# The Xi-transform (07frg138)

Arriving Sunday, March 25 and departing Sunday April 1, 2007

## Organizers

Maciej Dunajski (Cambridge University)
George Sparling (University of Pittsburgh)

## Objectives

Objectives of the FRG.

1. One focus will be on the understanding of the implications of
higher dimensions for the unification of quantum mechanics and
non-commutative geometry. Non-commutative string theory along the
lines of the work of Alain Connes and William Ugalde is expected to
play an important role; a unifying language will be sought
in the (coherent) topos theory of William Lawvere and Myles Tierney.

2. A second focus will be on the construction and further development
of specific examples of the Xi-transform.

3. A third focus will try to comprehend the relation of dynamical
theories to the structures, such as the fermionic fluids of Shou-Cheng
Zhang and Jiangping Hu, that arise in
the context where there is no preferred time, where the dynamics
occurs on the boundary.

4. A fourth focus will be to try to understand the relation between
the holomorphic quantum twistor theories of Witten, Mason and others
and the hyperbolic non-analytic theory of Sparling. Here the theory
of hypersurface twistors of Penrose, Sparling and Le Brun seems to

References

1. Shou-Cheng Zhang and Jiangping Hu, A Four-Dimensional
Generalization of the Quantum Hall Effect, Science 294(5543), 823-828,
2001.

2. Jonathan Bain, Condensed Matter Physics and the Nature of
Space-time, Polytechnic University, New York, preprint 2006.

3. Edward Witten, Perturbative Gauge Theory As A String Theory In
Twistor Space, Communications in Mathematical Physics, 252, 189-258,
2004.

4. Rutger Boels, Lionel Mason and David Skinner, Supersymmetric Gauge
Theories in Twistor Space, hep-th/0604040, 23 pages, 2006.

5. Mohab Abou-Zeid, Christopher Hull and Lionel Mason, Einstein
Supergravity and New Twistor String Theories, hep-th/0606272, 67
pages, 2006.

6. George Sparling and Philip Tillman, A primordial theory,
cond-mat/0401015, 55 pages, 2004.

7. George Sparling, Schizzo d'una sintesi, Urbino Seminar, July 2006;
this work will be submitted for publication and posted on the gr-qc
arxiv by October 12th 2006.

8. Dana Mihai, George Sparling and Philip Tillman, Non-Commutative
Time, the Quantum Hall Effect and Twistor Theory, cond-mat/0401224, 14
pages, 2004.

9. George Sparling, Twistor theory and the four-dimensional Quantum
Hall Effect of Zhang and Hu, cond-mat/0211679, 31 pages, 2002.

10. George Sparling, The twistor theory of hypersurfaces in
space-time}, in Further Advances in Twistor Theory,
Volume III, editors Lionel Mason, Lane Hughston, Piotr Kobak and
Klaus Pulverer, London: Pitman Press, 2001.

11. Devendra Kapadia and George Sparling, A class of conformally
Einstein metrics, Classical and Quantum
Gravity, 24, 4765-4776, 2000.

12. George Sparling, Zitterbewegung, Seminaires et Congres, Societe
Mathematique de
France, 4, 275-303, 2000.

13. Shou-Cheng Zhang, To see a world in a grain of sand,
hep-th/0210162, 14 pages, 2002.

14. John Baez, The Octonions, Bulletin of the American Mathematical
Society, 39, 145-205, 2002.