Decorated Marked Surfaces: Calabi-Yau categories and related topics

This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration $\Delta$ on a marked surfaces $\mathbf{S}$, to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.

[1]  Y. Qiu,et al.  Finite presentations for spherical/braid twist groups from decorated marked surfaces , 2017, Journal of Topology.

[2]  A. King,et al.  Cluster exchange groupoids and framed quadratic differentials , 2018, Inventiones mathematicae.

[3]  Y. Qiu,et al.  Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface , 2018, International Mathematics Research Notices.

[4]  Y. Qiu.  The braid group for a quiver with superpotential , 2017, Science China Mathematics.

[5]  Y. Qiu,et al.  $\mathbb{X}$-stability conditions on Calabi-Yau-$\mathbb{X}$ categories and twisted periods , 2018 .

[6]  Y. Qiu.  Topological structure of spaces of stability conditions and topological Fukaya type categories , 2018, 1806.00010.

[7]  Y. Qiu.  Decorated marked surfaces (part B): topological realizations , 2018 .

[8]  Y. Qiu,et al.  Decorated marked surfaces II: Intersection numbers and dimensions of Homs , 2014, Transactions of the American Mathematical Society.

[9]  Y. Qiu,et al.  Cluster categories for marked surfaces: punctured case , 2013, Compositio Mathematica.

[10]  Y. Qiu.  Decorated marked surfaces: spherical twists versus braid twists , 2014, 1407.0806.

[11]  M. Kontsevich,et al.  Dynamical systems and categories , 2013, 1307.8418.

[12]  I. Smith,et al.  Quadratic differentials as stability conditions , 2013, Publications mathématiques de l'IHÉS.

[13]  T. Brustle,et al.  On the cluster category of a marked surface without punctures , 2010, 1005.2422.

[14]  D. Thurston,et al.  Cluster algebras and triangulated surfaces. Part I: Cluster complexes , 2006, math/0608367.