsets
Abstract: We use a version of Rabinowitz-Floer homology for Legendrians defined using Symplectic Field Theory,
and its filtered invariance, to show that Legendrian lifts of Bohr—Sommerfeld Lagrangians have a well-defined
integer-valued bound on the displacement energy, which is invariant under Legendrian isotopy. This means that
such Legendrians cannot be "shrunk" by Legendrian isotopy. Applications includes contact non-squeezing results
for preimages of certain symplectic balls and cubes under the prequantization-bundle projection. This is joint work
in progress with M. Sullivan.
Maciej Dunajski
Title: Quasi-local mass, Kerr horizon, and causality
Abstract: I will discuss two approaches to mass in General Relativity. One quasi-local, and applicable to closed
surfaces in space times (like that of the Kerr, or Kerr de-Sitter horizon), and one global, based on causal properties
of space-times near space-like infinity.
Michael Entov
Title: Relative Symplectic Field Theory filtered by the action functional and Legendrian isotopies
Abstract: Relative Symplectic Field Theory associates to a (non-degenerate) pair formed by a Legendrian submanifold
and a contact form on a contact manifold a version of the Legendrian contact homology, and to an exact Lagrangian
cobordism between Legendrian submanifolds a morphism between the corresponding Legendrian contact homologies.
The Legendrian contact homology comes equipped with the action filtration, induced by the actions of the Reeb chords,
and thus gives rise to persistence modules; the exact Lagrangian cobordisms then induce morphisms between the
persistence modules.
I will discuss the information on various new geometric measurements of Legendrian isotopies, and, in particular,
of positive embedded Legendrian isotopies, provided by the machinery above and by the theory of persistence modules.
This is a work in progress, joint with L.Polterovich.
Jonny Evans
Title: Open problems around Lagrangian intersections
Abstract: Given a convex body K in n-dimensional Euclidean space, is there a point in K which lies on 2n normals
to the boundary? (This is known to be true if n\leq 4 but otherwise open) Is it possible for a Lagrangian torus in
CP^n which is Hamiltonian isotopic to the Clifford torus to have smaller volume than the Clifford torus?
(This is known to be impossible if n=1 and open otherwise)
Both of these long-open questions can be formulated in terms of intersection theory for Lagrangian submanifolds,
and both hint at persistent family intersection phenomena which go beyond what can be detected by Lagrangian
intersection Floer cohomology. I will explain this. Whilst I can't solve either question yet, I will explain how to use
these ideas to get a lower bound on the volume of any Hamiltonian deformation of the Chekanov torus in CP^2.
Charles Frances
Title: Conformal Singularities
Abstract: In this talk, we will consider conformal embeddings between pseudo-Riemannian manifolds, which are
only defined outside a "singular locus". We will try to understand under which circumstances this singular set is
removable, namely one can extend the embedding across it. In the Riemannian setting (and dimension at least 3),
the situation is fairly well understood when the singular locus has small Hausdorff dimension.
We will also consider the problem in Lorentzian signature, and present some rigidity results in this framework.
Leonardo García Heveling
Title: Causal boundaries, conformal symmetries, and spacetime splitting
Abstract: In geometry, one often adds a boundary to a space consisting of the "points at infinity".
For the spacetimes of general relativity, this is accomplished by the causal boundary. In this talk,
we will introduce a new technique that helps us, among other things, to prove that the causal boundary
is particularly simple when the spacetime has a certain kind of conformal symmetry. We then discuss
some applications motivated by Bartnik's splitting conjecture.
Liang Jin
Title: Contact Hamiltonian systems via a variational approach
Abstract: Let M be a closed manifold. Contact Hamiltonian systems on the 1-jet space of M are natural extensions
of Hamiltonian systems on the cotangent bundle of M from the viewpoint of characteristic method for first order PDE's.
By introducing variational principles for such contact Hamiltonian systems, one can apply the spirit of weak KAM theory
to understand the viscosity solutions to the corresponding Hamilton-Jacobi equations.
In this talk, I will summarise results on both PDE and dynamical aspects of contact Hamiltonian systems obtained so
far by the above mentioned approach. If time permits, I will also discuss some conjectures concerned while we are
dreaming to have a complete generalization of Aubry-Mather theory in this context. This talk is based on joint works
with Jun Yan (Fudan University) and Kai Zhao (Tongji University).
Michael Kunzinger
Title: Curvature Bounds in Lorentzian Pre-length Spaces
Abstract: We give an overview of the currently known notions of (sectional) curvature bounds in Lorentzian pre-length spaces.
In particular, we introduce two rather new approaches: On the one hand a four-point condition, which also works in
non-intrinsic settings (where distance realizers need not exist). On the other hand, we define curvature bounds via
convexity/concavity conditions on the (modified) time-separation function. We show that under mild assumptions,
all currently known notions of sectional curvature bounds are equivalent. We also demonstrate that in smooth
strongly causal spacetimes sectional curvature bounds in timelike planes (as introduced by Alexander and Bishop)
are equivalent to synthetic timelike curvature bounds.
[1] T. Beran, M. Kunzinger, F. Rott, On curvature bounds in Lorentzian length spaces, J. London Math. Soc., to appear.
[2] T. Beran, M. Kunzinger, A. Ohanyan, F. Rott, The equivalence of smooth and synthetic notions of timelike sectional curvature bounds, Preprint, 2024.
Lionel Mason
Title: Asymptotically de Sitter Einstein-Weyl structures and scattering maps
Abstract: We show that in 2+1 dimensions, for a solution to the Einstein-Weyl equations that is asymptotically anti-deSitter,
one can define a scattering map from points of past infinity to future infinity (both naturally isomorphic to the Riemann sphere)
by following light rays from one to the other. We show that, in a neighbourhood of the scattering map being the antipodal map,
every small deformation corresponds to a solution to an asymptotically de Sitter solution to the Einstein-Weyl equations.
The construction is driven by holomorphic discs in the complex quadric formed from the cartesian product of past and future infinity,
with boundary on the totally real 2-sphere determined by the scattering map.
This based on joint work with Claude Lebrun in Maths Res Lett from 2008.
Lukas Nakamura
Title: On the size of an overtwisted disk
Abstract: In this talk, we introduce a notion of size of an overtwisted disk in a strict contact manifold M, which in the
case of a standard contact form corresponds to the action of the simple closed Reeb orbit in the overtwisted disk.
In terms of the size, we give an upper bound on the Shelukhin-Chekanov-Hofer distance between Legendrians in
the complement of the overtwisted disk and a lower bound on the capacity of a standard symplectic ball that guarantees
that its product with M is overtwisted. This is joint work with Á. del Pino Gómez.
Paweł Nurowski
Title: Exceptional simple real Lie algebras f4 and e6 via contactifications
Abstract: In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra
of authomorphism is the split real form of the simple exceptional complex Lie algebra f4. Cartan's formula is written in the standard
Cartesian coordinates in R15. In the talk I will explain how to find analogous formula for the flat models of any bracket generating
distribution D whose symbol algebra n(D) is constant and 2-step graded, n(D)=n−2⊕n−1.
I will use the general formula to provide other distributions with symmetries being real forms of simple exceptional Lie algebras f4 and e6.
Miguel Sánchez Caja
Title: Marsden theorem and the completeness of left-invariant semi-Riemannian metrics on Lie groups
Abstract:The study of invariant Riemannian metrics in a Lie group G goes back to Euler's study of rigid solid motion.
When considering indefinite semi-Riemannian metrics, a noticeable possibility is geodesic incompleteness. Our main
aim is to give a sufficient condition for the geodesic completeness of all the left-invariant indefinite metrics on G,
which is fulfilled and generalizes all the known cases so far. This condition, which might have interest on its own right,
means that the norm of the adjoint representation grows at most affinely with respect to any auxiliary left invariant
positive definite metric. We will follow a heuristic approach starting at a celebrated theorem by Marsden
(compact homogeneous semi-Riemannian manifolds are geodesically complete) and providing a variant of his proof.
Based on recent research with A. Elshafei, A.C. Ferreira and A. Zeghib (arXiv:2308.16513).
Egor Shelukhin
Title: Contact non-orderability and the Hofer metric
Abstract: We study contact non-orderability, the existence of positive contractible loops of contactomorphisms, from the
point of view of the contact Hofer metric. This provides a conceptual way to recover the previous constructions of such
loops due to Eliashberg-Kim-Polterovich and to produce previously unknown examples. We will explain our approach
and describe additional applications. This talk is based on a joint work in progress with Jakob Hedicke.
Abdelghani Zeghib
Title: Scalar Invariants of Lorentz metrics
Abstract: To a Lorentz metric one associates the set of all its scalar curvature invariants, such as the scalar curvature,
the eigenvalues of the Ricci curvature, the norm of the Riemann tensor… We address the inverse problem question,
that is whether these invariants completely determine the metric?
Slides