LOGOS Seminar

I will introduce several notions of timelike curvature bounds in Lorentzian length spaces and prove their equivalence. I will also introduce the new four-point condition and prove its equivalence. Let's see ho far I will get...

Since the introduction of Ricci flow by Hamilton in 1982, it has been a fundamental problem to understand the evolution of metrics and their curvature properties under the flow. While positive scalar curvature and 2-positive curvature operator are preserved in all dimensions, there exist infinitely many dimensions where certain curvature conditions lying in between are not preserved. In this talk, I will present some examples which admit metrics with different curvature conditions and discuss the evolution of their metrics under the Ricci flow. This is based on joint works with David González- Álvaro.

In Schwarzschild spacetime with positive mass $M$, there exist (unstable) circular orbits of trapped null geodesics at the Schwarzschild radius $r=3M$, outside the black hole horizon at $r=2M$. These orbits fill a three-dimensional submanifold $S^2 imes mathbb R$ 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 $SO(3) imesmathbb R^2$. 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.

Thanks to Gromov’s pre-compactness theorem and the work of Cheeger and Colding, any complete n-manifold with nonnegative Ricci curvature and Euclidean volume growth is asymptotic to a family of cones at infinity in the pointed Gromov-Hausdorff sense. When $n=4$ a naive argument neglecting all the regularity issues suggests that the sections of these cones at infinity are positively Ricci curved and hence homeomorphic to spherical space forms, by Hamilton’s work. I will discuss joint work with Elia Bruè and Alessandro Pigati where we make this argument rigorous.

A symplectic manifold is a pair $left(X^{2n},omega ight)$, where $X^{2n}$ is a smooth manifold and $omega$ is a differential 2-form such that $operatorname{d}omega=0$ 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 $Ain H_2(X;mathbb Z_2)$, 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 $Ain H_2(X;mathbb Z_2)$ is realised by a non-orientable embedded Lagrangian surface $L$, 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.

A spacetime singularity is called Tipler strong if the volume form acting on independent Jacobi fields defined by causal geodesics vanishes as the singularity is approached. Such a singularity is accompanied by a notion of strong Ricci curvature growth along incomplete geodesics. On the other hand, a naked singularity is one that can be identified by the past-incompleteness of causal geodesics. A key question concerning the cosmic censorship debate is whether there exists "generic" initial data that can collapse to a naked singularity in finite time. This question was initially addressed in the well known works of Datt (1938) and Oppenheimer and Snyder (1939) for homogeneous dust data. An alternative approach was developed by Joshi and Dwivedi (1993) for inhomogeneous dust. The idea is to consider the geodesic equation in the limit of the singularity. This yields an algebraic equation, and the polarity of its roots indicates whether the singularity is naked or not. One then imposes the sufficient condition for the existence of a Tipler strong singularity as given by Clarke and Królak (1985), in order to guarantee that any body approaching the naked singularity will be crushed to zero size, and thereby rendering these singularities as physically interesting. For spherically symmetric collapse, the formation of strong curvature naked singularities can be characterized by a single parameter related to the mass profile and physical radius of the collapsing matter shell. I will present the generalizations of this line of work in the following directions: (i) generalization to higher dimensions, (ii) generalization to all type-I matter fields.

Under reasonable assumptions, the space $mathcal{C}$ 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 $mathcal{N}$ 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, $mathcal{M}$ is a conformal symplectic filling of $mathcal{N}$.

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.

We extend a singularity theorem by Galloway & Ling for spacetimes with expanding Cauchy surface (positive definite second fundamental form) to those with positive semi-definite second fundamental form. Although a seemingly minor loss of restriction it allows for a broader topological class of spacetimes to be complete. Using results from the topological study of 3-manifolds, minimal surface theory as well as Penrose's classical singularity theorem, we will determine a connection between the topology of the Cauchy surfaces and the completeness of spacetime.

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.

Consider a non-rotating spherically symmetric charged black hole with mass $M > 0$ and a charge $Q eq 0$, 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.

Impulsive gravitational waves were introduced by Penrose in the late 1960s and have been widely studied since. They are space times of low regularity described by either a Lipschitz continuous or even a distributional metric tensor.
We will start the talk with a brief introduction to Lorentzian geometry and then introduce impulsive gravitational waves with focus on the continuous form of the metric. We will study its distributional Ricci curvature and apply the theory on synthetic timelike Ricci curvature bounds introduced by Cavalletti-Mondino. This is joint work with Andrea Mondino (Oxford) and Clemens Sämann (Vienna).

In this talk, I will explore conditions for a timelike curvature-driven Gromov compactness theorem within the framework of Lorentzian length spaces (LLS).
We will begin by introducing the concept of the (pointed) Gromov-Hausdorff distance for metric spaces and discuss its adaptation to the Lorentzian context.
Following this, we will examine Gromov compactness results for length spaces and investigate their generalization to the Lorentzian setting. To conclude, I will outline a preliminary approach to proving a curvature-driven compactness theorem in the Lorentzian case, supported by examples of GH-diverging Lorentzian length spaces to illustrate key challenges and requirements.