Title:   Conical Deficits and Cosmic Strings
Abstract:   Conical deficits in a spacetime with an angular symmetry have long
been associated with the presence of a cosmic string in theoretical physics, but to what extent is this association accurate? I will review what a cosmic string is for the mathematical audience, and why the above association is generally accepted as correct. However, I will then show that while the association is broadly correct, there can be sub-leading order deviations. This will be demonstrated by looking at black holes threaded with a cosmic string, for which there are exact conical metrics, as well as solutions with a finite width vortex. Surprisingly, the apparently innocent sub-leading terms in the fall-off to a conical solution do have an impact, as shown by studying geodesics in the pierced black hole background.

Title:   Einstein-Fluid Initial Data
Abstract:   Fluids have been a standard matter source for gravitation since the beginning days of general relativity. Nevertheless, constructing initial data for this family of matter models is surprisingly nuanced. In this talk we describe a novel approach to building Einstein-fluid initial data based on a recently established phase-space technique for building non-vacuum initial data sets. Compared to prior approaches to working with fluids, the input parameters used here allow for more direct specification of physical quantities, such as the number of particles in any given region. We focus on perfect fluids but also discuss extensions of these ideas to viscous models.

Title:   Metric completeness in Riemannian and Lorentzian geometry
Abstract:   The Hopf-Rinow theorem is a fundamental result in Riemannian geometry relating metric completeness, properness, and geodesic completeness. We recall the many reasons why it does not hold in Lorentzian geometry. Then we present our joint work with García-Heveling relating global hyperbolicity and completeness of the null distance and show that and how this result can be viewed as an extension of the metric part of the Hopf-Rinow theorem to Lorentzian geometry. We also discuss our generalization to proper cone structures and end with some thoughts about the semi-Riemannian setting.

Title:   Cosmological Dynamics of Scalar Fields in String-Theoretic Dark Sectors
Abstract:   Scalar fields are a ubiquitous feature of string theory compactifications, typically appearing in axion–saxion pairs with non-trivial interactions. Motivated by this structure, I will use the dynamical systems approach to analyse how such scalars can play a role in the dark sector of the universe. First, I will present a model in which one scalar behaves as dark matter and the other as dark energy, providing a unified picture of the dark sector. I will then turn to multifield quintessence models, where accelerated expansion can arise even with steep potentials, thereby evading recent quantum gravity swampland constraints and offering a possible framework for evolving dark energy hinted at by observations.

Title:   A comparison theorem for the total mass of AE, ALE, AF, and ALF Riemannian Toric 4-Manifolds
Abstract:   One of the fundamental conjectures in mathematical relativity is the positivity of total mass for complete non-compact Riemannian manifolds assuming appropriate lower bounds on scalar curvature. This conjecture has been proved for AE manifolds using several techniques, starting with the celebrated results of Schoen-Yau and Witten. There are counterexamples to this conjecture in the AF, ALF, and ALE cases. In this talk, we will refine this conjecture and prove a comparison theorem for the mass of toric 4-manifolds. The proof is robust and can be extended to higher dimensions if additional assumptions are added. This is a joint work with Marcus Khuri and Hari Kunduri.

Title:   Dyon Loops and Abelian Instantons    

Abstract:   We construct Abelian gauge field configurations that carry non-zero instanton number. Each such ""Abelian instanton"" is generated by a closed magnetic worldline in four-dimensional Euclidean space, provided the Abelian gauge field has non-trivial winding along the closed worldline. The resulting field configuration corresponds to a Euclidean dyon loop featuring non-zero instanton number. We embed these dyon loops in a UV-complete theory using the Georgi-Glashow model and show that the full instanton charge is borne entirely by the unbroken U(1) sector. In this same model, using a numerical relaxation procedure, we show that Euclidean dyon loops are a continuous deformation of small BPST instantons.

Title:   Existence of CMC hypersurfaces in cosmological spacetimes
Abstract:   Constant mean curvature (CMC) spacelike hypersurfaces have played an important role in mathematical relativity, in particular in the study of the Einstein equations, both in terms of solving the initial data constraints, and in evolving CMC initial data. Further interest for us on this topic comes from the Bartnik splitting conjecture. In this talk we review some old, and present some new, existence results for CMC spacelike hypersurfaces in the class of cosmological (spatially closed) spacetimes. Some connections to the causal boundary of spacetime are also discussed. This involves joint work with Eric Ling.

Title:   How much null-energy-condition breaking can the Universe endure?
Abstract:   Quantum fields can notoriously violate the null energy condition (NEC). In a cosmological context, NEC violation can lead to, e.g., dark energy at late times with an equation-of-state parameter smaller than -1 and nonsingular bounces at early times. However, it is expected that there should still be a limit in semiclasssical gravity to how much “negative energy” can accumulate over time and in space as a result of quantum effects. In the course of formulating quantum-motivated energy conditions, the smeared null energy condition has emerged as a recent proposal. This condition conjectures the existence of a semilocal bound on negative energy along null geodesics, which is expected to hold in semiclassical gravity. In this work, we show how the smeared null energy condition translates into theoretical constraints on NEC-violating cosmologies. Specifically, we derive the implied bounds on dark energy equation-of-state parameters and an inequality between the duration of a bouncing phase and the growth rate of the Hubble parameter at the bounce. In the case of dark energy, we identify the parameter space over which the smeared null energy condition is consistent with the recent constraints from the Dark Energy Spectroscopic Instrument.

Title:   A double-copy picture of strings in AdS    

Abstract:   The Kawai–Lewellen–Tye (KLT) relations are a striking example of how string theory can reveal hidden structures and connect a web of, in principle, very different theories. Originally, they state that (tree-level) closed-string amplitudes can be expressed as quadratic combinations of open-string amplitudes. In the field theory limit, this gives rise to the celebrated double-copy between gluons and gravitons. In this talk, we will explore how much of this elegant structure persists when we move from flat space to AdS. As a first step, I will propose the fundamental building blocks of tree-level open- and closed-string amplitudes in AdS and show how they are related via an AdS version of the KLT relations. This structure not only significantly simplifies explicit calculations but also points to deeper algebraic and geometric structures that remain to be uncovered.

Title:   Volume estimates and Hawking's singularity theorem for Lipschitz metrics
Abstract:   The classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. One approach to proving Hawking's singularity theorem is via first studying the influence of Ricci curvature bounds on the volume of certain spacetime regions and spacelike hypersurfaces. In this talk I will present improved geometric estimates of this type based on a segment type inequality (joint work with E.-A. Kontou, A. Ohanyan and B. Schinnerl). As we will see these estimates have been instrumental in the recent proof of Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity assuming a distributional version of the strong energy condition and an almost everywhere mean curvature bound along a local flowout of the initial hypersurface (joint work with M. Calisti, E. Hafemann, M. Kunzinger and R. Steinbauer). This result directly extends known C^1 results and complements versions of
Hawking's theorem for timelike non-branching Lorentzian length spaces.

Title:   Spectral Flow, Eta Invariant and Llarull's Rigidity Theorem in Odd Dimensions
Abstract:   In this talk, I will present the application eta invariant and spectral flow on the proof of the odd-dimensional part of Llarull’s Theorem and two of its extensions. Generally speaking, Atiyah-Singer index theory is one of the major tools in the study of Riemannian metrics of positive scalar curvature. In odd dimensions, the spectral flow of a family of twisted Dirac operators on a compact spin manifold can be used to provide a direct proof of Llarull’s rigidity theorem and the so called “spin-area convex extremality theorem”. Furthermore, combining with the deformed Dirac operator introduced by Bismut and Cheeger, this method can be used to prove noncompact extension of Llarull’s theorem, which provides a final answer to a question by Gromov. This talk is based on joint works with Guangxiang Su, Xiangsheng Wang and Weiping Zhang.

Title:   Intrinsic rigidity of extremal horizons and black hole uniqueness
Abstract:   I will survey the classification of extremal horizons in vacuum spacetimes (including a cosmological constant) and present a recent rigidity theorem which shows that the intrinsic geometry of compact cross-sections of such horizons must admit a Killing vector field. In particular, this implies that the extremal Kerr horizon is the most general such horizon in four-dimensional General Relativity, completing their classification. I will also discuss the application of such horizon rigidity to the corresponding black hole classification, in particular, I will present a recent uniqueness theorem which shows that the extremal Schwarzschild de Sitter spacetime (or its near-horizon geometry) is the only analytic Einstein spacetime with positive cosmological constant that contains a static extremal horizon with a compact cross-section.

Title:   Part 1: Radiative spacetimes in asymptotically-flat and cosmological settings
Abstract:   We shall discuss radiative spacetimes of various types in asymptotically-flat as well as cosmological settings. Asymptotically-flat systems in General Relativity (GR) are solutions of the Einstein equations tending to Minkowski spacetime at infinity. They model stars, clusters of stars, galaxies and related situations in physics. By studying the Cauchy problem for the Einstein equations using mathematical tools from the analysis of partial differential equations as well as geometry, we aim at answering deep questions from physics. Crucial insights have been revealed into gravitational waves (observed for the first time by Advanced LIGO in 2015 and many times since then). These waves are produced during the mergers of black holes or neutron stars and in core-collapse supernovae. We shall discuss the Cauchy problem for the Einstein equations in this context, explain results on gravitational radiation and the memory effect of gravitational waves, the latter being a permanent change of the spacetime. Gravitational waves carry information about their sources and surrounding environments. It will be important how fast the geometric quantities (metric, curvature components) tend to Minkowski spacetime towards infinity. We shall investigate gravitational radiation for classes of spacetimes with a broad range of different fall-off behavior. Understanding radiation necessitates a focus on relevant asymptotics, given that insights into radiation can be gained by analyzing the behavior of asymptotically-flat spacetimes at future null infinity. Therefore, we will derive and discuss such null asymptotic structures.
The cosmological setting is very different from the previous scenarios: In cosmological spacetimes there is no `null infinity’. Rather we will investigate what happens in the `cosmological zone’. Our universe is expanding and on its largest scale is highly inhomogeneous. Thus, we expect expansion as well as large-scale structures to alter radiation and memory. We shall discuss mathematical results on gravitational radiation in ΛCDM cosmology, and mention what happens for de Sitter and Friedman–Lemaître–Robertson–Walker (FLRW). Students will gain an overview of fundamental questions and the mathematical tools to tackle them.