Vorträge in der Woche 26.06.2023 bis 02.07.2023

Montag, 26.06.2023: The Einstein Constraint Equations of General Relativity

Dr. Rodrigo Avalos

Mathematical general relativity (GR) has become a thrilling area of research connecting problems in theoretical physics with classical and recent problems in geometry and analysis. In this context, the Einstein constraint equations (ECE) play a central role in the initial value formulation of GR and the understanding of global properties of solutions. In this talk, we intend to introduce to a broad audience the origin of the ECE within the initial value formulation of GR and show how the ECE relate to geometric partial differential equations (PDE) associated to classical problems in geometric analysis. We furthermore intend to highlight how techniques and ideas associated to the analysis of the ECE serve as a source of new well motivated geometric questions. In this process, among other things, we shall review both classical and recent results on the parametrisation of the space solutions of the ECE, an area with several open problems and a subject of active research.

Dienstag, 27.06.2023: Brill--Noether theory over the Hurwitz space

Isabel Vogt (Brown University)

While in the 19th century an algebraic curve was synonymous with a one-dimensional subset of projective space specified by polynomial equations, the modern study of curves makes use of the definition of an abstract curve independent of a projective embedding. Brill--Noether theory is the bridge between these two perspectives. The fundamental question: given an abstract curve C, what is the geometry of the space of maps of C to projective space with certain invariants? As a crowning achievement of the modern study of linear series in the 1980s, this geometry is well-understood when the curve C is sufficiently generic. However, in nature, curves are often encountered via a realization specified by polynomial equations of relatively small degree, which might force the curve to be too special for the classic Brill--Noether theorem to apply. In this talk, I will discuss joint works with subsets of Kaelin Cook-Powell, Dave Jensen, Eric Larson and Hannah Larson which provide the first complete analogue of all of the main theorems of Brill--Noether theory when the curve is equipped with a low degree map to the line.

Dienstag, 27.06.2023: How to kick a black hole

Jörg Frauendiener (Otago)

It is well known that gravitational waves interact in a non-linear way. This makes it difficult to describe them rigorously. The cleanest description is based on certain conformal properties of the Einstein equations — first discovered by R. Penrose they were rigorously developed and used by H. Friedrich to prove several important global results for general relativistic space-times. The conformal field equations which implement this conformal framework provide various well-posed initial (boundary) value problems for use in many different situations. The talk will give a computational perspective on one particular application, the non-linear interaction of gravitational waves with an initially static (and spherically symmetric) black hole. We will show how to 'kick' the black hole and possibly how to spin it up.

Dienstag, 27.06.2023: Solid Groups

Victoria Schleis

Donnerstag, 29.06.2023: A new approach to Scattering: On the Asymptotic states of Nonlinear Dispersive and Hyperbolic equations with General Data

Prof. Avy Soffer (Rutgers University)

I will present a new approach to finding the asymptotic states of Nonlinear Wave Equations with general initial data. In particular, we show for a large class of equations, that all asymptotic states are linear combinations of free wave, localized parts (solitons, breathers..) and a possibility of self-similar solutions as well in some cases. These results hold for initial data for which the H^1 Sobolev norm (the energy norm) is uniformly bounded in time. This answers the question of Asymptotic Completeness to a large class of equations, including for the first time, equations with time dependent potentials. These are joint works with Baoping Liu (Peking Univ) and Xiaoxu Wu(Rutgers).