Current seminars

Already in 1916, in his first paper on gravitational waves, Einstein emphasized the necessity to treat gravitational waves quantum theoretically and concluded that “quantum theory must modify not only Maxwellian electrodynamics but also the new theory of gravitation”. However, at that time, simple estimates indicated that it would be impossible to ever experimentally observe any phenomena that meaningfully involved quantum dynamics of the gravitational field. Thus, the hope of testing ideas about quantum gravity with experiments languished for nearly a century.

In the past few decades, however, the notion that we might be able to gain experimental access to quantum gravitational phenomena has resurfaced in a wide variety of contexts, from cosmology to tabletop scales, the latter being driven mostly by the recent advent of ultra-sensitive quantum control and detection in mesoscopic and macroscopic systems.

How can we interpret these kinds of measurements? What, exactly, would each of them teach us about the nature of gravity, quantum or otherwise? Similar questions were asked in the past about the quantum nature of the electromagnetic field, and much of our current understanding about how to observe “non-classical”, or “quantum”, signatures originated in quantum optics experiments starting from the early 1970s.

Inspired by the lessons from quantum optics I will provide a few examples and looming challenges for future possible tests of quantum gravity. Time permitting I will also provide a brief summary on our current efforts in Vienna to realize and explore quantum sources of gravity in the laboratory..

From Ehrenfest’s paradox to Mach’s principle, the history of General Relativity is closely linked to the concept of rotation. However, this notion is sometimes a source of confusion and misconceptions, often caused by a lack of standardization in the language used in the scientific literature.

In this talk, we will review two non-equivalent definitions of rotation (local and “absolute” rotations) in the context of General Relativity, with particular attention to their operative definitions. After addressing the link between these two concepts and their connection to gravitation, we will provide an overview of the GINGER (Gyroscopes in General Relativity) experiment, whose main goal is to measure the Lense-Thirring effect generated by Earth's rotation using a system of ring laser gyroscopes.

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.

The motion of spinning test particles in a curved background is a classical topic of General relativity. Currently, it finds applications in the modelling of inspirals of spinning compact objects into massive black holes.
In recent years, we have made significant progress in the modelling of the spin contribution to the orbit and outgoing gravitational-wave flux of the compact object in the black-hole field. Technically, this involved a number of tools, ranging from the Hamiltonian formalism to Killing-Yano tensors and the hidden symmetry of black holes. In this talk, I will summarize these developments.

The standard cosmological models of a homogeneous and isotropic universe in general relativity are described by the Friedmann--Lemaître--Robertson--Walker (FLRW) spacetimes. A natural step toward understanding the dynamics of wave-type equations---including the Einstein equations---near these backgrounds is to study the scalar wave equation on FLRW spacetimes and establish robust energy estimates.
In this talk, after outlining the stability problem for FLRW spacetimes, we consider the wave equation---as a proxy for the Einstein equations---on decelerated FLRW spacetimes with R^3 spatial sections. We demonstrate how dispersion and expansion affect the long-time behaviour of waves. In particular, we present uniform energy bounds and integrated local energy decay estimates across the full decelerated expansion range. Moreover, we describe a hierarchy of r^p-weighted energy estimates, in the spirit of the Dafermos--Rodnianski r^p-method, which lead to energy decay estimates.

Summer semester 2025

Time-periodic solutions to nonlinear dispersive equations have been the subject of many investigations over the years. The classic works prove the existence of small amplitude solutions with frequencies belonging to nowhere dense sets.

In this talk I will show numerical evidence suggesting existence of a completely new class of solutions for one-dimensional cubic wave equation on an interval with Dirichlet boundary conditions. Solutions belonging to it are characterised by large energies, have complicated mode compositions, and form intricate fractal-like patterns. Then I will show how these numerical results can be used to rigorously construct exact solutions belonging to this new class. Finally, I will demonstrate a systematic approach to analysing complex structures formed by these solutions.

This is a joint work with Maciej Maliborski.

This talk investigates the dynamics of cosmological Einstein-scalar field-fluid solutions near the Big Bang singularity, moving beyond assumptions of spatial homogeneity and isotropy. Over the past decade, significant progress has been made in understanding these dynamics through advances in the theory of nonlinear partial differential equations, though many questions remain unresolved.

I will provide an overview of these developments focusing on recent results from my collaboration with Todd Oliynyk.

In last week's talk, Walter Simon presented singularity theorems for manifolds with spacelike Cauchy surfaces satifying certain convexity conditions.

The main theorem read:
If a globally hyperbolic spacetime admitting a closed, spacelike, 2-convex Cauchy surface is geodesically complete, then the Cauchy surface is either a spherical space or finitely covered by a surface bundle over the circle with totally geodesic fibres.

We will recall these results and focus on the proof of the main theorem.

The proof is divided into three parts. First, we identify a suitably embedded minimal surface. Second, assuming geodesic completeness, we construct a neighborhood foliated by minimal surfaces. Finally, we apply compactness theorems to extend this foliation to the entire manifold.

Our approach relies on several recent developments, including the positive resolution of the virtually Haken conjecture. In particular, we will observe that in certain special cases - such as when the Cauchy surface is Haken - our result admits a natural strengthening.

Inspired by the classical singularity theorems in General Relativity, Galloway and Ling (Commun. Math. Phys. 2017, 2024) have shown the following: If a globally hyperbolic spacetime satisfying the null energy condition contains a closed, spacelike Cauchy surface N which is strictly 2-convex (meaning that the sum of the lowest two eigenvalues of the future extrinsic curvature is positive), then N is either a spherical space or past null geodesically incomplete.

In recent work (with Eric Ling, Carl Rossdeutscher and Roland Steinbauer) we have relaxed the above convexity condition in essentially two respects. Firstly, we admit 2-convex extrinsic curvatures (for which the sum of the lowest two eigenvalues is non-negative). Secondly, if N admits a U(1) isometry group, we can impose even less restricive convexity conditions. In addition to spherical spaces and past null geodesically incompleteness, this allows for certain classes of surface bundles for N or a finite cover therof, which we classify. While our results do not use Einstein's equations, we provide several classes of vacuum solutions (with our without cosmological constant) of these equations as examples.

In my talk I will give a brief overview of singularity theorems and then focus on explaining the above results via examples. The proofs will only be outlined.

Preview: In his talk on 21.5. on the same subject Carl Rossdeutscher will focus on selected aspects of the proofs.

The character and status of very simple sounding questions in manifold theory depends crucially on the dimension of the manifold. An illustrative example is the famous Jordan curve theorem which roughly speaking states that a topological embedding of S^1 into R^2 separates R^2 into a bounded component (which is homeomorphic to a disk) and an unbounded one. Naively speaking this theorem seems obvious, if annoying to prove. In reality, it is a miracle it is true in the first place as corresponding statements in dimension 3 and higher are simply wrong. In this talk I will try to give an overview over questions in low dimensional topology (such as existence and uniques of smooth structures, classification of manifolds) and how these depend on the dimension.

The Standard Model (SM) is one of the greatest successes of modern theoretical physics. Despite this, mathematical references studying the full SM (rather than just its sectors in isolation) on curved spacetimes are somewhat scarce, even though it seems important to understand the theory at a classical level in a more geometric setting. In my talk, I will first briefly review the mathematical structure of the SM Lagrangian, the corresponding Euler-Lagrange equations, and some of their basic properties, particularly related to conformality. Then, I will present a global existence result for the SM equations on four-dimensional spacetimes of expanding type. The talk is focused on a geometrically intrinsic approach to the theory, and the main ingredient for the proof is a gauge-invariant energy estimate. This is joint work with Volker Branding.

The first half concerns Gregory—Laflamme (GL) instabilities which occur at the level of linearised gravity. Heuristic and numerical evidence suggests that GL instabilities plague black holes in dimensions greater than 4 which have an event horizon that has one direction that is large compared to all others. Sam will discuss a direct mathematical proof of the Gregory—Laflamme instability for the 5D Schwarzschild black string. The proof relies upon reducing the linearised vacuum Einstein equation to a Schrödinger equation to which direct variational methods can be applied.   The second half of the talk concerns the blue-shift instability in the interior of rotating Kerr black holes at the level of linearized gravity. This instability is intimately connected to Penrose’s strong cosmic censorship conjecture. In contrast to the GL instability, this instability is weak in the sense that the C^0 norms of metric perturbations do not grow. Jan will discuss a mathematical result on the blue-shift instability, illustrate the mechanism by a toy example, and, if time permits, discuss some elements of the proof.

Recently it was proposed that GME experiments may probe linearized gravity only as a quantum controlled field and not as a quantum field theory. I will argue that the concept of a relativistic quantum controlled field is fundamentally flawed because of causality violation. Quantum control is also implicit in a recent path-integral description of those experiments, which casts doubt on its validity beyond the Newtonian approximation.