The program can be found below. For access information please register.

Schedule

Times are Central European Summer Time (UTC+2)

Lecture courses

• Spectral convergence in geometric quantization
  Kota Hattori

  In this series of lectures I explain geometric quantization from the viewpoint of spectral convergence. We take a Lagrangian fibration on a symplectic manifold and a family of compatible complex structures tending to the real polarization given by the fibration, and show a spectral convergence of the d-bar Laplacian on the prequantum line bundle to the spectral structure related to the set of Bohr–Sommerfeld fibers. This talk is based on the joint work with Mayuko Yamashita.

  1. Bohr–Sommerfeld fibers and the convergence of holomorphic sections (video, notes)
  2. Spectral structures and the measured Gromov–Hausdorff convergence (video, notes)
  3. Convergence of the spectral structures of d-bar Laplacians on the prequantum bundle (video, notes)
  4. Convergence of the frame bundle of the prequantum bundles (video, notes)

  Additional resources. Detailed plan of lectures, notes of exercise session 1, notes of exercise session 2.

• Quantization via integrable systems
  Eva Miranda

  Quantization seeks to associate a quantum system to a classical Hamiltonian system replacing functions by operators and Poisson brackets of functions by brackets of operators. Several paths have been traced for this passionate journey from geometry and analysis into Physics: geometric quantization, formal quantization, BRST quantization and semi-classical quantization to cite a few. All of them supply taylor-made master formulas to the day dreamer mathematicians who are looking into the quantum world through their quantization mirror.
  In this course we focus on the Geometric Quantization approach and we provide several a model that brings us closer to the role of quantization as a mathematical tamer of quantum physics. The almost metaphysical questions still waft in the air: Can this be achieved? Do these methods depend on additional data? Can we find a universal model?
  Geometric quantization and integrable systems are common mathematical objects on the interface of Geometry and Physics. Integrable systems represent a class of Hamiltonian systems which can be associated to an extra number of functions (first integrals). And geometric quantization meets integrable systems when these systems are used as data attached to the Geometric quantization process (a polarization). In this minicourse we will examine the quantization problem considering as polarizations the real polarizations associated to an integrable system and analyze the role of singularities in the quantization picture suggested by Kostant.

  1. Introduction to Symplectic Geometry, integrable systems and action-angle systems. Non-degenerate singularities of integrable systems. (video, slides, notes)
  2. Integrable systems in physics. (Problem session by Pau Mir)
  3. Basics on Geometric Quantization. Polarizations. Integrable systems as polarizations. Bohr–Sommerfeld leaves. Some computations. (video, slides, annotated slides)
  4. Sheaf cohomology computations. (Problem session by Pau Mir)
  5. Singularities everywhere! How to handle with singularities of integrable systems. How to handle singularities of the symplectic structures. (video, slides)

  Additional resources. Detailed plan of lectures, lecture notes.

• BV formalism in perturbative algebraic quantum field theory
  Katarzyna Rejzner

  Perturbative algebraic quantum field theory (pAQFT) is a rigorous approach to perturbative quantum field theory that combines the idea of locality coming from the Haag–Kastler axioms with methods of perturbation theory and homological algebra. The latter enter through the application of the BV (Batalin–Vilkovisky) formalism. In this series of lectures I will outline the main ideas of pAQFT and show how the BV formalism can be applied to perturbatively quantize gauge theories.

  1. Introduction (video, notes)
  2. Classical dynamics (video, notes)
  3. Symmetries (video, notes)
  4. Quantization of the free theory (video, notes)
  5. Quantum BV operator and quantum master equation (video, notes)

  Additional resources. Notes on global hyperbolicity.

• Convergence and continuity of star products
  Stefan Waldmann

  In my lecture series I will discuss the need of convergence for formal star products if one is interested in an honest quantum mechanical application of deformation quantization. Several competing approaches will be compared. In a series of examples I will advocate to take the series serious. Convergence and continuity of the star products can be shown in several important cases.

  1. Deformation quantization and formal star products (video)
  2. First example: Convergence in the flat case (video)
  3. An application to field theory (video)
  4. Second example: Linear Poisson structures and the Gutt star product (video)
  5. More examples: Cotangent bundle of a Lie group, Poincaré disc, and beyond (video)

  There will also be two exercise sessions held by Michael Heins.

  Additional resources. Notes.