Topics in noncommutative geometry
Student No.:50
Time:Mon 10:00-11:50, 2017-02-20~ 2017-06-05 (except for public holidays)
Instructor:Si Li  [Tsinghua University]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2017-2-20
Ending Date:2017-6-5

Student Presentations:


Apr 1,  XingXing Tang, About the Sign conventions. 


May 8, Zhenping Gui, Lie algebra cohomology of matrices and cyclic homology


May 15, Kai Xu, Kontsevich formality and deformation quantization


May 22, Yifan Li, String topology and cyclic homology


June 6, Keyou Zeng, Kontsevich-Shoikhet-Tsygan Formality, higher spin theories and AdS/CFT





Quantum world is noncommutative. This course aims to discuss certain aspects of homological methods in noncommutative geometry. We introduce basics on Hochschild, cyclic (co)homology and noncommutative Chern-Weil theory. As applications, we discuss several examples that arise from quantum field theories, including Kontsevich’s formality theorem, Ginzburg’s Calabi-Yau algebra, and Batalin-Vilkovisky’s quantization. At the end of the course, I will discuss my formalism with Costello on the B-twisted open-closed string field theory and its relation with homological mirror symmetry.





Basics on homological algebra, algebraic geometry and differential geometry. Some knowledge on quantum field theory will be helpful, but not required.





JL Loday, Cyclic homology. Grundlehren Math.Wiss. 301, Springer (1998)


V Ginzburg, Lectures on Noncommutative Geometry. arXiv:math.AG/0506603


B Zwiebach, Oriented open-closed string theory revisited. arXiv:hep-th/9705241


M Kontsevich, Deformation quantization of Poisson manifolds. arXiv:math.QA/9709040


K Costello and S Li, Quantization of open-closed BCOV theory, I. arXiv:hep-th/1505.06703.


Other references used will be mentioned in the course.