Working seminar: Bridgeland's stability conditions for meromorphic differentials
Winter semester 2020-21
JProf. Dr. Maria Beatrice Pozzetti
- Monday 17:00-18:00, on zoom (email us to get the link!)
Please, register for the seminar on MÜSLI, or send us an e-mail (firstname.lastname@example.org, email@example.com).
Bridgeland introduced in [Bri07] a general framework to understand the geometry behind the myriad notions of stability which appear in the construction of various types of moduli spaces. In short, given a triangulated category, such as the derived category of coherent sheaves on a complex manifold, he defines a notion of stability condition, and shows that the set of all stability conditions has the structure of a manifold.
Motivated by the work of Gaiotto-Moore-Neitzke [GMN13], Bridgeland and Smith [BS15] turned their attention to studying the geometry of meromorphic quadratic differentials on a Riemann surface with marked points, and applied the machinery of [Bri07] to realize the moduli space of meromorphic quadratic differentials as the space on stability conditions on an explicit triangulated category built from an associated quiver with potential.
Later, Smith continued the study initiated in [BS15], and in the papers [Smi15] and [Smi20] he identified an explicit Calabi-Yau 3-fold whose Fukaya category is closely related to the triangulated category.The goal of this seminar will be to understand the basic machinery behind these constructions, with a strong emphasis on how the explicit geometry of meromorphic quadratic differentials leads to concrete examples of spaces of stability conditions on triangulated categories. We will assume no prior knowledge of derived/Fukaya/triangulated categories.
This seminar is a working seminar aimed at PhD students and postdocs. Master students are welcome, but, apart from exceptional cases, we do not expect to assign credits for it. Talks will be given in English.
|13.11.2020||Bea||Talk 0: Organizational meeting|
|23.11.2020||Arnaud||Talk 1: Symplectic mapping class groups|
|30.11.2020||Levin||Talk 2: Fukaya categories|
|7.12.2020||Johannes||Talk 3: Stability conditions|
|14.12.2020||Johannes||Talk 4: Stability conditions II|
|11.01.2021||Tobias||Talk 5: Triangulated Categories and the Space of Stability Conditions|
|18.01.2021||Luca||Talk 6:Bridgeland stability conditions, curves (and quivers)|
|25.01.2021||Bea||Talk 7: Quadratic differentials and flat structures|
|01.02.2021||Bea||Talk 8:Period coordinates and triangulations|
|12.02.2021||Menelaos||Talk 9: Derived Equivalences and Bridgeland-Smith Correspondence|
|19.02.2021||Discussion||Talk 10: Final discussion|
- [Bay11] Bayer, Arend. A tour to stability conditions on derived categories. pdf
- [Bri07] Bridgeland, Tom. Stability conditions on triangulated categories. Ann. of Math. (2) 166 (2007). pdf
- [BS15] Bridgeland, Tom; Smith, Ivan. Quadratic differentials as stability con- ditions. Publ. Math. Inst. Hautes Etudes Sci. 121 (2015). pdf
- [GMN13] Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew. Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234 (2013), 239?403. pdf
- [Smi14] Smith, Ivan. A symplectic prolegomenon. arXiv:1401.0269
- [Smi15] I. Smith,Quiver algebras as Fukaya categories, Geom. Topol. 19 (2015), 2557--2617. pdf
- [Smi17] Smith, Ivan. Stability conditions in symplectic topology. arXiv:1711.04263
- [Smi20] Smith, Ivan. Floer theory of higher rank quiver 3-folds. arXiv:2002.10735v1
- [Smi] I. Smith, Symplectic mapping class groups and flat surfaces Slides from online talk