For a third time, a sizable portion of the gravity community gathered in Warsaw to discuss recent advances. Participation of excellent physicists working on exciting topics, not limited just to the field of canonical or quantum gravity, created a vibrant atmosphere in the meeting. The workshop was sponsored by Banach Center of Mathematics of Polish Academy of Sciences (PAN). It was organized by Jerzy Lewandowski, Jacek Jezierski (Warsaw University), Jerzy Kijowski (Center for Theoretical Physics PAN, Warsaw) and Abhay Ashtekar (Penn State) who served as a scientific advisor. Over 90 participants from 12 countries attended about 60 talks. Workshop was divided into two parts. First week was devoted to problems in classical general relativity; its title was "Null Structures and other Aspects of Classical Gravity". Second week was devoted to problems of ``Quantum Gravity''. In between the two parts was a one day celebration of Ted Newman's birthday, with talks by Aichelburg, Ashtekar, Penrose, Stachel and Trautman as well as interesting after dinner reminiscences in the evening, in Palac Staszica. The organization of that day was directed by Bialynicki-Birula, with the help of Demianski, Nurowski, Tafel and Trautman.
By far the most extensive application of the null-cone structures being the subject of the first part of the meeting is the Null Surface Formulation (NSF) of the Einstein's theory started about 10 years ago by Newman and his collaborators. According to this theory, the space-time is a secondary object defined as the set of solutions of certain 3rd order ODE. The recent development indicates relations with Cartan's theory of differential equations (Newman, Nurowski, Kozameh) and applications to the gravitational lensing (Fritelli, Tod). NSF is a relative of the Twistor program, the advantage of the NSF being that it applies to real, not necessarily analytic space-time of the Lorentz signature. However, in one of his three talks Penrose reported on his recent attempt to construct the twistor space for a generic curved space-time of the signature! An exciting application of the twistors that bridges the classical and the quantum theories is the Bialynicki-Birula twistor Wigner-function introduced for the participants of the workshop by its inventor. Twistor spaces corresponding to anti-self-dual metrics in the signature with covariantly parallel spinor were characterized before the audience by Dunajski. A recent discovery of Damour, Henneaux, Julia and Nicolai traces the roots of the BKL BKL behavior near space-time singularities (in the dimensions greater/equal 4) to the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac-Moody algebra. This intriguing result and its consequences were presented in a comprehensive lecture by Henneaux, one of the three talks on singularities (Bizon, Aichelburg). Another major topic was the novel, quasi-local generalization of the black hole theory, provided by ``isolated horizons'' (IHs). The mechanics and geometric invariants including geometric conditions that distinguish the Kerr horizon among all IHs were discussed (Beetle, Krishnan, Lewandowski, Pawlowski). An interesting result shown by Racz was his proof of the existence of a Killing vector in the case of the bifurcate IHs. Related formulations of the mechanics of the null shells and scri were explored by Chrusciel, Kijowski and Tafel. In the area of the traditional black hole theory, Jacobson argued that ``black hole entropy is not about black holes''. The recent progress in understanding of the Penrose inequality was discussed by Frauendiener. The ``canonical'' theme of the workshop was underlined by the Beig's talk on the motion of the point particles in general relativity and constraint equations. Other subject covered were ``32 Double Coverings of for '' (Trautman), ``The Hopf fibration - five times in physics'' (Urbantke) and ``Real Sources of Holomorphic Coulomb Fields'' (Kaiser).
The main topic discussed in the second, quantum, part of the Workshop was ``quantum geometry''. Recent advances within this approach in meeting that challenges of quantum gravity were reviewed by Ashtekar in his lecture on the Newman day. A focal point of research in the canonical approach during last several years has been the semi-classical sector of the theory. Thiemann and his collaborators (Winkler, Sahlmann) explored the idea of construction of semi-classical states by gluing the coherent states defined on SU(2). Another approach follows from Varadarajan's embedding of the free Maxwell theory Fock space into the U(1) analog of the polymer-like excitations Hilbert space of the quantum geometry. A generalization to the SU(2) theory described in the second of Ashtekar's talk provides a natural candidate for the Fock flat space-time vacuum and a starting point to bridge the background independent, non-perturbative approach and perturbative results. A third way to extract a semi-classical information from the non-perturbative sector was the subject of Bojowald's talk. By a quantum symmetry reduction, and by exploiting discreteness of volume in quantum geometry, he obtained a substitute for the familiar Wheeler-DeWitt equation that naturally resolves the big-bang singularity. A second focal point was provided by the lively discussions on ``spin foam models'', which provide a path integral approach to the quantum gravity, based again on quantum geometry. The idea initiated by Reisenberger and Rovelli and has drawn a great deal of attention because of the recent finiteness results by Perez, Crane and Rovelli which, roughly speaking, are analogous to the finiteness claims of perturbative string theory. All principal researchers (except Baez and Crane) in the area reported on the status of their work. The promising idea of providing the space of the spin-foams with the Hopf algebra structure was explained by Markopoulo. The issue of observables in quantum geometry was discussed by Pullin. A third focal point to the workshop was provided by simplicial Lorentzian gravity. Many of the frequently asked questions were exhaustively answered by Loll, Ambjorn and Jurkiewicz.
To provide a balance, there were several talks on the nearby areas, particularly quantum groups (Woronowicz, Zapata, Kowalski-Glikman), branes (Meisner, Pawelczyk, Louko), 2+1-gravity (Bengtsson, Freidel, Wisniewski), the theta functions (Mourao), as well as the talks on the status of the other issues of the quantum theory, such as general covariance (Fredenhagen), gravitational quantum state reduction (Penrose), gravitational collapse (Hajicek), technical and conceptual issues in approaches based on histories (Dasgupta, Kuchar), QCD on the lattice (Kijowski), pre-canonical quantization (Kanatchikov). Especially instructive was the lecture by Woronowicz on the representations of the quantum Lorentz group. Those of us who apply the quantum groups in everyday work, could ask the master about some subtleties and other possible ways of q-deforming.
In summary, the atmosphere of the meeting was most stimulating due
to active participation of both, experienced as well as young
researchers. Interactions between different areas of research in
mathematical/quantum gravity and at the same time avoiding the
overload of big conferences seemed to be an advantage. Hopefully,
the Warsaw workshops CQG have already become a tradition and
future ones will again bring excellent researchers and lecturers.