In relativistic quantum mechanics, the discrete spectrum of the Dirac hamiltonian with a Coulomb potential famously agrees with Sommerfeld’s fine structure formula for the hydrogen atom. In the Coulomb approximation, the proton is assumed to only have a positive electric charge. However, the physical proton also appears to have a magnetic moment which yields a hyperfine structure of the hydrogen atom that’s normally computed perturbatively. Aiming towards a non-perturbative approach, Pekeris in 1987 proposed taking the Kerr-Newman spacetime with its ring singularity as a source for the proton’s electric charge and magnetic moment. Given the proton’s mass and electric charge, the resulting Kerr-Newman spacetime lies well within the naked singularity sector which possess closed timelike loops. In 2014 Tahvildar-Zadeh showed that the zero-gravity limit of the Kerr-Newman spacetime (zGKN) produces a flat but topologically nontrivial spacetime that’s no longer plagued by closed timelike loops. In 2015 Tahvildar-Zadeh and Kiessling studied the hydrogen problem with Dirac’s equation on the zGKN spacetime and found that the hamiltonian is essentially self-adjoint and contains a nonempty discrete spectrum. In this talk, we show how their ideas can be extended to classify the discrete spectrum completely and relate it back to the known hydrogenic Dirac spectrum but yielding hyperfine-like and Lamb shift-like effects.

Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

The “memory effect” is the permanent relative displacement of test particles after the passage of gravitational radiation. It is associated with both the propagation of massive bodies out to timelike infinity (“ordinary memory”) or the propagation of radiation out to null infinity (“null memory”). The memory effect can be characterized by the failure of the shear tensor at order $1/r$ to return to zero at late times, even though it is “pure gauge.” Closely analogous effects occur in electromagnetism, where the vector potential at order $1/r$ fails to return to zero even though it is “pure gauge.” In both cases, the Fourier transform of the radiative field has divergent behavior at low frequencies. This gives rise to infrared divergences (i.e., infinite numbers of “soft” gravitons/photons) in the quantum field theory description if one attempts to describe these states as vectors in the usual Fock Hilbert space representation. To obtain a mathematically sensible quantum scattering theory, one must allow states with nonvanishing memory in the “in” and “out” Hilbert spaces. An elegant solution to this problem in massive quantum electrodynamics was given by Kulish and Fadeev, who constructed a Hilbert space of incoming/outgoing charged particle states that are “dressed” with radiative fields of corresponding memory, so as to yield vanishing large gauge charges at spatial infinity. However, we show that this type of construction fails in quantum gravity. The primary underlying reason is that the “dressing” contributes to null memory, thereby invalidating the construction of eigenstates of large gauge charges. In quantum gravity, there does not appear to be any choice of (separable) Hilbert space of incoming/outgoing states that can accommodate all scattering states. Thus, we argue that scattering should be described at the level of algebraic incoming/outgoing states rather than attempting to artificially restrict states to a particular Hilbert space.

The Strichartz estimates are a family of inequalities for the wave equation, which quantify its dispersive nature by bounding the "size" of a solution in terms of an appropriate "energy" at initial time. Typically these estimates admit *maximizers*, that is, solutions which attain the maximal size per unit energy, but such maximizers are rarely known. Most notably, Foschi (2004) found that solutions with a Poisson kernel initial data maximize the conformal Strichartz estimate on the Minkowski spacetime $\mathbb{R}^{1+3}$, and conjectured that the same should hold in arbitrary dimension $\mathbb{R}^{1+d}$. We discuss this conjecture via the Penrose conformal compactification, and we prove that it fails for $d=2, 4, 6,...$

We single out a notion of staticity for non-compact spaces which encompasses several known examples including any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For a (time-symmetric) initial data set modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both on the interior and along the boundary) whose rigidity statement in particular implies that no such umbilical hypersurface admits a compactly supported deformation keeping the original lower bound for the mean curvature. Joint work with S. Almaraz.

One of the most effective tools for proving that Strong Cosmic Censorship holds for a family of cosmological solutions of Einstein’s equations is to verify that these solutions all exhibit Asymptotically Velocity Term Dominated (AVTD) behavior in a neighborhood of their Big Bang singularities. After presenting some background history of known results and conjectures concerning the behavior of the gravitational field near the Big Bang in cosmological solutions of Einstein’s equations, we discuss recent work with Ellery Ames, Florian Beyer, and Todd Oliynyk in which we prove that polarized $T^2$-Symmetric vacuum solutions in a neighborhood of Kasner solutions all exhibit AVTD behavior close to the initial singularity. We also discuss our very recent extension of these results to the case of non-vanishing cosmological constant.

We obtain necessary and sufficient conditions for an initial data set for the vacuum conformal Einstein field equations to give rise to a spacetime development in possession of a Killing spinor. The fact that the conformal Einstein field equations are used in our derivation allows for the possibility of the initial hypersurface intersecting non-trivially with (or even being a subset of) null infinity. For conciseness, these conditions are derived assuming that the initial hypersurface is spacelike. Hence, in particular, these conformal Killing spinor initial data equations encode necessary and sufficient conditions for the existence of a Killing spinor in the development of asymptotic initial data on spacelike components of null infinity.

I will present our theory of the Regularity Transformation (RT-)equations, an elliptic system of partial differential equations which determines coordinate and gauge transformations that remove apparent singularities in spacetime by establishing optimal regularity for general connections. This gain of one derivative for the connections above their $L^p$ curvature then suffices to establish Uhlenbeck compactness. By developing an existence theory for the RT-equations we prove optimal regularity and Uhlenbeck compactness in Lorentzian geometry, including general affine connections and connections on vector bundles with both compact and non-compact gauge groups. As an application in General Relativity, our optimal regularity result implies that the Lorentzian metrics of shock wave solutions of the Einstein-Euler equations are non-singular — geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense —, resolving a longstanding open problem in the field.

The Hawking mass of an area-constrained Willmore sphere is a useful quasi-local measure for the strength of the gravitational field of an initial data set for the Einstein field equations. If the scalar curvature satisfies certain asymptotic assumptions, the asymptotic region of such an initial data set is foliated by large area-constrained Willmore spheres with non-negative Hawking mass. It has been conjectured that these are the only such spheres. In this talk, I will present a proof of this conjecture for all large area-constrained Willmore spheres with non-negative Hawking mass and outer radius less than the logarithm of the inner radius. This is joint work with Michael Eichmair, Jan Metzger, and Felix Schulze.

In this talk I will summarize recent progress and open mathematical problems in understanding black hole mergers quasi-locally, i.e. by using marginally trapped surfaces instead of event horizons.

We will review recent developments concerning the use of level set techniques associated with solutions to elliptic equations, in particular spacetime harmonic functions, and their application to positive mass theorems and comparison geometry.

After reviewing the basics of relativistic elasticity theory, I will introduce a general framework to study spherically symmetric self-gravitating elastic bodies systematically within general relativity, and apply it to investigate compactness bounds in this context.

Most works on the analysis of the wave equation on Kerr black holes rely on a combination of the vector field method and Fourier decomposition, with the notable exception of a generalized vector field method introduced by Andersson-Blue. Their method allows for commutation with second order differential operators entirely in physical-space by supplementing the Killing vector fields with the Carter operator of Kerr to obtain a local energy decay identity at the level of three derivatives of the solution for sufficiently small |a|. I this talk I will describe the main ideas of Andersson-Blue’s method and explain its advantages in two recent applications where physical space-estimates have been crucial: the linear stability of Kerr-Newman black hole to coupled gravitational-electromagnetic perturbations and our proof of the non-linear stability of the slowly rotating Kerr family with Klainerman-Szeftel.