A new Geometry for Einstein’s Theory of Relativity & Beyond

Talks elsewhere

2026

  • Karim Mosani, \(C^0\)-inextendibility of black hole spacetimes, Theoretical Physics seminar, Theoretical Physics seminar, Raman Research Institute, Bangalore, IN, July 8, 2026.
    Abstract

    One of the central questions in causality theory is whether spacetime singularities signal a breakdown in the regularity of the metric tensor of the spacetime. This leads naturally to the next question: Can a given spacetime be extended further beyond the singularity by lowering the regularity of the metric tensor? More precisely, can it be isometrically embedded as a proper subset of a larger spacetime equipped with a less regular, say continuous, Lorentzian metric? If such an extension exists, the spacetime is called $C^0$-extendible; otherwise, it is called $C^0$-inextendible. The problem of $C^0$-(in)extendibility is closely connected to two major themes in general relativity: the geometric meaning of spacetime singularities and the formulation of the strong cosmic censorship conjecture. A breakthrough in this direction was Sbierski’s proof of the $C^{0}$-inextendibility of the maximal analytic Schwarzschild spacetime. In this talk, we will discuss how Sbierski’s method can be adapted to a broad class of warped-product black hole spacetimes with a static exterior region. These spacetimes are globally hyperbolic, have codimension-two Riemannian fibre, and possess a curvature singularity. Assuming moreover that the fibre has certain topological properties, precisely compactness, connectedness, and homogeneity, we will sketch the proof of $C^{0}$-inextendibility of such a class of spacetimes.

    This talk is based on joint work with Clemens Saemann.

  • Karim Mosani, \(C^0\)-inextendibility of warped-product black hole spacetimes, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul 01, 2026.
    Abstract

    We investigate how far Sbierski’s proof of \(C^0\)-inextendibility for Schwarzschild spacetime can be adapted to a much broader class of static warped-product black hole spacetimes with a curvature singularity as r\to 0. This class includes models with matter fields, and is therefore not restricted to the vacuum setting, while also going beyond spherical symmetry.

    This is an ongoing joint work with Clemens Saemann.

  • Inés Vega González, Singularity theorems below Lipschitz regularity, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    The singularity theorems of General Relativity establish the occurrence of spacetime “singularities”, in the sense of causal geodesic incompleteness of the spacetime manifold under certain physically reasonable conditions. These results were formulated for smooth metrics, however for the theorems to make physically meaningful predictions, one needs to extend their validity to lower regularities. In recent years there have been efforts in finding low regularity versions of the Penrose and Hawking singularity theorems, getting as low as Lipschitz Lorentzian metrics. The main goal of my research is lowering the regularity threshold of the Hawking theorem to metrics of Sovolev regularity \(W^{1,p}\), which is lower that Lipschitz regularity, with \( L^p\) bounded curvature. Our main tool is the use of RT-equations, a set of elliptic equations which allow us to raise the regularity of the metric by one derivative on a \(W^{2,p}\) related atlas. A particular issue is to find a reasonable definition of mean curvature for that specific scenario.

  • Sebastian Gieger, A synthetic Lorentzian Cartan-Hadamard theorem, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    The Cartan-Hadamard theorem is a classical result in Riemannian geometry stating that on simply connected complete Riemannian manifolds with nonpositive sectional curvature, the exponential map at any point is a global diffeomorphism. In this talk we prove a version of the Cartan-Hadamard theorem for Lorentzian length spaces with timelike curvature bounded above by 0 that are future one-connected. Along the way, we establish a conjugate point estimate and show that in this setting any timelike curve can be deformed into a timelike geodesic with the same endpoints and give some applications. The results presented are part of joint work with Darius Erös, Joe Barton, Tobias Beran, Mauricio Che, Felix Rott and Jona Röhrig

  • Miguel Prados Abad, Timelike ideal boundary of a non-positively curved Lorentzian Length Spaces, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    We study the notion of timelike ideal boundary of a Lorentzian Length Space with non-positive curvature. We establish metric curvature bounds of such an object and consider the case of generalized cones, studying the relation between the timelike ideal boundary and the metric ideal boundary of the fiber.

  • Simone Vincini, Causally convex functions and gradient flows on Lorentzian manifolds, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    Convex functions and their gradient flows are central concepts in nonsmooth analysis and nonsmooth geometry, often allowing to bypass smoothness assumptions. In positive signature, they play a big role in the study of Alexandrov spaces and spaces with Ricci curvature bounded from below; by contrast, in Lorentzian geometry no analogue seems to be available. We introduce a notion of convexity, causal convexity, tailored to Lorentzian signature and we study the analytical properties of causally convex functions on globally hyperbolic spacetimes. We then formulate two notions of gradient flow for these functions, one akin to the energy dissipation inequality in positive signature, the other encoding a differential inclusion, similar to the EVI notion. Finally, we prove general stability, existence and uniqueness results for these flows and clarify the relationship between the two notions. This is a work in collaboration with Mathias Braun, Nicola Gigli, Robert McCann and Matteo Zanardini.

  • Leonardo Garcia Heveling, Conformal transformations of spacetimes without observer horizons, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    The “no observer horizons” (NOH) condition on a spacetime, inspired by cosmology, means that there is a single point in the future/past causal boundary. Conformal transformations of such spacetimes are quite rigid. In a recent preprint, we classified them into two types: those that send (necessarily all) points towards infinity, and those that preserve compact sets. We will discuss this result and its implications for the Einstein static universe (which has a large conformal group), and formulate a version of the Lichnerowicz conjecture for NOH spacetimes. The latter states that, up to conformal equivalence, the only NOH spacetime with conformal group strictly larger than its isometry group is the Einstein static universe.

    Based on joint work with Abdelghani Zeghib.

  • Marta Sálamo Candal, Comparison theory for Lipschitz spacetimes, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    In this talk, I will present comparison theorems for Lipschitz spacetimes in sharp form: d’Alembert, timelike Brunn–Minkowski, timelike Bishop–Gromov, and timelike Bonnet–Myers. The three first results are obtained as a consequence of our main theorem, in which we show that a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The timelike Bonnet–Myers is obtained parallely using the localization technique from convex geometry. Our framework covers a remarkable class of examples of spacetimes, including impulsive gravity waves, thin shells, and matched spacetimes. This talk is based on joint work with Mathias Braun.

  • Davide Manini, On the geometry of synthetic null hypersurfaces and the Null Energy Condition, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    In this talk, I will present a joint work with Fabio Cavalletti (Milan) and Andrea Mondino (Oxford), where we develop a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple (H, G, m): H is a closed achronal set in a topological causal space, G is a gauge function encoding affine parametrizations along null generators, and m is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the synthetic null energy condition (NC^e(N )), defined via the concavity of an entropy power functional along optimal transport, with parameterization given by the gauge G. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of (\(NC^e(N )\) is the stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov–Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking’s area theorem. Moreover, we extend the celebrated Penrose singularity theorem to continuous spacetimes and we prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic null energy condition.

  • Roland Steinbauer, Singularity Theorems Beyond \(C^2\) : A Status Report, XII International Meeting in Lorentzian Geometry, Instituto Superior Técnico, Lisboa, PT, Jul XX, 2026.
    Abstract

    With their singularity theorems, Penrose and Hawking have firmly established singularity formation as a generic feature of General Relativity. However, the original proofs rely on the metric tensor being at least \(C^2\)-regular, which undermines the physical significance of the results: incompleteness might be avoided by a physically harmless drop in regularity, and the theorems might merely predict a rough, but otherwise nonsingular, geometry. While this caveat has long been known, it is only during the last decade that real progress has been made in lowering the regularity in the singularity theorems. In this talk we summarize these developments, which build on (a) extensions of causality theory and (b) extensions of focusing estimates for causal geodesics under (c) distributional or synthetic curvature bounds / energy conditions. Our focus will be on two recently established versions of the Hawking theorem. The first, for Lipschitz metrics, rests on a worldvolume estimate derived from a segment-type inequality, which bounds the volume of the set of points on a spacelike surface from which long maximisers emanate. The second, for \(W^{1,p}\)-metrics with \(L^p\)-bounded curvature, relies on the RT-regularisation, which uses elliptic theory to raise the regularity of the metric by one derivative. Both approaches employ manifold convolution to regularise the metric, combined with Friedrichs-type estimates controlling the curvature of the regularised metric in terms of the distributional curvature.

  • Omar Zoghlami, Geometric and causal properties of the sub-Lorentzian Heisenberg group, Math Seminar, University of Trento, Math department, May 19, 2026.
    Abstract

    In this talk we present the sub-Lorentzian geometry of the Heisenberg group \(\mathbb{H}\), the simplest non-holonomic analogue of a spacetime. After outlining motivations that show why such structure is a compelling object of study, we analyze the causal structure and present a characterization of geodesics of the space via a planar Lorentzian isoperimetric problem. We further explain geometrically why this space does not admit standard synthetic Ricci curvature bounds. If time permits, we will also discuss its Lorentzian Hausdorff dimension which we compute to be \(4\), analogously to the usual Hausdorff dimension of the space and therefore in contrast with the topological dimension of the manifold which is \(3\).

    This talk is based on joint work with Chiara Rigoni and Samuël Borza.

  • Samuël Borza, Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory, Analysis on Metric Space Seminar, Okinawa Institute of Science and Technology Graduate University (OIST), JP, Feb 25, 2026.
    Abstract

    This talk will explore some aspects of non-smooth Lorentzian geometry, the mathematical framework underlying Einstein’s general relativity, which is currently being developed. Just as metric length spaces provide a synthetic generalisation of smooth Riemannian manifolds, the time-separation function plays the role of a “distance” in Lorentzian geometry. The need for a non-smooth Lorentzian framework appeared early on, most famously with Penrose’s singularity theorems. After introducing the basic concepts and some initial results in this synthetic setting, we will turn to causal set theory, a radical approach to quantum gravity in which spacetime is modelled as a discrete causal graph. I will formulate a new notion of curvature, inspired by Ollivier-Ricci curvature on metric graphs, using optimal transport between causal diamonds. We will see that it does recover Ricci curvature on smooth Lorentzian manifolds, and numerical examples will be presented.

  • Samuël Borza, Ollivier-Ricci Curvature in Non-Smooth Lorentzian Geometry and Causal Set Theory, Causal Set Seminar, Dublin Institute for Advanced Studies, IE (online), Feb 5, 2026.
    Abstract

    This talk will explore some aspects of non-smooth Lorentzian geometry, the mathematical framework underlying Einstein’s general relativity, which is currently being developed. Just as metric length spaces provide a synthetic generalisation of smooth Riemannian manifolds, the time-separation function plays the role of a ‘distance’ in Lorentzian geometry. The need for a non-smooth Lorentzian framework appeared early on, most famously with Penrose’s singularity theorems. After introducing the basic concepts and some initial results in this synthetic setting, we will turn to Causal Set Theory, a radical approach to quantum gravity in which spacetime is modelled as a discrete causal graph. I will formulate a new notion of curvature, inspired by Ollivier-Ricci curvature on metric graphs, using optimal transport between causal diamonds. We will show that it does recover Ricci curvature on smooth Lorentzian manifolds. If time permits, we will also present numerical examples.

    This is a joint work with Jona Röhrig and Joe Barton.

2025

  • Raquel Perales, Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into \(ℝ\)n, Analysis Seminar, University of Pisa, Dec 10, 2025.
  • Samuël Borza, Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory, Department of Mathematics & Statistics Colloquia 2025/26, Maynooth University, IE, Dec 10, 2025.
    Abstract

    This talk will explore some aspects of non-smooth Lorentzian geometry, the mathematical
    framework underlying Einstein’s general relativity, which is currently being developed. Just
    as metric length spaces provide a synthetic generalisation of smooth Riemannian manifolds,
    the time-separation function plays the role of a ‘distance’ in Lorentzian geometry. The need
    for a non-smooth Lorentzian framework appeared early on, most famously with Penrose’s
    singularity theorems. After introducing the basic concepts and some initial results in this
    synthetic setting, we will turn to Causal Set Theory, a radical approach to quantum gravity in
    which spacetime is modelled as a discrete causal graph. I will formulate a new notion of
    curvature in the spirit of Ollivier-Ricci curvature, using optimal transport between causal
    diamonds (Alexandrov intervals).

  • Raquel Perales, A Compactness Theorem for the Intrinsic Timed-Hausdorff Distance, UK Metric Geometry & Analysis Network, Cardiff University, Dec 3, 2025.
  • Samuël Borza, Ollivier-Ricci curvature in non-smooth Lorentzian geometry and causal set theory, Geometry and Topology Seminar, Durham University, UK, Dec 4, 2025.
  • Davide Manini, On the geometry of synthetic null hypersurfaces and the Null Energy Condition, , University of Hamburg, Nov 19, 2025. Slides
    Abstract

    In the talk, I will present a joint work with Fabio Cavalletti (Milan) and Andrea Mondino (Oxford), where we develop a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple \((H, G, m)\): \(H\) is a closed achronal set in a topological causal space, \(G\) is a gauge function encoding affine parametrizations along null generators, and m is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the synthetic null energy condition (\(NC^e(N )\)), defined via the concavity of an entropy power functional along optimal transport, with parameterization given by the gauge \(G\). This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of \(NC^e(N )\) is the stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov–Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking’s area theorem. Moreover, we extend the celebrated Penrose singularity theorem to continuous spacetimes and we prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic null energy condition.

  • Davide Carazzato, A strong quantitative isoperimetric inequality for a capillarity problem, GSSI Analysis & PDE, Gran Sasso Science Institute (GSSI), L’Aquila, Nov 25, 2025.
    Abstract

    During the seminar, we will introduce a stronger version of the quantitative isoperimetric inequality, originally developed by Fusco and Julin. Building on that, we will arrive to the analogous inequality for a capillarity problem using the so-called selection principle, based on the regularity theory for the perimeter functional. We will also highlight the difficulties that arise when we apply Fusco and Julin’s method to our situation. This result was obtained in collaboration with Giulio Pascale and Marco Pozzetta.

  • Karim Mosani, Geometry and Topology of Trapped Photon Region in Kerr-Newman and Kerr-Sen Spacetime, Programme Geometry and Convergence in Mathematical General Relativity, Simons Center for Geometry and Physics, Stony Brook, NY, US, Oct 2, 2025. Video
    Abstract

    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)

  • Raquel Perales, Rigidity of mass-preserving 1–Lipschitz maps from integral current spaces into Euclidean space, Programme Geometry and Convergence in Mathematical General Relativity, Simons Center for Geometry and Physics, Stony Brook, NY, US, Sep 15 — 19, 2025.
    Abstract

    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.

  • Roland Steinbauer, Generalizing the Penrose Cut-and-Paste Method: Null Shells with Pressure and Energy Flux., Programme Geometry and Convergence in Mathematical General Relativity, Simons Center for Geometry and Physics, Stony Brook, NY, US, Sep 9, 2025. Video
    Abstract

    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.

  • Raquel Perales, Mini-course: General Relativity and Intrinsic Flat Convergence: Almost Rigidity of the Positive Mass Theorem (Part 2), Programme Geometry and Convergence in Mathematical General Relativity, Simons Center for Geometry and Physics, Stony Brook, NY, US, Sep 4 2025. Slides – Video
    Abstract

    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.

  • Davide Carazzato, Characterization of the maximizers of an exterior optimal transport problem, Minisymposium “Optimal Transport and Applications” at the “ÖMG-DMV Annual Meeting” (Linz), Johannes Kepler University Linz, Sep 03, 2025.
    Abstract

    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.

  • Raquel Perales, Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into \(ℝ\)n, Analysis Seminar, University of Warwick, Jun 19, 2025.
  • Samuël Borza, Measure contraction properties for sub-Riemannian structures beyond step 2, The London Geometry and Topology seminar, Imperial College London, UK, May 2, 2025.
    Abstract

    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.

  • Samuël Borza, Measure contraction properties for sub-Riemannian structures beyond step 2, Analysis seminars, Loughborough University, UK, Apr 30, 2025.
    Abstract

    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.

  • Davide Carazzato, A partial optimal transport functional in an isoperimetric problem, Seminari di Matematica Applicata, University of Pavia, Mar 19, 2025.
    Abstract

    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.

2024

  • Roland Steinbauer, Synthetic curvature for GR and beyond, 5th EPS Conference on Gravity, Prague, Dec 10, 2024. Slides
    Abstract

    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.

  • Karim Mosani, Geometry and topology of trapped photon region in Kerr-Newman and Kerr-Sen spacetimes, The 33rd Workshop on General Relativity and Gravitation in Japan, Osaka, Dec 2–6, 2024.
    Abstract

    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).

  • Samuël Borza, Failure of the Measure Contraction Property on the Martinet Flat structure, OxPDE Lunchtime Seminar, University of Oxford, UK, Nov 21, 2024.
    Abstract

    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.