Workshop on the

November 26 - 27, 2010

University of Cologne

University of Cologne

Purpose

- 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 (Göteborg)

- Semyon
Klevtsov (Bruxelles)

- Xiaonan Ma (Paris)

João P. Nunes (Lisbon)

- Siye Wu (Hong Kong)

- Talks
- C.-Y. Hsiao, Szegö 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 Szegö 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.

- Organizers
- William D. Kirwin and George Marinescu

- Program

- 26 November 2010
15h00 - 16h00

Klevtsov

Random Bergman metrics 16h30 - 17h30

Wu

Colloquium: Twisted analytic torsion

- 27 November 2010
10h00 - 11h00

Hsiao

Szegö kernel asymptotics and Morse inequalities on CR manifolds 11h30 - 12h30

Ma

Bergman kernel and geometric quantization 14h00 - 15h00

Wu

Projective flatness in the geometric quantisation of bosons and fermions 15h30 - 16h30

Nunes

Examples of degenerating Kahler polarizations in geometric quantization

- *All talks will be held in the Hörsaal des Mathematischen Instituts.