Jun 19 – 23, 2017
Europe/Rome timezone

Decorated character varieties, Painlevé equations and quantization

Not scheduled
3h 20m
Room D (basement) (SISSA)

Room D (basement)


Via Beirut 2 - 4, 34151, Trieste, Italy (note that this is not SISSA's current main building but the old building in the Miramare park)


Marta Mazzocco (University of Loughborough, UK)


In this lecture course I will introduce some notions in (quantum)Teichmuller theory for orientable non compact Riemann surfaces. We will then discuss colliding boundary components and discuss the Painlevé equations as an example. Finally we will link with the theory of Cherednik algebra.

Tentative schedule:

Lecture 1: Teichmuller theory for orientable non compact Riemann surfaces: Thurston shear coordinates, Fock decomposition, Goldman bracket and quantisation. Complexification and $SL_2$--character varieties.

Lecture 2: Colliding holes in orientable non compact Riemann surfaces. Bordered cusps, Teichmuller theory for Riemann surfaces with bordered cusps. Example: confluence of the Painlevé equations as hole collision. Quantisation.

Lecture 3: Cherednik algebra associated to the root system $C_1$. Basic representation. Spherical sub-algebra and Askey--Wilson algebra. Askey--Wilson polynomials and q-Askey scheme.

Lecture 4. $C_{1}$ Cherednik algebra and sixth Painlevé equation. Confluence of the Painlevé equation as Whittaker degeneration of the $C_{1}$ Cherednik algebra. Open problems.

Presentation materials

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