Department of Mathematics

Forschungsinteressen / Research interests

Mathematische Relativitätstheorie / Mathematical General Relativity
Geometrische Analysis / Geometric Analysis
Differentialgeometrie / Differential Geometry

ORCID

 

Aktuelles / New

see t3://page?uid=171912#1785058

 

Alumni of the Group

see t3://page?uid=167343#1031439

 

Forschungsziele / Research goals

The main goal of my research is to obtain a deeper understanding of geometric, analytic, and physical properties of initial data sets with prescribed asymptotic behavior in General Relativity. Many of the questions arising in this context are of independent interest from a (semi-)Riemannian or Geometric Analysis perspective.

At the moment, I work towards obtaining a consistent geometric definition of center of mass for asymptotically Euclidean/hyperbolic and otherwise completely general initial data sets of non-zero mass (collaborators: Anna Sakovich, Julien Cortier, and Christopher Nerz). Our work extends and/or reinterprets work on CMC-foliations by Huisken--Yau (as. Euclidean) and by Neves--Tian (as. hyperbolic), respectively.

I am also interested in characterizing the geometric and topological properties of photon spheres and photon regions in static and stationary asymptotically flat spacetimes (with Gregory Galloway and with Sophia Jahns). In the static setting, we have proved photon sphere uniqueness theorems in the context of vacuum and electro-vacuum solutions of the Einstein equations. These have implications for the static n-body problem. Our results have been generalized to various matter fields (Yazadjiev, Yazadjiev--Lazov, etc.).

Fruthermore, I am fascinated by Bartnik's quasi-local mass and static metric extension conjectures and work on several aspects of these conjectures (with Armando Cabrera Pacheco, Stephen McCormick, and Pengzi Miao and with Oliver Rinne and Markus Strehlau).

In my thesis, I studied geometrostatic systems, i.e. static vacuum asymptotically flat solutions of the Einstein equations. My aim was to obtain a deeper understanding of their asymptotic analysis and to gain more insight into their physical interpretation (mass, center of mass, behaviour of test bodies, Newtonian limit à la Ehlers, etc.). In particular, I proved consistency results showing that certain physical properties like relativistic mass and center of mass converge to their Newtonian counterparts. I am now working towards extending my techniques to more general settings.

Together with Marcus Ansorg and Jörg Hennig, I have studied a geometric inequality between horizon area and angular momentum for stationary and axisymmetric black hole solutions. Our work has interesting applications in proving non-existence of multiple black hole horizons (Hennig--Neugebauer). It has been extended to general axisymmetric spacetimes containing (marginally) stable marginally outer trapped surfaces (Gabach-Clément, Jaramillo). Geometric inequalities of this type are attracting more and more attention and many different techniques have been introduced to the field (e.g. by Sergio Dain).

Wissenschaftliche Veröffentlichungen / Publications

see google scholar

 

Ausgewählte Vorträge / Selected invited talks

  1. On CMC-foliations of asymptotically flat manifolds, Online Seminar Geometric Analysis, 2020 [video]
  2. On CMC-foliations of asymptotically flat manifolds, DMV-Tagung, Sektion Differentialgeometrie, globale Analysis und Anwendungen, Karlsruhe, 2019
  3. Static black hole uniqueness theorems, ICTP School on Geometry and Gravity, Trieste, 2019 [poster, videos]
  4. On CMC-foliations of asymptotically flat manifolds, Geometric Analysis and General Relativity. A conference in honour of Gerhard Huisken, ETH Zürich, 2019
  5. On special hypersurfaces of the Schwarzschild spacetime and related uniqueness theorems, Geometric Analysis meets Geometric Topology, Heidelberg, 2019
  6. Wo liegt der Schwerpunkt eines Sterns -- und was hat das mit Mathematik zu tun?, Open Salzburg Mathematics Colloquium, 2019 [poster]
  7. On special hypersurfaces of the Schwarzschild spacetime and related uniqueness theorems, A Celebration of Mathematical Relativity in Miami, 2018 [poster]
  8. On foliations related to the center of mass in General Relativity, International Congress on Mathematical Physics, Montréal, 2018
  9. On extensions of CMC Bartnik data, Relativity Seminar, Vienna, 2018
  10. On the center of mass of asymptotically hyperbolic initial data sets, Asymptotically hyperbolic manifolds, Banff Research Station (BIRS), 2018 [video]
  11. On foliations related to the center of mass in General Relativity, Brussels-London geometry seminar, 2018
  12. On foliations related to the center of mass in General Relativity, Field equations on Lorentzian spacetimes, Hamburg, 2018 [poster]
  13. AWM Distinguished Speaker series, University of Oregon, 2018
  14. Rigidity properties of the Schwarzschild manifold in all dimensions, Advances in General Relativity, Erwin-Schrödinger-Institut, Wien, 2017 [conference website]
  15. A geometric boundary value problem related to the static equations in General Relativity, Advances in Geometric Analysis, ETH Zürich, 2017 [conference website]
  16. Service Learning im Lehramtsstudium Mathematik, mit Dr. Stefan Keppeler, Arbeitsgemeinschaft Mathematik zwischen Schule und Hochschule, Universität Tübingen, 2016 [Ankündigung, Folien, Blog]
  17. On foliations related to the center of mass, Conference in Mathematical General Relativity, Tsinghua Sanya International Mathematics Forum, 2016 [poster]
  18. Mathematik lehren -- oder lehren, Mathematik zu lernen (und lehren)? Kolloquium über Mathematik, Informatik und Unterricht, ETH Zürich, 2015 [Ankündigung, Folien]
  19. Scientific (r)evolution: a mathematical perspective, Gravity and Geometry: Centenary Perspectives on General Relativity, Rotman Institute of Philosophy, London, Ontario, 2015 [poster, conference website]
  20. From Schwarzschild to General Relativity: modeling physical phenomena with the help of geometry, New directions in Mathematical Physics and beyond, Jena, 2015 [conference website]
  21. Mass in Newtonian gravity and general relativity (Colloquium delivered at Monash University, 2014) [slides]
  22. The geometry of static spacetimes in General Relativity (Delivered at Stanford University, 2013) [slides]
  23. The geometry of static spacetimes in General Relativity (Delivered at the Mathematical Sciences Research Institute MSRI 2013) [video]
  24. The Newtonian limit of geometrostatics (Delivered at the Centre International the Rencontres Mathématiques CIRM 2011) [video]
  25. From Newton to Einstein: A guided tour through space and time (Delivered at Geometry Festival @ Duke University 2012; Junior Colloquium @ University of Tennessee, Knoxville 2012; Undergraduate Lecture Series @ CUNY 2012; Cross program lecture @ Park City Mathematics Institute PCMI 2013; undergraduate lecture @ Lewis & Clark College 2013, Heidelberger Life Science Lab 2013 etc.) [video, slides, Folien deutsch]