Workshop on the

Geometry of Quantization

November 26 - 27, 2010
University of Cologne

A "quantization" is some process by which the mathematical structure of a quantum mechanical system is determined from a given classical system. In the 1920s, Dirac enunciated what one should hope to achieve with any reasonable notion of "quantization". It was quickly realized that in most settings (including even simple, real-world examples like the harmonic oscillator), it is actually not possible to achieve all of Dirac's requirements.

The attempts to develop notions of quantization which approach what Dirac prescribed have yielded many deep advances in both mathematics and physics, and there remain, even 100 years later, many unanswered questions. The purpose of this workshop is to bring together leading mathematicians working in branches of mathematics related to problems of quantization and thus promote interaction and progress in the field. There will be ample time for private discussions and spontaneous working sessions.

Confirmed Speakers
Chin-Yu Hsiao (G÷teborg)
Semyon Klevtsov (Bruxelles)
Xiaonan Ma (Paris)
JoŃo P. Nunes (Lisbon)
Siye Wu (Hong Kong)

C.-Y. Hsiao, Szeg÷ kernel asymptotics and Morse inequalities on CR manifolds
We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies Kohn's condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg÷ kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds

S. Klevtsov, Random Bergman metrics
The problem of defining 'random geometry' appears naturally in physics. In particular, in the Polyakov's 2d gravity the ensemble of conformal 2d metrics is considered with the Liouville weight. On the other hand, ensemble of random metrics can be also naturally considered on the space of Kahler metrics, e.g. using Bergman embeddings. We introduce the respective physical and mathematical definitions, and explain relationships between the two pictures.

X. Ma, Bergman kernel and geometric quantization

J. P. Nunes, Examples of degenerating Kahler polarizations in geometric quantization

S. Wu,  Projective flatness in the geometric quantisation of bosons and fermions
Geometric quantisation requires choosing a real or complex polarisation. Quantum physics is independent of the choice if there exists a projectively flat connection on the vector bundle of Hilbert spaces over the space of polarisations. In this talk, I begin with symplectic vector spaces and explain the geometry of projectively flat vector bundles. I will explain quantisation of fermionic systems and its relation to Clifford algebra and spinor representations. Finally, systems with symmetries will be considered.
William D. Kirwin and George Marinescu


26 November 2010
15h00 - 16h00
Random Bergman metrics
16h30 - 17h30
Colloquium: Twisted analytic torsion

27 November 2010
10h00 - 11h00
Szeg÷ kernel asymptotics and Morse inequalities on CR manifolds
11h30 - 12h30
Bergman kernel and geometric quantization
14h00 - 15h00
Projective flatness in the geometric quantisation of bosons and fermions
15h30 - 16h30
Examples of degenerating Kahler polarizations in geometric quantization

*All talks will be held in the H÷rsaal des Mathematischen Instituts.
Supported by Graduiertenkolleg Globale Strukturen in Geometrie und Analysis