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**1. Speaker : Luc Hillairet (University of Nantes)**

**Title :** Spectral variations of translation surfaces.

**Abstract :**

We investigate how different spectral quantities may vary along paths in the moduli space of translation surfaces. We relate it to finite rank operators accounting for the presence of conical singularities. (Joint with A. Kokotov)

**2. Speaker : Takashi Ichikawa(Saga University)**

**Title:** Modular forms and invariants for algebraic curves.

**Abstract:**

We consider the relationship between modular forms and invariants on the moduli of algebraic curves (with additional data). Especially, we relate certain Teichm\"{u}ller modular forms with Klein's amazing formula, and Mumford forms with Selberg (type) zeta values of Schottky groups. We use universal 1-forms and periods to determine proportionality constants in these relations.

**3. Speaker: Alexey Kokotov (Concordia University)**

**Title:** Determinants of Laplacians and polyhedral surfaces

**Abstract:**

In the joint paper with D. Korotkin we found an explicit holomorphic factorization formula for the determinant of the Friedrichs extension of the Laplacian on a compact polyhedral surface with trivial holonomy. We will discuss generalizations of this result for

1)Arbitrary compact polyhedral surfaces

2)Other self-adjoint extensions of the symmetric Laplace operator

3)Non compact flat surfaces with cylindrical ends (more precisely, so-called Mandelstam diagrams).

The talk is based on the joint works with L. Hillairet and V. Kalvin.

**4. Speaker: Dmitry Korotkin (Concordia University)**

**Title:** Baker-Ahhiezer spinor kernel and tau-function on moduli spaces of meromorphic differentials.

**Abstract: **

In this talk we discuss the relationship between spinor Baker-Akhiezer kernel, Bergman and Baker-Akhiezer tau-functions on moduli spaces of meromorphic differentials on Riemann surfaces. The Ahlfors-Rauch type variational formulas (written with respect to homological coordinates) describe both dependence on KP times (for fixed Riemann surface) and moduli of the Riemann surface. Analysis of global properties of the Bergman and Baker-Akhiezer tau-functions allows to establish new relationships between various divisor classes on these spaces.

**5. Speaker: Andrew McIntyre (Bennington College)**

**Title:** Chern-Simons invariants, determinant of Laplacian, and tau functions

**Abstract:**

Suppose X is a compact 2-manifold, of fixed genus 2 or more, with hyperbolic metric. It is known (Belavin-Knizhnik, Bost, Takhtajan-Zograf) that the determinant of the Laplacian on X is the modulus squared of a holomorphic function F on the Teichmuller space of such X, times a "conformal anomaly". It has been gradually understood (Polyakov, Krasnov, Takhtajan-Teo, Schlenker) that the conformal anomaly is the exponential of a regularized volume of a certain infinite-volume hyperbolic 3-manifold M whose conformal boundary is X. It is a result of Zograf that the function F may be written as a Selberg zeta-like product for the 3-manifold M. (These results are a baby case of physicists' conjectured "holography".) This raises the question of the meaning of the phase of F. Park realized that the phase of F may be interpreted in terms of a regularized Atiyah-Patodi-Singer eta invariant of M.

In this joint work with Jinsung Park, KIAS, we generalize constructions of Meyerhoff and Yoshida to define a regularized Chern-Simons invariant for M, which forms a natural complexification of the regularized volume. We relate it to the regularized eta invariant. The definition involves a framing that is singular along curves in M, reminiscent of Witten's work on the Jones polynomial. The Bergman tau function, introduced and studied by Kokotov-Korotkin, makes a surprise appearance.

**6. Speaker: Shu Oi (Waseda University)**

**Title:** The KZ equation on the moduli space $\mathcal{M}_{0,n}$ and

the Riemann-Hilbert problem II

**Abstract: **

This talk is the second part of lectures given by me and Professor Ueno. In this talk, we introduce a characterization of the multiple polylogarithms of one variable by using a recursive Riemann-Hilbert problem of additive

This Riemann-Hilbert problem corresponds to the inverse problem of the connection problem of the KZ equation of one variable. We also talk about the five term relation for the dilogarithm, which gives another characterization of the dilogarithm. The five term relation is considered as the part of the connection problem of the KZ equation on $\mathcal{M}_{0,5}$ and suggests how to formulate the Riemann-Hilbert problem for the KZ equation of the two variables.

**7. Speaker: Yuji Terashima (Tokyo Institute for Technology)**

**Title****：** Torsion functions on moduli spaces

**Abstract****：**

In this talk, we explain a new method to compute explicitly torsion functions on moduli spaces of representations of 3-manifolds by using the theory of cluster transformations.

**8. Speaker: Kimio Ueno (Waseda University)**

**Title:** The KZ equation on the moduli space $\mathcal{M}_{0,n}$ and the Riemann-Hilbert problem, I

**Abstract : **

My talk is the first part of lectures given by me and Dr Oi.

In my talk, I will discuss about the outline of the connection problem of the KZ equation on $\mathcal{M}_{0,4}$, which corresponds to the KZ equation of one variable, and the connection problem of the KZ equation on $\mathcal{M}_{0,5}$, which corresponds to the KZ equation of two variables will show that the Drienfeld associator gives the connection matrix for the KZ equaitons, and that it satisfies the duality relation, hexagon relations, and the pentagon relation.

**9. Speaker : Lin Weng (Kyushu University)**

**Title :** General Uniformity of Zeta Functions

**Abstract:**

In the paper of Atiyah-Bott on 'Yang-Mills Equations over Riemann Surfaces', parallel structures in geometry and arithmetic were exposed in terms of Yang-Mills equations and stable bundles. Later on, in studies related to the Verlinder formula, Witten, resp. Zagier, obtained a volume formula, resp. a mass formula, for moduli spaces of stable bundles on Riemann surfaces, resp. on curves defined over finite fields. In this talk, we expose yet another parallel structure from number theory. To be more precise, we will explain a pair of formulas relating the volumes of fundamental domains and that of moduli spaces of stable lattices. Among this twin relations, the first, due to Kontsevich-Soibelman, is obtained in their works on the wall-crossing, and the second, our own, is obtained using non-abelian zeta functions. Based on this, we would finally try to unify all three theories under the framework of zeta functions.

**10. Speaker: Richard Wentworth (University of Maryland)**

**Title:** Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces

**11. Speaker : Anton Zorich (University of Paris 7)**

**Title:** Lyapunov exponents of the Hodge bundle (joint work with A. Eskin and M. Kontsevich)

**Abstract:**

Various properties of dynamical systems on Riemann surfaces, of billiards in polygons, of measured foliations can be described in the language of the associated flat metric with conical singularities and with trivial holonomy. Such metric naturally defines a complex structure and a holomorphic 1-form on the Riemann surface. I will try to show how sophisticated geometric properties of the individual flat surface are related to simpler properties of the complex Teichmuller geodesic (or, more precisely, of its closure) in the moduli space of Abelian differentials.