Final Report NSF Grant INT-9979435

U.S.-Argentina Workshop on Quantum symmetries

in theoretical physics and mathematics

Bariloche, Patagonia, Argentina January 10-21, 2000

The workshop took place at the Instituto Balseiro of the Centro Atomico of Bariloche from January 10 to 21, 2000. There were 68 participants from 15 countries, the majority (40) from Latin America, 16 from Europe, 4 from the US, 3 from Marocco, 2 from Australia and 2 from Japan. There were five women, well above average for a meeting of this sort. The scientific range of the participants ran from graduate students to post-docs to young researchers on up to leading theoreticians.

The workshop received major support from the CIMPA (France), the NSF and the ANPCyT (Argentina) and also from the ICTP (Trieste, Italy), the CLAF (Brazil) and the CNRS (France). This provided accomodations for all participants and travel support for 8 lecturers and 38 other participants. The facilities at the Centro Atomico of Bariloche provided accomodations at bargain rates.

The theme, ``Quantum symmetries in theoretical physics and mathematics'', is at the cutting edge of research in both mathematics and physics. The discovery some 15 years ago of quantum groups (a special class of Hopf algebras) has led to an ongoing torrent of activity in algebra but also in topology (manifold and knot theories), mechanics, dynamical systems, operator theory and even number theory. In physics, qunatum groups first appeared in integrable systems but now appear also in statistical mechanics, conformal field theory, quantum field theory and more.

The organizers invited international specialists to give a course each on the basics of relevant aspects of quantum group theory leading up to their recent results. The workshop was thus in major part a `school' providing exposition for the training of the next generation of researchers. The various lecturers provided significantly different points of view: from analysis, geometry, algebra and physical models, especially conformal.

The lecturers were N. Andruskiewitsch (Argentina), M. Dubois Violette (France), D. Evans (UK), A. Ocneanu (USA), O. Ogievetsky (France), N. Reshetikin (USA), M.Rosso (France), A. Varchenko (USA), S. Woronowicz (Poland), J.B. Zuber (France).

Each gave a series of lectures, 5 hours in all, as follows:

N. Andruskiewitsch: Finite dimensional Hopf algebras

M. Dubois Violette: Homological and cohomological properties of non-commutative algebras

D. Evans: Interpretation of conformal field theories in terms of sub-factors of von Neumann algebras.

A. Ocneanu: Classification of qunatum symmetries of type SL3

O. Ogievetsky: Perturbative and non-perturbative R-matrices

N. Reshetikin: Classical and quantum integrable systems

M.Rosso: The combinatorial point of view on quantum groups

A. Varchenko: Dynamical quantum groups

S. Woronowicz: Non-compact quantum groups and multiplicative unitary operators

J.B. Zuber: CFT, BCFT, ADE and all that

The lectures will be published in the AMS series `Contemporary Mathematics'; the editors will be the directors of the workshop: R. Coquereaux (CPT - Marseille) and R. Trinchero (CAB - Bariloche).