Director: Prof. Dr. Gerhard Huisken
The division "Geometric Analysis and Gravitation" is concerned with the role
of mathematics in the description of gravitation. Mathematical concepts
of central importance in this context are geometric evolution equations and
geometric variational problems. Topics studied in the division include
the choice of natural coordinates, cosmology and symmetry, asymptotically
flat spacetimes, numerical methods, and Einstein's equations with matter.
The description of astronomical objects by means of mathematics is one
of the oldest human endeavours. To forecast the motion and appearance of
the moon, the planets and the stars Greek mathematicians developed the
first basic concepts of geometry still valid today, their theory of
conic sections was used centuries later by Kepler to formulate his laws
of motion for the planets. Following the discovery of infinitesimal
calculus in the 17th century the development of analysis made it
possible to formulate laws of equilibrium and laws of motion in terms of
variational principles, culminating in particular in the variational
interpretation of the Einstein field equations in General Relativity by
David Hilbert.
The second half of the 20th century has seen tremendous progress both in
gravitational physics and in mathematics: On the side of physics black
holes have developed from a theoretical and mathematical curiosity to
very real objects of observation, gravitational lensing has become an
important observational tool, cosmological models interact with particle
physics, and gravitational waves are within reach of modern detectors.
On the other hand mathematical analysis has developed a deep
understanding of nonlinear elliptic and parabolic partial differential
equations arising from geometric variational problems, differential
geometry has developed methods to link local curvature properties of
surfaces and spaces to global properties of their shape, and numerical
simulations allow detailed quantitative predictions from complex
mathematical models. A main task for the Division "Geometric Analysis
and Gravitation" is the pursuit of basic research on mathematical
methods relevant for the modelling of gravitation and the investigation
of specific models for concrete physical phenomena using modern
mathematical techniques.
Since Galilei, Brahe and Kepler the theory of gravitational attraction
between celestial bodies has benefited from lively interaction between
astronomical observation, theoretical physics and mathematical
modelling. In recent times the search for unified theories and new
developments in String Theory and M-Theory have suggested many new
connections between mathematics and physics, the interpretation of
observations in astrophysics and gravitational wave experiments require
new methods in theoretical and numerical analysis. The other divisions
of the Albert Einstein Institute provide continuous exposure to these
developments and benefit in turn from the mathematical expertise offered
here.
Current research topics
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