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Working Areas of the IMPRS for Geometric Analysis, Gravitation and String Theory
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Theoretical research in gravitational physics nowadays covers a broad range of topics, ranging from pure and applied mathematics (differential geometry, the theory of partial differential equations and numerical analysis) over the physics of black holes, gravitational radiation and cosmological applications of Einstein's theory to the most ambitious attempts to unify gravity with the other fundamental forces in nature and thereby reconcile Einstein's theory of general relativity with quantum theory (we summarily refer to these attempts as "string theory"). All these topics are represented under one roof at the AEI, and the school will aim to disseminate and promote research on them in collaboration with universities in Potsdam and Berlin. While in the area of classical general relativity, the proposed research school is concentrating more on the mathematical and conceptual foundations as outlined above, it will have close contact with the research groups at the AEI involved in experimental aspects of general relativity as well as the numerical simulation of the relevant models, with the possibility of assigning up to two PhD positions to problems in this area of research.
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General Relativity
During the last decade major progress has been made on
many foundational aspects of general relativity: Non-linear wave equations
such as the Einstein equations or wave maps have been much better understood,
first global existence and stability results for data without symmetries have
become known and the asymptotic behaviour of solutions could be clarified in
several cases.
The interplay between physical matter content and the
geometric structure of spacetime has been a further area of successful
research:
New energy inequalities have been established and different concepts
of quasilocal energy have been related to each other, specific models such as
perfect fluid, the Einstein-Maxwell or the Einstein-Vlasov system have become
understood in great detail, even in the vicinity of spacetime
singularities. Concerning the Einstein field equations, the time decay of
solutions has often been controlled by expansions in quasinormal modes, in
close analogy to "resonance expansions" in Schrödinger theory.
The theory of
nonlinear elliptic and parabolic equations for hypersurfaces and submanifolds
has been developed to a stage where hypersurface foliations can be constructed
in a broad variety for the purpose of measuring energy contents of a region or
for the construction of suitable gauges in numerical relativity. An ambitious
goal would be to describe all classical physical quantities associated with an
isolated gravitational system in purely geometric terms using a natural family
of hypersurfaces contained in spacetime. Elliptic techniques have also been
used to prove rigidity theorems and uniqueness results for several models in
relativity.
In this section the aim will be to apply advanced mathematical
techniques to concrete questions in General Relativity and to identify links
between newly available mathematical techniques and physical problems. It is
hoped also that a better understanding of the foundations of the above
problems in general relativity will provide justification and fruitful
theoretical input for numerical simulations, just as in turn numerical
simulations in neighbouring research groups can draw attention to interesting
new phenomena.
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Geometrical Analysis
The theoretical foundations of General Relativity bring together geometry and
analysis in many of their particular research directions, such as Riemannian
and Lorentzian Geometry, Lie Groups, partial differential equations and the
calculus of variations, even geometric measure theory. One major challenge in
the foundations of General Relativity is to identify or create those
mathematical structures which are best suited to the modelling of classical
physical concepts such as mass, momentum or angular momentum in the nonlinear
context of a curved spacetime satisfying the Einstein equations. In view of
the new developments discussed above several mathematical research areas can
be identified in this context:
Exciting breakthroughs on nonlinear wave maps
were recently made concerning global existence of solutions. For the first
time this opens the door to further studies on the even more difficult
equations governing gravitation in relativity or branes in string theory. It
also creates hope towards a better understanding of singular behaviour in such
equations as it occurs for example in models of gravitational collapse.
The mathematical theory of hypersurfaces and submanifolds of prescribed curvature
has made progress almost to a level where it has become possible to engineer
suitable nonlinear equations for these objects that ensure favourable
properties such as monotonicity identities, making them so interesting in
applications such as relativity. It seems therefore important to further
develop the theory of nonlinear elliptic and parabolic equations related to
curvature to obtain an even more precise picture of solutions and their local
and global behaviour.
In this section research projects would often aim at
refining and creating the mathematical tools and structures in such a way that
they become suitable for the applications above in general
relativity. Concrete projects would concern the construction and analysis of
foliations of prescribed curvature in manifolds and spacetimes, the existence
and behaviour of solutions to a wide range of elliptic and parabolic equations
for submanifolds as well as the behaviour near singularities. These topics
are also related to open problems in the calculus of variations with
applications to isoperimetric and energy inequalities. Numerical analysis in
relativistic hydrodynamics has already successfully used smoothed particle
methods in modelling phenomena with free boundaries. Wavelet analysis may be a
successful tool also in resolving fractal structures. A further major area of
research will be the asymptotic behaviour of solutions to the Einstein
equations and the constraint equations, with a view towards the derivation of
a priori estimates which control the behaviour of solutions and give
justification to numerical approximations.
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Quantum Gravity, String Theory and the Unification of Fundamental Interactions
The search for a consistent theory of quantum gravity and its
unification with the fundamental interactions of particle physics constitutes
the biggest challenge for theoretical physics at the turn of the century. The
conceptual and mathematical difficulties that must be overcome are
enormous. There is at this time no theory that can pretend to be "the" theory
of quantum gravity. Rather, there are promising ansaetze, and it is hoped that
these idea - together with unmistakable signals of new physics in
observational astrophysics and particle accelerator experiments - will
eventually pave the way to a better understanding. Current activities in the
quantum gravity division at the AEI reflect the diversity of the major
approaches.
The approach to quantum gravity which is presently favored by the
majority of researchers is based on superstring theory. This ansatz tries to
cure the shortcomings of perturbatively quantized general relativity by a
radical modification of Einstein's theory at very short distances (of the
order of the Planck length). The fundamental "object" of the theory is a one-
dimensional extended object, a relativistic string (or superstring, if
fermions are included). The point-particles of conventional quantum field
theory are identified with the quantized excitations of this string, and in
the most optimistic scenarios the massless excitations should then correspond
to the known matter described so well by the standard model of elementary
particle physics. Remarkably, (closed) string theory predicts the existence of
a massless spin- two particle, the graviton, whose self-interactions coincide
with those of Einstein's theory at the lowest non-trivial order—in this way
the unification of gravity with the other fundamental forces in nature becomes
a prediction of string theory. Superstring theory is at this time the only
ansatz that succeeds in removing the inconsistencies of perturbatively
quantized Einstein's theory. The main task is now to find a non-perturbative
formulation of the theory (sometimes dubbed "M Theory"). One of the candidates
is supermembrane theory and M(atrix) theory.
An important question concerns
the moduli space of consistent compactifications of string theory. To obtain
information on non-perturbative relations between different compactifications
one establishes and exploits (non)perturbative duality symmetries. This leads
to a better understanding of the global structure of the parameter space of
string theory and provides powerful tools to compute various quantities
(e.g. correlation functions) of the compactified theories.
Alternative approaches to quantum gravity, which are also represented at the
Albert Einstein Institute, are based on canonical and path integral
methods. Although the canonical quantization of Einstein's theory leads
directly to a kind of Schrödinger equation for quantum gravity, the so-called
Wheeler-DeWitt equation (actually a constraint equation), this approach had
yielded little fruit until Ashtekar's discovery of new variables in 1986. The
aim of this approach is to implement the guiding principles of Einstein's
theory directly in a background independent theory of quantum general
relativity; in particular, it has led to completely new ideas about the
structure of space and time at very short distances.
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Areas of common interest and joint projects
The school will foster research in all these areas, and in addition put
special emphasis on those aspects of the research program which link the
topics described above. Below is a selection of topics of common interest
between several of the participating institutions, where synergy effects are
expected to be particularly important:
- Extremal surfaces in spaces with Lorentzian signature
- Quantization of (super)membrane theories
- Minimal surfaces and the initial value problem forEinstein's field equations
- Exceptional Lie groups and exceptional geometries
- Energy inequalities and supersymmetry
- Calibrations and variational problems
- PDEs in Yang Mills theories
- Structure of the Geroch group
- BKL dynamics and symmetries in dimensionally reduced models of gravity and supergravity
- Manifolds with special holonomy
- Superstring compactifications and brane models
- Field and string theories in non-commutative spaces
- String theory and algebraic geometry
- String theory and black holes
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A Selection of possible PhD Projects
General Relativity:
- The dynamics of constant mean curvature foliations
- Gauge conditions for the conformal field equations
- Existence of AdS type solutions in higher dimensions
- Asymptotics of cosmological models in a phase of unlimited expansion
Geometric Analysis:
- Weak solutions of mean curvature type equations
- Singularities of geometric nonlinear parabolic PDEs
- Qualitative behaviour of asymptotically flat branes
- Action-angle variables in spectral theory
- Wavelet analysis in the distribution of stars
- Asymptotics of solutions to elliptic and parabolic equations on manifolds with singularities
Quantum Gravity and String Theory:
- Higher order curvature corrections from the fundamental supermembrane
- Higher order corrections to Wilson loops in supersymmetric Yang Mills theories
- Gauged supergravities in three dimensions and the AdS/CFT correspondence
- Conformal quantum mechanics, black holes and minimal representations
- Mirror symmetry in N=1 string compactifications
- Compactifications on manifolds with G2 holonomy and their phenomenology
- String compatifications with branes
- Field theories in non-commutatative spaces and string theory
- Yang Mills theories, matrix models and non-perturbative quantum supergravity
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© 2012, Max Planck Institute for Gravitational Physics, Potsdam |
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