LOGOS Seminar

Who wants to give talks when?

In this talk I will present a joint work with Mathias Braun, in which we prove a synthetic version of the Gannon-Lee incompleteness theorem. We assume the recent synthetic characterization of trappedness and of the null energy condition by Ketterer, instead of the classical hypoteses. I will start with an introduction on the incompleteness theorems of general relativity and the classical Gannon-Lee Theorem, and then I will describe Ketterer's conditions and state our synthetic Gannon-Lee Theorem. If time allows, I will briefly discuss where our proof differs from the classical one and how we apply the synthetic conditions. password for the recording @4!S&N2k

A Proof Assistant is a software environment that allows the composition of formalized mathematical theorems and proofs in a format that makes the proofs mechanically verifiable. Recently, a spotlight has been shining on the Lean Proof Assistant and its project Mathlib, aiming to build a unified library of formalized mathematics at the level of current research. I will give a quick and informal introduction to Lean’s formal language (Dependent Type Theory) and how it encodes logical statements/proofs, guided by the questions I myself had when first learning it. I will follow it up by a small practical demonstration. Along the way, I will try to highlight the opportunities and challenges that formalized proofs offer to mathematicians today.

We examine a class of semi-Riemannian manifolds that undergo smooth metric signature change from - Riemannian to Lorentzian — across a hypersurface with a transverse radical. This class includes physically motivated cosmological models such as the Hartle-Hawking “no-boundary” proposal, in which the universe transitions smoothly from a Euclidean to a Lorentzian phase. We show that these manifolds admit isometric embeddings into higher-dimensional pseudo Euclidean spaces and, in particular, prove the existence of global isometric embeddings of the canonical model into both Minkowski and Misner spaces. This framework provides a mathematical setting for studying smooth signature change and its role in higher-dimensional and cosmological models.

The uncertainty principle in harmonic analysis states that a non-zero function and its Fourier transform cannot be simultaneously too sharply localized. There are numerous precise mathematical formulations of this meta-theorem. Beurling’s own version of the uncertainty principle for Fourier transforms is the following clean and elegant statement: given $f \in L^{1}(\mathbb{R})$, $$ \iint_{\mathbb{R}^{2}} \bigl|f(x)\,\widehat{f}(\xi)\bigr|\, e^{\,|x\cdot \xi|}\, dx\, d\xi < \infty \;\Rightarrow\; f=0. $$ The goal of this talk is to present a general quantified version of Beurling's uncertainty principle. We will characterize in several ways those functions $f \in L^{1}(\mathbb{R}^{n})$ such that $$ \iint_{\mathbb{R}^{2n}} \frac{\bigl|f(x)\,\widehat{f}(\xi)\bigr|\, e^{\,|x\cdot \xi|}}{W\!\bigl(|x|+|\xi|\bigr)} \, dx\, d\xi < \infty, $$ where $W:[0,\infty)\to[1,\infty)$ is an unbounded non-decreasing function subject to certain natural regularity conditions. The talk is based on collaborative work with Lenny Neyt.

Motivated by the strong cosmic censorship conjecture in general relativity, it is desirable to understand and classify the strength of gravitational singularities. $C^0$-inextendibility results account for the strongest form of a gravitational singularity. Recently, Sbierski proved the $C^0$-inextendibility of a class of spatially spherical and hyperbolic FLRW spacetimes without particle horizons. In this talk, we show how his techniques in the spatially hyperbolic setting can also be applied to the spatially flat setting, yielding $C^0$-inextendibility results for spatially flat FLRW spacetimes also without particle horizons.

We discuss noncommutative versions of optimal transport theory, motivated by applications to dissipative quantum systems. In particular, we present a gradient flow formulation for Lindblad equations, a noncommutative notion of Ricci curvature, and functional inequalities related to convergence to equilibrium for quantum Markov semigroups. This is based on joint works with Eric Carlen.

Tartar's defect measure has introduced a new approach to the analysis of linear and nonlinear equations, leading to significant progress both in theory and in applications. An important contribution was made by Gérard, whose work motivates us to further investigate the relationship between weakly and strongly convergent sequences in Sobolev spaces, in a way analogous to microlocalization via the Sobolev-type wave front set $WF^s(f)$ (as well as the classical wave front set $WF(f)$ of a distribution $f$), introduced by Hörmander. We analyze the pull-backs and products within the framework of Sobolev spaces. In addition, Mikhlin-type multipliers acting from $L^p_{\mathrm{comp}}$ to $L^p_{\mathrm{loc}}$ are considered, in cases $p=1$ and $p=\infty$.

In his celebrated paper [3], Nash showed that every $C^{\infty}$ compact $m$-dimensional manifolds can be isometrically embedded into $\mathbb{R}^{m(3m+11)/2}$, while if the manifold is noncomapct then it can be isometrically imbedded into $\mathbb{R}^{(m+1)m(3m+11)/2+2m+2}$; much later, Günther [2] lowered the dimension of the target Euclidian space. For real-analytic compact manifolds, the analytic isometric embedding into $\mathbb{R}^{m(3m+11)}$ was shown by Greene and Jacobowitz in [1]. In this talk we consider the case when the manifold and the metric are of Gevrey regularity $p!\, s$, $s > 1$. By employing some of the ideas of Günther [2], we show that if the manifold is compact then it admits a Gevrey isometric embedding (of the same class) into $\mathbb{R}^{m(3m+11)/2}$, while if the manifold is noncomapct than it can be isometrically imbedded into $\mathbb{R}^{(m+1)m(3m+11)/2+2m+2}$ (again, the embedding has the same Gevrey regularity as the manifold). The talk is based on collaborative works with Andreas Debrouwere. References: [1] R. E. Greene, H. Jacobowitz, Analytic isometric embeddings, Ann. of Math. (2), 189–204, 1971 [2] M. Günther, Zum Einbettungssatz von J. Nash, Math. Nachr. 144 (1989), 165–187. [3] J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20–63.

The coarea inequality for Hausdorff measures plays a fundamental role in geometric measure theory and in the analysis of metric spaces. In this talk, we present a Lorentzian counterpart of the coarea inequality for Lorentzian Hausdorff measures, recently established by Robert McCann and Clemens Sämann. To formulate this result, we introduce the notion of uniformly $d$-controlling maps together with two new local geometric conditions: the local causal enlargement property and the causal estimation property. We discuss how they lead to the desired coarea inequality. If time permits, we will also discuss potential applications of the inequality and directions for future research.

José gives a talk at the weekly Kolloquium in the Sky Lounge, 12. OG, OMP 1. Further info here.

The aim of this talk is to explore a few open problems in general relativity — all revolving around the notion of mass. I will motivate these questions through examples and results illustrating how the study of mass can both reveal and detect geometric properties of space. In particular, I will highlight how classical results, such as the positive mass theorem and the Penrose inequality, fit within this perspective, and how the ideas underlying them point towards further directions in which our understanding remains incomplete.

Just a few years after Einstein's seminal 1915 work introducing general relativity, Kaluza proposed in 1921 a unification of gravity and electromagnetism by adding a hidden fifth dimension to spacetime. This idea, later refined by Klein in 1926, that forces might be understood as geometry in extra dimensions is still very much pursued today, for example in string theory. In this talk, we will try to understand Kaluza–Klein theory mathematically through the language of submersions and principal fibre bundles, and we will explore several issues it raises. I will then present sub-Lorentzian geometry, the spacetime analogue of sub-Riemannian geometry, as an alternative geometric framework. Instead of introducing a full higher-dimensional Lorentzian metric as in Kaluza–Klein theory, sub-Lorentzian geometry works with nonholonomic constraints and a degenerate metric. I will discuss how this point of view might address some of the challenges encountered early on in Kaluza–Klein theory.

Convexity is one of the most simple regularity properties outside of smoothness requirements. Its apparent ubiquity makes convex functions an invaluable tool in many settings. For example, the relation between convexity and curvature is a source of deep insights. One of principal aspects, which has concrete consequences in metric measure geometry, is the theory of gradient flows: despite the possible nonsmoothness, gradient flows of convex functions are often well posed. In this talk I will introduce the notion of causally convex functions, a proposal for a Lorentzian counterpart to convex functions, and some natural notions of gradient flow. I will sketch the basic regularity properties of those functions and the well posedness of their gradient flows, comparing them with what happens in positive signature. This is a work in collaboration with Mathias Braun, Nicola Gigli, Robert McCann and Matteo Zanardini.

In Riemannian geometry, the classical Toponogov and Cheeger–Gromoll splitting theorems describe the geometry of manifolds with non-negative curvature that contain a complete line, that is, a globally minimizing geodesic defined on all of R: such manifolds split as Cartesian products of the form R x N, where N is a manifold with non-negative curvature. In Lorentzian signature, analogous results were established by Beem–Ehrlich–Markvorsen–Galloway and Galloway. On the other hand, in metric geometry, there is an analogous theorem for CAT(0) spaces, namely metric spaces with global non-positive sectional curvature in the triangle-comparison sense. This theorem, due to Eberlein–O'Neill, states that if a space with non-positive curvature is foliated by asymptotic complete lines, then these lines are parallel, and the space splits as R x N, where N is again a CAT(0) space. In particular, the theorem applies to complete, simply connected Riemannian manifolds with non-positive sectional curvature. In this talk, I will discuss an analogue of the Eberlein–O'Neill splitting theorem for Lorentzian pre-length spaces whose timelike curvature is globally bounded above by zero, namely the Lorentzian analogues of CAT(0) spaces. This is joint work with Joe Barton, Tobias Beran, Sebastian Gieger, Jona Röhrig, and Felix Rott.