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

Talks in Vienna

Most of the talks of our members take place in the framework of the DIANA seminar.

  • Stefano Vincini, Wasserstein gradient flows, DIANA seminar, University of Vienna, Nov 21, 2025.
    Abstract

    “Felix Otto’s pioneering work “The geometry of dissipative evolutions” introduced a geometric perspective on the porous medium equation, interpreting it as a gradient flow in the space of absolutely continuous probability measures equipped with the Wasserstein metric. This insight led to the rigorous framework developed by Ambrosio, Gigli, and Savaré, which formalizes Wasserstein gradient flows and extends Otto’s asymptotic estimates to a broader class of dissipative equations.

    This talk is meant to be an overview of the key aspects of this theory, starting with gradient flows in Hilbert spaces as a motivation and heuristics. The presentation will focus on the three necessary ingredients to define Wasserstein gradient flows: the notion of “tangent” to an (absolutely continuous) curve of measures, the displacement convexity of the functional and the “Wasserstein” subdifferential calculus. The aim is to revisit Otto’s estimates from this abstract framework. Time permiting, we will discuss the Benamou-Brenier formula, and its link to the characterization of tangent vectors in the Wasserstein space.”

  • Roland Steinbauer, Synthetic Lorentzian geometry, Gravitational Physics Literature Seminar, University of Vienna, Nov 19, 2025.
    Abstract

    Spacetimes arising in general relativity often exhibit non-smooth features that challenge traditional differential geometric approaches to Lorentzian geometry. In Riemannian geometry, synthetic approaches, especially triangle comparison and optimal transport methods, have extended curvature concepts beyond smooth manifolds. Recent work has established the foundations for an analogous synthetic Lorentzian geometry based on the fundamental notion of Lorentzian length spaces. These spaces capture the essential causal structure of spacetime without requiring smoothness or a manifold structure at all. In this talk, we explain the basics of this new geometry, outline initial results including comparison theorems and convergence results, and explore potential applications in general relativity and discrete approaches to quantum gravity.

  • Roland Steinbauer, A new Lorentzian geometry, Mathematisches Kolloquium, inaugural lecture, University of Vienna, Nov 12, 2025.
    Abstract

    Lorentzian geometry is the mathematical language of General Relativity (GR), Einstein’s theory of space, time, and gravity. However, the geometries arising in GR often exhibit non-smooth features that challenge traditional differential geometric approaches to Lorentzian geometry. In its sister theory, Riemannian geometry, synthetic approaches using triangle comparison and optimal transport methods, have led to defining curvature bounds in a very general setting, allowing for non-smoothness or even the absence of a differential structure at all. However, in the Lorentzian setting, the absence of a metric structure has for a long time blocked a similar development. Only recently, the foundations for an analogous synthetic Lorentzian geometry, based on the fundamental notion of Lorentzian length spaces, have emerged. These spaces capture the essential causal structure of spacetime without requiring smoothness or a manifold structure at all. In this talk, we explain the basics of this new geometry, outline initial results including comparison theorems and
    convergence results, and explore potential applications in GR and discrete approaches to quantum gravity.

  • Davide Manini, On the geometry of synthetic null hypersurfaces and the Null Energy Condition, DIANA seminar, University of Vienna, Nov 7, 2025.
    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 (\(N\))\(C^{e}\)(\(N\)), 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 (\(N\))\(C^{e}\)(\(N\)) 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.

  • Tobias Beran, Coordinates for Lorentzian CBB – an overview, DIANA seminar, University of Vienna, Oct 24, 2025.
    Abstract

    (joint work with John Harvey, Felix Rott and Clemens Sämann) I will define strainers and the corresponding coordinate map, and show it is continuous and open. If this map is not a local homeomorphism, a way of increasing of the dimension of the strainer is presented. This then gives a coordinate theorem for finite dimensional LLS with CBB: near each point there is either an open set homeomorphic to \(\mathbb{R}^n\), or a nested sequence of open sets and corresponding sequence of strainers (which one should interpret as the space being infinite dimensional). If time permits, I will show that the time separation function lies between two flat time separation functions, making the coordinate map weakly bi-Lipschitz.

  • Inés Vega González, Introduction to Lorentzian Geometry, KISS Seminar Institut für Diskrete Mathematik und Geometrie, Institut für Diskrete Mathematik und Geometrie, TU Wien, Oct 8, 2025.
  • Luca Mrini, Non-perturbative spectral gap bounds from synthetic geometry in Hamiltonian systems, DIANA seminar, University of Vienna, Jun 27, 2025.
    Abstract

    From the simple harmonic oscillator to the Standard Model of particle physics, many quantum systems exhibit a finite gap between the ground state energy and that of the first excited state. This spectral gap can be traced back to the Bakry-Émery Laplacian—the natural self-adjoint, second-order differential operator on weighted Riemannian manifolds. The existence of such a gap reflects deep non-perturbative features of a quantum theory, including vacuum stability, exponential suppression of forces at large distances, and the absence of arbitrarily light particle excitations. It also plays a central role in the unsolved Millennium Prize problem: Yang–Mills & the Mass Gap.
    In recent years, multiple authors (Moncrief, Marini, Maitra 2019; Mondal 2023) have pursued a geometric approach to spectral gap estimates by establishing lower bounds on the Bakry–Émery Ricci curvature. However, many physically significant systems—including Yang–Mills theory—involve non-smooth geometries arising from quotienting by symmetry group actions. In this talk, I explore how synthetic geometric techniques can be used to study the spectral gap in such non-smooth, symmetry-reduced systems. I begin with the classical Lichnerowicz bound and the harmonic oscillator as motivating examples, then explain the procedure of symmetry reduction in Hamiltonian systems, and conclude by reviewing synthetic results on spectral gap estimates and quotient spaces.

  • Jona Röhrig, Space of directions in Lorentzian length space under CBA, DIANA seminar, University of Vienna, Jun 13, 2025.
    Abstract

    In my seminar talk, I will begin by providing a brief recap of some essential definitions from the field of Lorentzian length spaces, including angles and triangle and quadruple curvature bounds. Subsequently, I will introduce the concept of the space of directions and the tangent cone. We will then demonstrate that under upper curvature conditions, the space of directions is a geodesic length space with a curvature bound above of -1. If time permits, we will also explore an example to illustrate the consequences of imposing lower curvature bounds instead of upper ones.

  • Davide Carazzo, A strong quantitative isoperimetric inequality for a capillarity problem, Vienna Geometric Analysis Seminar, University of Vienna, Jun 11, 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. The central theme of the seminar is the so-called selection-principle, that is 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 is part of an ongoing collaboration with Giulio Pascale and Marco Pozzetta.

  • Miguel Prados Abad, Causal completions, DIANA seminar, University of Vienna, Jun 6, 2025.
    Abstract

    The original idea behind the causal completion of a spacetime is to attach “ideal points” so that every timelike curve has an endpoint. In their seminal 1972 paper, Geroch, Kronheimer and Penrose constructed such a completion using the set of all indecomposable past and future sets, along with certain natural identifications. However, their proposal exhibited undesirable properties in specific spacetimes and thus could not be considered definitive. Since then, various authors have introduced refinements—some in synthetic settings—to address these shortcomings. In this talk, we will survey these developments, focusing on the motivations behind each successive modification.

  • Karim Mosani, C^0-inextendibility, DIANA seminar, University of Vienna, Jun 6, 2025.
    Abstract

    We will discuss a subclass of FLRW spacetimes with negative spatial curvature \(K = -1\) and specific conditions on the scale factor. We will prove that under these conditions, the spacetime cannot be continuously extended beyond the Big Bang singularity. This establishes the \(C^0\)-inextendibility of the spacetime to the past. The result and its proof are due to Jan Sbierski.

  • Sebastian Gieger, A synthetic Lorentzian Cartan-Hadamard Theorem, DIANA seminar, University of Vienna, May 23, 2025.
    Abstract

    I will present a synthetic-Lorentzian analog of the generalized Cartan-Hadamard theorem for locally convex metric spaces established by S. Alexander and R. Bishop. This entails introducing an appropriate definition of local concavity for Lorentzian pre-length spaces as well as proving the existence and uniqueness of timelike geodesics between any pair of timelike-related points under the additional assumptions of global hyperbolicity and future one-connectedness. As a consequence we get globalization results for this notion of concavity and for upper timelike curvature bounds by \( k \geq 0\).

  • Luca Benatti, The Riemannian Penrose Inequality. A unified perspective on two approaches, Vienna Geometric Analysis Seminar, University of Vienna, May 21, 2025.
    Abstract

    The Penrose inequality is a conjecture stating that the total mass of an asymptotically flat spacetime is at least as large as the mass of the black holes it contains. In the Riemannian setting — that is, for time-symmetric spacetimes — this inequality has been proven through various methods. A notable approach by Huisken and Ilmanen (2001) relies on the monotonicity of Hawking mass along inverse mean curvature flow. More recently, Agostiniani, Mantegazza, Mazzieri, and Oronzio (2022) provided a proof using monotonicity formulas in the framework of nonlinear potential theory. However, both techniques require stronger assumptions on the metric than those needed to state the problem. In this talk, I will present a unified perspective on the two approaches. This synergy allows to extend the applicability of this inequality to more general settings.
    This talk is based on a series of joint papers with with M. Fogagnolo (UNIPD), L. Mazzieri (UNITN), A. Pluda (UNIPI) and M. Pozzetta (UNIMI).

  • Davide Carazzo, A strong quantitative isoperimetric inequality for a capillarity problem, DIANA semiar, University of Vienna, May 16, 2025.
    Abstract

    Quantitative inequalities have been intensively studied in the past years since they provide refined information about some relevant functionals. 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. The central theme of the seminar is the so-called selection-principle, that is a rather general approach which could be applied to different variational problems. This result is part of an ongoing collaboration with Giulio Pascale and Marco Pozzetta.

  • Samuël Borza, Sub-Lorentzian geometry on the Martinet structure, DIANA seminar, University of Vienna, May 9, 2025.
    Abstract

    I will review the basics of sub-Lorentzian geometry with a particular focus on abnormal extremals arising from Pontryagin’s Maximum Principle. I will explain how these appear quite naturally in the Lorentzian setting, even in Minkowski spaces. Then we will study the Martinet distribution in more detail, the first example of a sub-Lorentzian manifold in which abnormal curves play an especially significant role.

  • Miguel Manzano Rodríguez, The Formalism of Hypersurface Data: A New Approach to Spacetime Matching, Diana seminar, University of Vienna, Apr 11, 2025.
    Abstract

    The Formalism of Hypersurface Data, originally introduced by M. Mars (10.1007/s10714-013-1579-9, 10.1007/s00009-020-01608-1) is a framework for studying the geometry of hypersurfaces as independent manifolds, i.e., without requiring them to be embedded in any ambient space. This unconventional approach has turned out to be extremely useful in a variety of contexts, including initial value problems (such as the characteristic problem or the Killing initial data problem), the matching of spacetimes, the geometry of horizons, and perturbations of hypersurfaces. In this talk, I will introduce the fundamental concepts and key aspects of the hypersurface data formalism, and discuss how its application to the issue of spacetime matching has led to significant results, particularly in the case of matching across null boundaries.

  • Omar Zoghlami, Isoperimetric characterization of geodesics in the sub-Lorentzian Heisenberg group, DIANA seminar, University of Vienna, Apr 4, 2025.
    Abstract

    After a brief overview of the Heisenberg group and its sub-Lorentzian structure, I will give a description of how to compute the geodesics of this space by means of an isoperimetric problem in the Minkowski plane. Such problem is completely analogous to Dido’s problem in the Riemannian setting and allows us to find geodesics without relying on the Pontryagin’s Maximum Principle.

  • Inés Vega González, Integrated local energy decay estimates for solutions to the wave equation in the black hole exterior of sub-extremal Reissner-Nordström-de Sitter spacetimes – part 2, DIANA seminar, University of Vienna, Mar 28, 2025.
    Abstract

    Consider a non-rotating spherically symmetric charged black hole with mass \(M > 0\) and a charge \(Q \neq 0\), in a de Sitter background of positive curvature \(\Lambda\). Taking solutions to geometric wave equations on the exterior region of this black hole, we use a physical-space-based method for deriving the leading-order late-time behaviour of integrated local energy decay estimates of solutions.
    These estimates could be used for deriving the precise leading-order late-time behaviour of asymptotics and energy decay. Our method relies on exploiting the spatial decay properties of time integrals of solutions. With them, we are able to derive the existence and precise genericity properties of energy of the solutions and obtain uniform decay estimates of local energy in time.

  • Luca Benatti, Taming geometric inequalities using partial differential equations, DIANA seminar, University of Vienna, Mar 21, 2025.
    Abstract

    “The ball is the only domain in Euclidean space that minimizes area for a given volume.” This simple statement encapsulates the full essence and power of geometric inequalities: the (scale-invariant) ratio between an object’s area and volume is always bounded from below by a dimensional constant. By measuring it, I can determine whether the object is a ball.
    But what if the space around us is curved? And what if we consider other geometric quantities, such as the mean curvature? Through some selected examples, I will introduce a powerful PDE-based technique that has been successfully applied to address these questions.

  • Alessio Vardabasso, Distributional and synthetic curvature bounds for non-smooth Riemannian metrics, DIANA seminar, University of Vienna, Mar 14, 2025.
    Abstract

    he Cartan-Alexandrov-Toponogov Theorem (sometimes just Toponogov’s Theorem) is a cardinal theorem in comparison theory for smooth Riemannian manifolds. In short, it relates uniform bounds on the sectional curvature of a Riemannian manifold to specific inequalities on the internal distances and angle widths of geodesic triangles when compared to similar triangles in two-dimensional “model spaces” – spheres, hyperbolic spaces and the euclidean plane. While most of the theory of smooth Riemannian geometry comfortably works for metrics of regularity \(C^2\), below that threshold one must resort to distributional derivatives and other non-smooth tools to define important objects such as curvature tensors. A natural question then arises: which theorems still hold at these lower regularities? In this talk, after an introduction on distribution theory on manifolds, we will see how we can partially generalize the C.A.T. Theorem to metrics of \(C^1\) and Lipschitz regularity.

  • Inés Vega González, Integrated local energy decay estimates for solutions to the wave equation in the black hole exterior of sub-extremal Reissner-Nordström-de Sitter spacetimes, DIANA seminar, University of Vienna, Jan 17, 2025.
    Abstract

    Consider a non-rotating spherically symmetric charged black hole with mass \(M > 0\) and a charge \(Q \neq 0\), in a de Sitter background of positive curvature \(\Lambda > 0\). Taking solutions to geometric wave equations on the exterior region of this black hole, we use a physical-space-based method for deriving the leading-order late-time behaviour of integrated local energy decay estimates of solutions.
    These estimates could be used for deriving the precise leading-order late-time behaviour of asymptotics and energy decay. Our method relies on exploiting the spatial decay properties of time integrals of solutions. With them, we are able to derive the existence and precise genericity properties of energy of the solutions and obtain uniform decay estimates of local energy in time.

  • Joe Barton, The Circle Method, DIANA seminar, University of Vienna, Jan 1, 2025.
    Abstract

    The circle method is a powerful tool of analytic number theory developed by Hardy, Littlewood and Ramanujan in the early 20th century. In this talk, I will develop the ideas of the circle method through Waring’s problem. In a letter to Lagrange, Edward Waring claimed that ‘every positive integer can be represented as the sum of 4 squares, 9 cubes, 19 fourth powers, and so on…’ . Hardy and Littlewood used the circle method to find an asymptotic formula for the number of solutions Waring’s problem by turning the additive problem into an analytical problem.

  • Samuël Borza, Sub-Lorentzian geometry and optimal transport on the Heisenberg group, DIANA seminar, University of Vienna, Dec 12, 2024.
    Abstract

    I will explain the basics of sub-Lorentzian geometry, a little-studied theory, through one of the simplest examples: the three-dimensional Heisenberg group. Roughly speaking, the geometry of this group is controlled by curves that are allowed to travel only in two out of three directions, and a Lorentzian metric defined on these preferred directions allows us to compute the time-separation between events. We will particularly focus on placing the Heisenberg group within the broader context of non-smooth Lorentzian length spaces. Finally, I will formulate the Lorentzian optimal transport problem and present a version of Brenier’s theorem. This talk will be based on a joint work with Wilhelm Klingenberg and Patrick Wood.

  • Miguel Prados Abad, Spaces of geodesics of a spacetime: contact and symplectic structures, DIANA seminar, University of Vienna, Nov 29, 2024.
    Abstract

    Under reasonable assumptions, the space \(C\) of causal geodesics of a spacetime (\(M, g\)) of dimension \(n\) inherits from \(T^*M\) a natural structure of smooth manifold with boundary whose interior \(\mathrm{int}\,\mathcal{C}=\mathcal{M}\), the set of timelike geodesics, has a natural symplectic structure, and whose boundary \(\partial \mathcal{C}=\mathcal{N}\), the space of lightlike geodesics, is a conformal object that has a natural structure of contact manifold. The interest on these spaces is partly due to some results which enable to study causality in \(M\) in terms of skies in \(N\) by means of the so-called Legendrian linking. The purpose of this talk will be to introduce these symplectic and contact structures and, time permitting, explore the relationship between them. To that end, we will first show that the sets concerned are actually manifolds, using certain classes of Jacobi fields to describe their tangent spaces. Next, we will build the canonical symplectic and contact structures on them and show that, for strongly causal spacetimes, \(M\) is a conformal symplectic filling of \(N\).

  • Marta Sálamo Candal, Lagrangian surfaces, DIANA seminar, University of Vienna, Nov 15, 2025.
    Abstract

    A symplectic manifold is a pair (\(X^{2n}\), \(\omega\)), where \(X^{2n}\) is a smooth manifold and \(\omega\) is a differential 2-form such that \(d\)\(\omega\) = 0 and \(\omega\)\(^{n}\) > 0, known as the symplectic form. This simple definition gives rise to a broad area in geometry and topology with many connections to other disciplines such as classical mechanics, low-dimensional topology or algebraic and complex geometry. Among the many objects that one can study in this setting, we find the Lagrangian submanifolds. These are those submanifolds \(L\) of half the ambient dimension on which the symplectic form vanishes identically on each tangent space of \(L\). The study of Lagrangian submanifolds is a central topic in symplectic topology that can tell us a great deal about the symplectic manifold (\(X\), \(\omega\)). There are many interesting questions one can ask about Lagrangian submanifolds. In this work, we will study one of these.
    We study the minimal genus question for a symplectic rational 4-manifold (\(X\), \(\omega\)), which asks, for a given \(A\) \(\in\) \(H\)\(_2\)(\(X\);\(\mathbb{N}\)\(_2\)), what are the possible topological types of non-orientable Lagrangian surfaces in the class \(A\); and specially, what is the maximal Euler number, or, equivalently, the minimal genus. We start by ensuring that 2-homology classes can be represented by a non-orientable surface. Next, we are able to proof that, when having a symplectic structure in our manifold, these surfaces representing the homology classes can be taken to be non-orientable embedded Lagrangians. In this setup, the minimal genus question arises, and we study a partial answer to this question for rational 4-manifolds. We will see that if a homology class \(A\) \(\in\) \(H\)\(_2\)(\(X\);\(\mathbb{N}\)\(_2\)) is realised by a non-orientable embedded Lagrangian surface , then \(\mathcal{P}(A)=\chi(L)\ \operatorname{mod}\ 4\), where \(\mathcal{P}(A)\) is the Pontrjagin square of \(A\). We will briefly discuss the problem for the zero class, and prove the main result of the essay for non-zero classes, which states the reciprocate for some symplectic structures in the case of rational 4-manifolds.

  • Karim Mosani, Geometry and topology of trapped photon region in stationary axisymmetric black hole spacetimes, DIANA seminar, University of Vienna, Oct. 25, 2024.
    Abstract

    In Schwarzschild spacetime with positive mass \(M\) , there exist (unstable) circular orbits of trapped null geodesics at the Schwarzschild radius \(r\) = 3\(M\), outside the black hole horizon radius \(r\) = 2\(M\), at . These orbits fill a three-dimensional submanifold \(S^{2}\) x \(\mathbb{R}\) called the photon sphere of the Schwarzschild spacetime. In general, a region in spacetime that is a union of all trapped null geodesics is called the Trapped Photon Region (TPR) of spacetime. In this seminar, we will consider three models of stationary, axisymmetric (sub-extremal and extremal) black hole spacetimes: Kerr, Kerr-Newman, and Kerr-Sen. We will see that, unlike the TPR of Schwarzschild spacetime, the TPR in such spacetimes is not a submanifold of the spacetime in general. However, its canonical projection in the (co-)tangent bundle is a five-dimensional submanifold of topology \(S\)\(0\)(3) x \(\mathbb{R}^{2}\) . This result has potential applications in various problems in mathematical relativity. The talk is based on the paper by Cederbaum and Jahns (2019), where they prove the result in Kerr spacetime, and by Cederbaum and myself (under preparation), where we extend this result to the remaining two abovementioned spacetimes.