HIGHER DIMENSIONAL GENERAL RELATIVITY

In a recent paper P.Bizon and T.Chmaj (P. Bizon, T. Chmaj, and B.G. Schmidt, Critical Behavior in Vacuum Gravitational Collapse in 4+1 Dimensions, Phys. Rev. Lett. 95, 071102 (2005)) put forward an idea which opens up new horizons in the studies of higher dimensional gravity. Namely, they showed that in five spacetime dimensions one can perform a consistent cohomogeneity-two symmetry reduction of the vacuum Einstein equations which - in contrast to the spherically symmetric reduction - admits time dependent asymptotically flat solutions. This model provides a simple theoretical setting for studying the dynamics of gravitational collapse in vacuum, both numerically and analytically. Similar models can be formulated in higher dimensions as long as the corresponding sphere admits a non-round homogeneous metric. We plan to investigate how the dynamics of such models depends on the dimension and the coset space. In particular we are interested in the role of self-similar and stationary solutions in singularity formation. We plan to study qualitatively new phenomena, which appear in higher dimensions, for example, in nine dimensions where the Hawking mass is not monotone, suggesting a possibility of violating the weak cosmic censorship in the evolution of initial data having locally negative energy density.

Further subtopic concerns Calabi Yau manifolds, and their recent generalizations, which appear in studies of string compactifications, where some of their geometrical properties are related to appropriate low-energy 4D physics. In a second line of investigation certain Calabi-Yau geometries appear also in detailed studies of the entropy of certain black holes which can be calculated both from the gravitational (geometric) point of view and using a microscopic framework. We are mostly interested in the complex Monge-Ampere equation, especially on Kähler manifolds. The main reason why this equation is so important in geometry is the fact that the Ricci curvature of a given Kähler metric is given by the Monge-Ampere operator. Therefore many important problems like solving the Calabi conjecture or finding a Kähler-Einstein metric boil down to solving this equation (this was achieved by Yau in 1978).

GRAVITY AND GAUGE THEORIES – GEOMETRY AND INTEGRABILITY

In a very recent investigation (A. Janik, R. Peschanski, `Asymptotic perfect fluid dynamics as a consequence of AdS/CFT', hep-th/0512162), we showed that the properties of a gauge theory matter system were mapped into geometrical properties of 5 dimensional solutions of Einstein equations on the string/gravity side. It is extremely interesting to consider various extensions to other systems with less symmetries, to study the interrelations between the appearance of horizons on the dual string/gravity side with properties of the gauge theory matter in such a dynamical setting. Another problem where geometrical notions appear (now in the most general sense) is the study of the AdS⁵ ×S⁵ superstring on the quantum level. A lot of evidence has been accumulated that the worldsheet theory is integrable, i.e. it possesses a symmetry structure most probably of a Hopf algebra type (Yangians have been argued to play a role here), and key properties are determined by the solutions of the Yang-Baxter equation. It is very interesting to examine in detail these solutions, which are not yet complete, their precise symmetry structure, as it seems that various features are different from the ones considered before. Moreover it would be then fascinating to employ techniques from integrable field theories to study the model in the fully quantum regime.

EVOLUTION EQUATIONS AND TOPOLOGICAL DEFECTS

The main limitation of the analytical methods methods proposed in the literature is that they do not work in the case of global topological defects, where certain fields (the ones related to massless particles) have 'bad' asymptotic behaviour in the directions perpendicular to the defect. An elegant singular perturbation method, based on asymptotic expansion of the appropriate solutions of the pertinent nonlinear partial differential equations with respect to extrinsic curvature of the brane, has been worked out for local topological defects. Our aim is to extend this method to the global defects. The problem has nontrivial geometrical part due to generally non-vanishing curvature of the pertinent branes.
Evolution equations for the p-branes, obtained in the case of local topological defects, lead to flow equations for the metric as well as for the extrinsic curvature of the branes. We would like to investigate properties of such flows, in particular, we would like to find the relations with the Ricci flow, which has played the important role in the recent progress on the Poincare conjecture.

NONCOMMUTATIVE GEOMETRY: EQUATIONS OF MOTIONS

Although many examples noncommutative manifolds have been studied, it was only recently that significant results concerning their geometry (Dirac operator) were discovered. Still, we have no knowledge of their Riemannian geometry. The classical (commutative) objects like Riemann or Ricci curvature or scalar curvature have yet no precise counterpart in the noncommutative description. Moreover, the action makes sense only in the Euclidean framework thus leaving the physically relevant Lorentzian cases without mathematical model. Finally, even with the action we still cannot not even pose the equation of motions, which would replace in the noncommutative setup the Einstein equations. It is a major task and a big endeavour to cure these problems: first to make sense of the Lorentzian action and then to construct noncommutative geometrical objects, which would appear in the Einstein-type equations of motions. However, the task is not at all hopeless: due to the detailed construction of spectral geometries for several noncommutative manifolds, including quantum spheres (with respective quantum symmetries) and the extension of the constructions to the case of Lorentzian metrics, we are able to investigate all details of this particular geometries.