Graduate seminar Ubiquity of root systems (Fall 2017)

Root systems made their first appearance in the classification of simple Lie algebras. They later showed up in nearby areas such as the classification of semisimple Lie groups, symmetric spaces, reductive algebraic groups over fields (not necessarily closed, and not necessarily of characteristic zero), in singularity theory, the theory of quivers, and in the McKay correspondence. The goal of the seminar is to explore this.

Literature (to be expanded):

N. Bourbaki:  Lie groups and Lie Algebras, Ch. 4-6. Springer Verlag (2002).

T. Bridgeland, A. King, and M. Reid: The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535–554.

I.V. Dolgachev: McKay’s correspondence for cocompact discrete subgroups of SU(1,1), Groups and symmetries, 111–133, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009.

R. Laza: The moduli space of cubic fourfolds, J. of Algebraic Geometry 18 (2009), 511-545.

R. Laza: The moduli space of cubic fourfolds via the period map, Ann. of Math. 172 (2010),  673-711.

E. Looijenga: Isolated singular points on complete intersections (2nd edition)Surveys of Modern Mathematics, Volume V, Higher Education Press, Bejing China (2013). 

J.A. Wolf: On the classification of hermitian spaces, Journal of Mathematics and Mechanics 13 (1964), 489-496.

Time and Place
Wednesdays 19:00-21:00, except on public holidays in Lecture Room 1 (1st floor) of the Jin Chun Yuan West Building. 

  Sept 20: Zheng Zhiwei: Root systems, Dynkin diagrams and Classification I.
  Sept 27: 
Zheng Zhiwei: Root systems, Dynkin diagrams and Classification II.
   Oct 11: Wang Bin: 
Classification of semisimple Lie algebras I.
Oct 18: Wang Bin: Classification of semisimple Lie algebras II.
   Oct 25: 
Zheng Zhiwei: Del Pezzo Surfaces and Construction of Root Systems of E-type.
    Nov 1: Zi Yunpeng: Classification of Hermitian Symmetric Domains I.
    Nov 8: 
Zi Yunpeng: Classification of Hermitian Symmetric Domains II.
Nov 15: Zi Yunpeng: Classification of Hermitian Symmetric Domains III.
Nov 22: Chen Binyi: Mckay Correspondences I.
  Nov 29: Chen Binyi: Mckay Correspondences I (cont.), Wu Zhixiang: Mckay Correspondences II.
                We will use theories of quivers and Hilbert schemes to explain Mckay correspondences which
                  relates Kleinian singularities and Mckay diagrams of finite subgroups in SU(2). The minimal resolution
                  of singularities will be given by certain moduli space of representations of quivers, which will be
                  identified to G-Hilbert schemes of A2.

    Dec 6:  Zhong Yiming: Isolated singular points on complete intersections I.
                First we introduce some basic notions and then give the classification of the simple hypersurface singularities.
                 Then we discuss the notions of vanishing lattice and monodromy group in a geometric way.

   Dec 13: Zhong Yiming: Isolated singular points on complete intersections II.
   Dec 20: Eduard Looijenga: 
Simultaneous resolution of a versal deformation of a Kleinian singularity.
                This is another connection between Kleinian singularities and the simple Lie algebras of type A, D and E. We
                  review Grothendieck's conjectures and their proof by Slodowy and Brieskorn.

Contact and in charge of the organization: 郑志伟