Geometric Analysis and Gravitation
The division on 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.