Talks elsewhere

Consider the trapped photon region in the domain of outer communication of sub-extremal Kerr spacetime. Cederbaum and Jahns proved that the canonical projection of this trapped photon region in the (co-)tangent bundle is a five-dimensional submanifold of topology \(SO(3)\times \mathbb{R}^2\). By adapting the latter’s methodology, we generalize this result to two stationary axisymmetric classes of spacetimes admitting black hole horizon, namely Kerr-Newman spacetime and Kerr-Sen spacetime. The former is a solution of Einstein-Maxwell equations, while the latter is a solution of Einstein-Maxwell-Dilaton-Axion equations. These include both sub-extremal and extremal cases. The result has potential applications in various areas of mathematical relativity, like black hole uniqueness theorems, black hole dynamical stability, gravitational lensing, and black hole shadows. (This is a joint work with Carla Cederbaum)

The cut-and-paste method is a procedure for constructing null thin shells by matching
two regions of the same spacetime across a null hypersurface. Originally proposed by Penrose, it
has so far allowed to describe purely gravitational and null-dust shells in constant-curvature
backgrounds. In this paper, we extend the cut-and-paste method to null shells with arbitrary
gravitational/matter content. To that aim, we first derive a locally Lipschitz continuous form of
the metric of the spacetime resulting from the most general matching of two constant-curvature
spacetimes with totally geodesic null boundaries, and then obtain the coordinate transformation
that turns this metric into the cut-and-paste form with a Dirac-delta term. The paper includes an
example of a null shell with non-trivial energy density, energy flux and pressure in Minkowski
space. https://arxiv.org/abs/2508.00231 This is joint work with Miguel Manzano and Argam
Ohanyan.

We will survey applications of intrinsic flat convergence to questions arising in
mathematical general relativity with a focus on an open conjecture of Lee-Sormani on the Geometric
Stability of the Positive Mass Theorem. We will survey progress towards this conjecture in work of
Huang, Lee, Stavrov, Allen, Perales, and others.

We study an energy defined through an optimal transport problem between the uni-
form density on a set and the uniform density on the complement of that set. This
quantity captures how much the set is concentrated, and we prove that the maxi-
mizers coincide with a ball in many relevant cases. This is achieved by showing the
monotonicity of that functional with respect to a certain symmetrization procedure.
This talk is based on a joint work with Almut Burchard and Ihsan Topaloglu.

I will introduce Carnot groups, and their quotients, as metric measure spaces. These are examples of Carnot–Carathéodory spaces, or sub-Riemannian manifolds, and include the Heisenberg group, the Engel group, and the Martinet flat structure, to name but a few. Once we have a good grasp of these geometric structures, I will outline some open problems in the field before shifting focus to the study of curvature and the so-called metric contraction properties. These analytic inequalities aim to define, in a synthetic way, a lower bound on the Ricci curvature. The new results I will present show how these properties can be preserved by taking quotients, and how this affects their validity or failure. This is joint work with Luca Rizzi from SISSA.

In this talk, I will introduce Carnot groups and their quotients as metric measure spaces. These are examples of Carnot–Carathéodory spaces, or sub-Riemannian manifolds, and include the Heisenberg group, the Engel group, and the Martinet flat structure, to name but a few. Once we have a good grasp of these geometric structures, I will outlinesome open problems in the field before shifting focus to the study of curvature and the so-called metric contraction properties. These analytic inequalities aim to define, in a synthetic way, a lower bound on Ricci curvature. The new results I will present show how these properties can be preserved by taking quotients, and how this affects their validity or failure. This is joint work with Luca Rizzi from SISSA.

We consider a shape optimization problem of isoperimetric type, containing a repulsive term defined in term of an optimal transport problem. We provide a brief introduction about the basic geometric properties of the partial optimal transport term, and we focus on its maximizers to sustain the intuition about its repulsive nature. In fact, through a symmetrization technique, we show that the unique maximizer of the optimal transport term is the ball, in complete competition with the perimeter term. This is based on a joint work with Almut Burchard and Ihsan Topaloglu.

Synthetic methods have profoundly transformed Riemannian geometry in recent decades, extending core concepts and results beyond smooth manifolds. More precisely, bounds on sectional and Ricci curvature, using triangle comparison and optimal transport respectively, have proven to be so robust as to be independent of any differentiable structure. Recently, the foundations for an analogous synthetic Lorentzian geometry have been laid based on the core notion of Lorentzian length spaces. This talk explains the basics of this new geometry, outlines initial results, and explores potential applications in general relativity and discrete approaches to quantum gravity.

Consider the trapped photon region in the domain of outer communication of sub-extremal Kerr spacetime. Cederbaum and Jahns (Gen.Relativ.Gravit, 51, 79, 2019) proved that the canonical projection of this trapped photon region in the (co-)tangent bundle is a five-dimensional submanifold of topology \(SO(3)\times \mathbb{R}^2\). By adapting the latter’s methodology, we generalize this result to two stationary axisymmetric classes of spacetimes admitting black hole horizon, namely Kerr- Newman spacetime and Kerr-Sen spacetime. The former is a solution to Einstein-Maxwell equations, while the latter is a solution to Einstein-Maxwell-Dilaton-Axion equations. These include both sub-extremal and extremal cases. The result has potential applications in various areas of mathematical relativity, like black hole uniqueness theorems, black hole dynamical stability, gravitational lensing, black hole shadows, and cosmic censorship conjecture. (This work is in collaboration with Cederbaum).

The Martinet flat structure is one of the simplest sub-Riemannian manifolds that has many non-Riemannian features: it is not equiregular, it has abnormal geodesics, and the Carnot-Carathéodory sphere is not sub-analytic. I will review how the geometry of the Martinet flat structure is tied to the equations of the pendulum. Surprisingly, the Measure Contraction Property (a weak synthetic formulation of Ricci curvature bounds in non-smooth spaces) fails, and we will try to understand why. If time permits, I will also discuss how this can be generalised to some Carnot groups that have abnormal extremals. This is a joint work in progress with Luca Rizzi.