For ten weeks this winter, a diverse collection of gravitational
physicists gathered at Santa Barbara's Kavli Institute of
Theoretical Physics to discuss myriad phenomena related to gravity
in 
spacetime dimensions. Though there was also
interest in the case 
, the program (organized by Luis Lehner,
Rob Myers, and myself) largely focused on the case 
. Why
study gravity outside of four dimensions? In my opinion, the most
basic reason is that the dimension is a parameter one can dial to
learn more about the deep nature of gravitational phenomena.
Recall, for example, that via Kaluza-Klein reduction a theory with
can in some cases be regarded as a 4-dimensional theory
with complicated matter fields. Thus, one might expect any truly
universal property of gravity to apply equally well to both high
and low dimensions. The Bekenstein-Hawking area law for black
hole entropy is perhaps a prime example such such a
dimension-independent phenomenon, and as such is widely regarded
as a deep principle of gravitational physics. One aim of this
program was to discover what other phenomena are similarly
universal, and which phenomena are not. Other motivations for
studying higher dimensional gravity include string theory, various
`large extra dimension' scenarios for our universe, and general
fun with mathematical physics.
Gravity in higher dimensions exhibits a number of striking features which stretch our 3+1 intuition. For this reason, a primary goal of the program was to bring together higher-dimensional physicists with specialists (e.g. mathematical physicists or numerical physicists) who usually work in 3+1 dimensions. The hope was that by bringing to bear sophisticated tools, progress could readily be made on a number of higher dimensional issues, mostly centered on the physics of black holes. As usual in general relativity, these issues focused on existence, uniqueness, thermodynamics, stability, and dynamics.
This fusion was quite successful, and the spectrum of results which came out of the workshop is too broad to fully summarize here. With apologies to those whose work I will not mention (and to those for whom I mention only a small part of their work), I'd like to quickly review a few areas which were the focus of much discussion and where progress was especially significant, and/or there is great potential for further input from inspired readers of this article. I emphasize that only a small fraction of the interesting results obtained are mentioned below.
Existence, uniqueness, and thermodynamics
What sorts of black objects exist in higher dimensions? Recently, we have learned that higher dimensions host a rich spectrum of black objects, including stationary black rings for which cross-sections of the horizon are not spheres; see [1] for a recent review. In 3+1 dimensions, Hawking's theorem guarantees that the horizon topology is spherical. Using similar methods, some results on the higher dimensional case were already known. However, a forthcoming set of papers ([2] and others to appear) by Lars Andersson, Greg Galloway, Jan Metzger, and Rick Schoen will close an important loophole related to the possibility of Ricci-flat metrics on the horizon, further narrowing the possibilities.
As with Hawking's theorem, it can be quite unclear how a given result will generalize to higher dimensions. A particularly interesting example is that of the black hole rigidity theorem, which states that every stationary black hole is axisymmetric; i.e., that it has a rotational Killing field. An important corollary of this result is that the event horizon of a stationary black hole is a Killing horizon, and thus that it has a well-defined surface gravity which is constant over the horizon. In this way, the rigidity theorem is deeply connected to black hole thermodynamics, and one would expect it to generalize readily to all dimensions. However, the standard proof in 3+1 dimensions [3] relies on the fact that cross sections of the horizon have topology 
. The generalization to higher dimensions is highly nontrivial, and has only recently been established by Hollands, Ishibashi and Wald [4], in part as a result of the KITP program and interaction with Jim Isenberg and Vince Moncrief, from whom a related paper is expected soon.
Stability
Stationary black rings exist, but are they dynamically stable? A number of potential instabilities have been discussed in the literature: radial instabilities, Gregory-Laflamme Instabilities (see below), super-radiant instabilities, and potential instabilities associated with absorption or emission of radiation. Our program saw significant progress in establishing that black rings do suffer from such instabilities. First, a work by Jordan Hovdebo and Rob Myers established [5] that very large black brings are unstable to a Gregory-Laflamme type instability. In addition, Henriette Elvang, Roberto Emparan and Amitabh Virmani provided evidence [6] that all neutral black rings (at least, in 4+1 dimensions) are unstable in some way. For the branch of the black ring solutions with the largest entropy, they show that such black rings suffer from a radial instability as well as a Gregory-Laflamme instability. Furthermore, they give evidence that the Gregory-Laflamme instability should lead the black ring to break up into a set of black holes with large orbital angular momentum. Finally, Oscar Dias showed [7] that if `doubly spinning' black rings exist, then they will necessarily have a super-radiant instability.
Dynamics
A new dynamical issue that arises in dimensions is the
Gregory-Laflamme instability [8] of thin black strings, membranes,
etc. It is known that many black string solutions are linearly
unstable to perturbations that break translational symmetry along
the string but which preserve rotational symmetry around the
string. The instability causes the string to become `lumpy,'
thickening in some places while thinning at certain `necks.' The
endpoint of this instability has been a subject of much debate and
discussion. The original work [8] conjectured that the thin necks
might shrink to zero size and then `break,' so that the endpoint
is a set of separated black holes. Much of the interest in this
work revolves around the fact that such a bifurcation would
violate certain forms of Cosmic Censorship. Because the
interesting question involves the non-linear regime, it is natural
to explore this question numerically. Indeed, the 2003 numerical
simulation [9] was the focus of much discussion at the program. In
particular, when plotted against their asymptotic time coordinate,
the evolution of many quantities in the spacetime shows signs of
slowing significantly near the point where their code crashes. One
might take this as evidence in favor of a scenario (see e.g. [10])
in which the endpoint is simply a static lumpy black string. It
goes without saying that better numerical simulations are needed
(and the authors of [9] are making progress in this direction).
However, our discussions also produced the conclusion that more
physics could be obtained by analyzing the data of [9] in terms of
a more physical time coordinate; e.g., the retarded time along
past null infinity. Because this retarded time might differ
substantially from the coordinate time of [9], such a new analysis
might suggest very different results more in line with the
original bifurcation suggestion of [8]. Interested individuals
may wish to consult [10,11,12] for more detailed discussions of
possible endpoints.
Summary
The 2006 KITP program `Scanning new horizons: GR Beyond 4
dimensions' was a period of intense interaction and discussion
between a diverse array of physicists which led to a number of
exciting new results. Yet much remains to be done, and in
particular there is much room for input from both mathematical and
numerical relativists. Although the physics concerns large numbers
of dimensions, interesting special cases often have sufficient
symmetry to reduce problems to either 3 or 2+1 dimensions or less.
Examples of such problems include the stability of
`ultra-rotating' higher dimensional black holes, the existence of
stationary black rings in 
dimensions, and the existence
and stability of `braneworld' black holes. In such cases,
numerical analyses can be especially useful. I will be only too
happy if this short summary encourages others to enter this
exciting field and to further develop existence, uniqueness,
thermodynamic, stability, and dynamic results in gravity beyond 4
dimensions. Links to the program talks and discussions can be
found at http://online.itp.ucsb.edu/online/highdgr06/, and provide
a useful introduction to many such topics.
References
[1] R. Emparan and H. S. Reall, ``Black Rings,'' hep-th/0608012 .
[2] G. J. Galloway, ``Rigidity of outer horizons and the topology of black holes,'' gr-qc/0608118 .
[3] Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152-166 (1972)
[4] S. Hollands, A. Ishibashi and R. M. Wald, ``A higher dimensional stationary rotating black hole must be axisymmetric,'' gr-qc/o605106 .
[5] J. L. Hovdebo and R. C. Myers, ``Black rings, boosted strings and Gregory-Laflamme,'' Phys. Rev. D 73, 084013 (2006) hep-th/0601079 .
[6] H. Elvang, R. Emparan and A. Virmani, ``Dynamics and stability of black rings,'' hep-th/0608076 .
[7] O. J. C. Dias, ``Superradiant instability of large radius doubly spinning black rings,'' Phys. Rev. D 73, 124035 (2006) hep-th/0602064 .
[8] R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837 (1993) hep-th/9301052 .
[9] M. W. Choptuik, L. Lehner, I. Olabarrieta, R. Petryk, F. Pretorius and H. Villegas, ``Towards the final fate of an unstable black string,'' Phys. Rev. D 68, 044001 (2003) gr-qc/o304085 .
[10] G. T. Horowitz and K. Maeda, ``Fate of the black string instability,'' Phys. Rev. Lett. 87, 131301 (2001) hep-th/0105111 .
[11] D. Garfinkle, L. Lehner and F. Pretorius, ``A numerical examination of an evolving black string horizon,'' Phys. Rev. D 71, 064009 (2005) gr-qc/0412014 .
[12] D. Marolf, ``On the fate of black string instabilities: An observation,'' Phys. Rev. D 71, 127504 (2005) hep-th/0504045 .