Leon Takhtajan

Synopsis

Character varieties appear in many areas of mathematics. Character varieties of surface groups inlclude Teichmüller spaces, spaces of projective structures on surfaces, moduli spaces of flat connections, hoilomorphic vector bundles and Higgs bundles. In context of Hamiltonian action, character varieties are examples of symplectic reduction. In mathematical physics, character varieties appear in the context of Yang-Mills and Chern-Simons theories and are related to many interesting construction.

Lectures

TuTh 9:45am-11:05am, Math Tower 5-127. There will be a simultaneous Zoom broadcast of chalkboard lectures.

Instructor

Leon Takhtajan

Office hours: Wed & Th 2:30pm-4:00pm in Math Tower 5-111 and by Zoom personal meeting (by email request).

Course description

The course will cover the basics of Teichmüller theory and character varieties. Specifically, we plan to cover the following topics (if time permits):

• Character varieties as GIT quotients Hom(Γ,G)//G, where Γ is a surface group and G is a reductive Lie group (real or complex).
• Special examples: G=PSL(2,R) (Teichmüller theory), G=PSL(2,C) (moduli of projective structures) and G=SU(n) (moduli of flat connections/holomorphic vector bundles).
• Complex and symplectic structure of character varieties. Goldman symplectic form and Ralph Fox free differential calculus.
• Complex analytic structure of Teichmüller spaces according to Ahlfors-Bers (using quasiconformal mappings and Beltrami equation) and Weil (using Kodaira-Spencer deformation of complex structures theory).
• Weil-Petersson metric on Teichmüller spaces. Proof that it is a Käler metric and the study of its curvature properties.
• Goldman symplectic form as symplectic form of Weil-Petersson metric.
• Goldman symplectic form in case G=PSL(2,C) and canonical symplectic form on the cotangent bundle to the Teichmüller space.
• Moduli spaces of holomorphic vector bundles over a Riemann surface and Narasimhan-Seshadri theorem.
• Character varieties for orbifold surfaces in case G=PSL(2,C).

Textbook

There will be no assigned textbooks. Each week a reference will be given on the Blackboard to research papers and monographs related to the material covered.

Information for students with disabilities

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