Tame derived categories and their applications in algebraic geometry, representation theory, singularity theory and mathematical physcics
Overview
The technique of derived and triangulated categories is a universal language in which a synthesis of concepts and methods of various mathematical fields is accumulated. With its help one can apply concepts and results from algebraic geometry in compact formulation and solution of complicated analytic problems, a well-known example of which is the interpretation of complexes of coherent sheaves on Calabi-Yau varieties as D-branes in homological mirror symmetry and in string theory. In the case of derived categories on elliptic curves and their degeneracies, we intend to continue with applications to Yang-Baxter equations and integrable systems.The study of coherent sheaves on projective curves of arithmetic gender one is closely related to the representation theory of certain associative algebras. Another goal of the project is to transfer the geometric intuition of coherent sheaves to the noncommutative case.Various questions of birational geometry of three-dimensional singularities, such as the existence of a crepant resolution, can be formulated in the language of maximal Cohen-Macaulay moduli. Through the concept of cluster tilt theory, a connection is established between the stable category of CM moduli over Gorenstein local rings and representations of certain tame associative algebras.The following concrete topics will be addressed in the project:
- Derived categories of degeneracies of elliptic curves.
- Vector bundles on elliptic fibers and their applications in mathematical physics
- Tame derived categories of associative algebras and their geometry
- Cohen-Macaulay moduli over curve and surface singularities
DFG Proceedings Emmy Noether Junior Research Group
(3 PhD positions for 36 months and one position of a junior research group leader for 60 months)
More Information
Results
Maximal Cohen–Macaulay modules over surface singularities, Trends in Representations of Algebras and Related Topics, 101–160, EMS Publishing House (2008) I. Burban, Yu. Drozd Mutation–classes of diagrams via in?nite graphs, Math. Nachr. 284 (2011), no. 17-18, 2184–2205 T. Henrich Tilting on non-commutative rational projective curves, Mathematische Annalen 351, no. 3, 665–709 (2011) I. Burban, Yu. Drozd Frobenius morphism and vector bundles on cycles of projective lines, Comm. in Algebra 40, no. 8, 2983–2988 (2012) I. Burban On the Hall algebra of an elliptic curve I, Duke Math. Journal, vol. 161, no. 7, 1171–1231 (2012) I. Burban, O. Schi?mann Semi-stable vector bundles on elliptic curves and the associative Yang–Baxter equation, Journal of Geometry and Physics 62, no. 2, 312–329 (2012) I. Burban, T. Henrich Singularity category of a non-commutative resolution of singularities, Adv. in Math., vol. 231, no. 1, 414–435 (2012) I. Burban, M. Kalck Two descriptions of the quantum a?ne algebra Uv (sl2 ) via Hall algebra approach, Glasgow Journal of Math. 54, no.2, 283–307 (2012) I. Burban, O. Schi?mann Vector bundles on degenerations of elliptic curves and Yang–Baxter equations, Memoirs of the AMS 220, no. 1035 (2012) I. Burban, B. Kreu?ler Vector bundles on plane cubic curves and the classical Yang-Baxter equation, Journal of the European Math. Soc. I. Burban, T. Henrich (See online at https://arxiv.org/abs/1202.5738) Composition algebra of a weighted projective line, J. Reine Angew. Math, vol. 679, 75–124 (2013) I. Burban, O. Schi?mann Analytic moduli spaces of simple sheaves on families of integral curves, Math. Nachr., vol. 287, no. 2-3, 173–183 (2014) I. Burban, B. Kreu?ler (See online at https://doi.org/10.1002/mana.201200355)Project-related publications (selection)