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In this reading course we use methods of homotopy theory, such as L_\infty and G_\infty structures, to quantize
geometric structures, such as Poisson bivectors and r-matrices. Our main goal is to obtain insights into the
celebrated formality theorem of Kontsevich, together with related works of Dolgushev and Tamarkin on
deformation quantization. We further study the quantization of dynamical r-matrices via Drinfel'd associators,
following the work of Calaque.
Last update: Šmíd Dalibor, Mgr., Ph.D. (19.05.2025)
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The goal is to get familiar with the deformation problem arising from Poisson geometry, where Poisson bivectors constitute the semi-classical limit of star products. As a preparation, the necessary amount of differential geometry will be studied, specifically the Cartan calculus on smooth manifolds. The solution of the deformation problem, given by Kontsevich's formality theorem, further requires the notions of homotopy Lie algebra and the corresponding quasi-isomorphisms. On the algebraic side, the deformation theory of Gerstenhaber will give the formal background to understand the deformations of associative algebras from a general point of view. Globalization methods, in the sense of Dolgushev and Cattaneo-Felder, might be considered, as well. Last update: Weber Thomas, Ph.D. (25.09.2025)
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The participants are expected to study a subject related to the direction of the course and present it in the second half of the semester. This can also be done in joint work with other participants. For interested students there might be the possibility to continue the course project in form of a master thesis. Last update: Weber Thomas, Ph.D. (25.09.2025)
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C. Esposito: Formality theory: from Poisson structures to deformation quantization, Springer-Verlag Heidelberg, Berlin, New York (2015)
J. Schnitzer: Poisson Geometry and Deformation Quantization, Lecture Notes https://mate.unipv.it/schnitzer/ (2023)
S. Waldmann: Poisson-Geometrie und Deformationsquantisierung, Springer-Verlag Heidelberg, Berlin, New York (2007) Last update: Weber Thomas, Ph.D. (25.09.2025)
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Reading course, where the first lessons will be delivered by the lecturer and participants are supposed to contribute with short presentations in the second half of the semester. The course will also be streamed via Teams. Recordings, as well as lecture notes, will be made available on Teams and via email.
Last update: Weber Thomas, Ph.D. (25.09.2025)
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1) Smooth manifolds and Cartan calculus
2) Poisson manifolds
3) Lie groups and actions
4) Differential operators
5) Star products and deformation quantization
6) Gerstenhaber deformation theory
7) Fedosov and Kontsevich quantization
8) Globalization procedures Last update: Weber Thomas, Ph.D. (25.09.2025)
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