Neil Cornish, Cambridge

N.J.Cornish@damtp.cam.ac.uk

On a crisp fall weekend in Cleveland, an unlikely collection of mathematicians and physicists met at Case Western Reserve University to discuss the large scale topology of the universe. There was a certain irony to the location, as the meeting was being held just a short walk from where Michelson and Morely dispensed with the ether one hundred years earlier, while the cosmologies being discussed come with an absolute frame of reference.

The main aim of the meeting was to foster closer ties between geometers, cosmologists and theoretical physicists. Through this exchange of ideas and expertise, we hoped to arrive at a better understanding of the theoretical and observational characteristics of multi-connected cosmologies.

The meeting ran to a workshop format with a small number of talks providing a springboard for extensive and lively discussions. This meant that all 30 participants did indeed participate, even though only half the participants gave talks. One of the most active participants was Bill Thurston, who got things rolling by taking us on a tour of topology and geometry in dimensions 1 through 5. We learnt how topology becomes more flexible with increasing dimensionality while geometry becomes more rigid. The majority of Thurston's talk was devoted to 3-manifolds, where both topology and geometry find their optimal balance between flexibility and rigidity. Concepts such as the prime decomposition of 3-manifolds were made accessible to the physics audience by relating the underlying ball-gluing construction to wormholes. Thurston emphasised that most 3-manifolds are hyperbolic. Picking up on this lead, David Spergel reviewed the mounting observational evidence that we live in a sub-critical universe with hyperbolic spatial sections. Gary Gibbons took us back to the quantum gravity epoch and considered how the universe might arrive at a non-trivial topology. The mathematicians were introduced to the Euclidean path integral approach and semi-classical real tunnelling geometries. Steve Carlip continued in a similar vein, but argued that the density of topologies might dominate the gravitational action in the path integral. Later in the meeting John Freedman discussed multi-connected spacetimes from a Lorentzian quantum gravity perspective and Bai-Lok Hu described the Casimir and other finite size effects. Closely related to this was Jean-Philippe Uzan's description of the obstructions to forming topological defects, such as cosmic strings, in universes with non-trivial topology.

The majority of the workshop was devoted to observational searches for
topology. The meeting organizer, Glenn Starkman, introduced this topic
with a historical review of global topology in cosmology, starting in
1917 with de Sitter's variant to Einstein's static universe and
moving to the present. Since the best window on topology is provided
by the cosmic microwave background radiation, David Spergel provided a
review of CBMR physics and observations. We also heard how the MAP and
Planck satellites will transform our view of the CMBR early next
century. Speaking for the Toronto group, Turan Souradeep told us about
their efforts to model the CMBR power spectrum in multi-connected
hyperbolic universes. On the same topic, Janna Levin entertained us
with her quirky description of the work done by the Berkeley group,
showing how cusped manifolds lead to flat spots in the CMBR. Still on
this theme, I outlined the work Neil Turok and I have done to develop
a simple numerical method for finding the eigenmodes of arbitrary
compact manifolds. Moving to more direct detection methods, I
explained how Spergel, Starkman and I hope to use the MAP satellite to
search for topologically matched circles in the CMBR. Searching a lot
closer to home, Boud Roukema showed how matched quasar groupings could
be used to test for non-trivial topology. This method should be quite useful
when the Sloan and Quest digital sky surveys deliver millions of new
quasars positions. As a fitting testament to the cross-disciplinary
nature of the workshop, one of the most intriguing observational
prospects was described by the topologist Jeff Weeks. Using his *
SnapPea* computer program, Weeks showed how the size and position of
just a few matched circle pairs could be used to completely
reconstruct the topology of the universe. A similar procedure can
also be used with the quasar groupings.

All the talks stimulated enthusiastic discussion that brought in
the other participants, including the topologist Colin Adams, Rob
Meyerhoff, John Ratcliffe and Bill Goldman
and physicists Ted Jacobson, Tanmay Vachaspati and Rich Gott. By the
end of the workshop the topologists were arguing about the Sachs-Wolfe
effect and the physicists were arguing about Dehn surgery on cusps.
All the participants agreed that it was one of the best meetings they
had been to in years.
If you want to hear more about what went on at the meeting, keep an
eye out for the workshop proceedings that will be appearing as a
special issue of * Classical and Quantum Gravity*. Or you could do
the nineties thing and visit the workshop website at
http//theory5.phys.cwru.edu.
There you will find contributed talks, a copy of the Cleveland Plain
Dealer article covering the workshop (complete with a picture of Glenn
Starkman eating a hyperbolic potato chip) and perhaps an audio file of
the radio coverage by the CBC radio program ``Quips and Quarks'',
produced by Dan Falk.

Sun Feb 8 20:46:09 EST 1998