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
provide the needed link.

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.