Dg manifolds and higher structures
ZOOM: 815 762 8413
PASSWORD: BIMSA
DG manifolds are a useful geometric notion that unifies many important structures such as homotopy Lie algebras, foliations, and complex manifolds. In particular, the Todd class of DG manifolds extends both the Todd class of a complex manifold and the Duflo element in Lie theory. DG manifolds also play an important role in the study of the differential geometry of singular spaces. In this series of lectures, we will introduce DG manifolds and discuss various applications in geometry and higher structures. We will describe noncommutative calculi for DG manifolds. In particular, we will prove a Duflo-Kontsevich type theorem for DG manifolds. The classical Duflo theorem in Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds can both be derived as special cases of this Duflo-Kontsevich type theorem for DG manifolds. We will also discuss its application in derived differential geometry, which deals with singularities arising from zero loci or intersections of submanifolds. This is mainly based on joint work with Kai Behrend, Ruggero Bandiera, Zhuo Chen, Hsuan-Yi Liao, Rajan Mehta, Seokbong Seol, Mathieu Stienon, and Maosong Xiang.
PASSWORD: BIMSA
DG manifolds are a useful geometric notion that unifies many important structures such as homotopy Lie algebras, foliations, and complex manifolds. In particular, the Todd class of DG manifolds extends both the Todd class of a complex manifold and the Duflo element in Lie theory. DG manifolds also play an important role in the study of the differential geometry of singular spaces. In this series of lectures, we will introduce DG manifolds and discuss various applications in geometry and higher structures. We will describe noncommutative calculi for DG manifolds. In particular, we will prove a Duflo-Kontsevich type theorem for DG manifolds. The classical Duflo theorem in Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds can both be derived as special cases of this Duflo-Kontsevich type theorem for DG manifolds. We will also discuss its application in derived differential geometry, which deals with singularities arising from zero loci or intersections of submanifolds. This is mainly based on joint work with Kai Behrend, Ruggero Bandiera, Zhuo Chen, Hsuan-Yi Liao, Rajan Mehta, Seokbong Seol, Mathieu Stienon, and Maosong Xiang.
Lecturer
Date
22nd January ~ 17th February, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday,Friday | 00:00 - 00:00 | - | - | - |
Prerequisite
Lie algebra, Differential Geometry
Reference
[1] Mathieu Stiénon, Luca Vitagliano, and Ping Xu, A_∞ algebras from Lie pairs, arXiv:2210.16769.
[2] Kai Behrend, Hsuan-Yi Liao, and Ping Xu, On the structure of étale fibrations of L_∞-bundles, arXiv:2307.08179.
[3] Kai Behrend, Hsuan-Yi Liao, and Ping Xu, Differential graded manifolds of finite positive amplitude, Int. Math. Res. Not. IMRN. 2024, no. 8, 7160–7200. [4] Kai Behrend, Matt Peddie, and Ping Xu, Quantization of (-1)-shifted derived Poisson manifolds, Comm. Math. Phys. 402 (2023), 2301–2338.
[5] Zhuo Chen, Maosong Xiang, and Ping Xu, Hochschild cohomology of dg manifolds associated to integrable distributions, Comm. Math. Phys. 375 (2022), 647–684.
[6] Seokbong Seol, Mathieu Stiénon, and Ping Xu, Dg manifolds, formal exponential maps and homotopy Lie algebras, Comm. Math. Phys. 391 (2022), 33–76. [7] Mathieu Stiénon and Ping Xu, Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds, Banach Center Publications 123 (2021), 63–110.
[8] Ruggero Bandiera, Zhuo Chen, Mathieu Stiénon, and Ping Xu, Shifted Derived Poisson Manifolds Associated with Lie Pairs, Comm. Math. Phys. 375 (2020), 1717–1760.
[9] Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Formality and Kontsevich-Duflo type theorems for Lie pairs. Adv. Math., 352:406–482, 2019.
[10] Zhuo Chen, Maosong Xiang, and Ping Xu, Atiyah and Todd classes arising from integrable distributions, J. Geom. Phys. 136 (2019), 52–67.
[11] Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu, Formality theorem for differential graded manifolds, C. R. Math. Acad. Sci. Paris 356 (2018), 27–43.
[12] Rajan Amit Mehta, Mathieu Stiénon, and Ping Xu, The Atiyah class of a dg-vector bundle, C. R. Math. Acad. Sci. Paris 353 (2015), no. 4, 357–362.
[2] Kai Behrend, Hsuan-Yi Liao, and Ping Xu, On the structure of étale fibrations of L_∞-bundles, arXiv:2307.08179.
[3] Kai Behrend, Hsuan-Yi Liao, and Ping Xu, Differential graded manifolds of finite positive amplitude, Int. Math. Res. Not. IMRN. 2024, no. 8, 7160–7200. [4] Kai Behrend, Matt Peddie, and Ping Xu, Quantization of (-1)-shifted derived Poisson manifolds, Comm. Math. Phys. 402 (2023), 2301–2338.
[5] Zhuo Chen, Maosong Xiang, and Ping Xu, Hochschild cohomology of dg manifolds associated to integrable distributions, Comm. Math. Phys. 375 (2022), 647–684.
[6] Seokbong Seol, Mathieu Stiénon, and Ping Xu, Dg manifolds, formal exponential maps and homotopy Lie algebras, Comm. Math. Phys. 391 (2022), 33–76. [7] Mathieu Stiénon and Ping Xu, Atiyah classes and Kontsevich-Duflo type theorem for dg manifolds, Banach Center Publications 123 (2021), 63–110.
[8] Ruggero Bandiera, Zhuo Chen, Mathieu Stiénon, and Ping Xu, Shifted Derived Poisson Manifolds Associated with Lie Pairs, Comm. Math. Phys. 375 (2020), 1717–1760.
[9] Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu. Formality and Kontsevich-Duflo type theorems for Lie pairs. Adv. Math., 352:406–482, 2019.
[10] Zhuo Chen, Maosong Xiang, and Ping Xu, Atiyah and Todd classes arising from integrable distributions, J. Geom. Phys. 136 (2019), 52–67.
[11] Hsuan-Yi Liao, Mathieu Stiénon, and Ping Xu, Formality theorem for differential graded manifolds, C. R. Math. Acad. Sci. Paris 356 (2018), 27–43.
[12] Rajan Amit Mehta, Mathieu Stiénon, and Ping Xu, The Atiyah class of a dg-vector bundle, C. R. Math. Acad. Sci. Paris 353 (2015), no. 4, 357–362.
Audience
Graduate
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English