▷ M. Wolf
- Title:  Rays In Teichmuller Space
- Abstract: If Teichmuller space is viewed as a deformation space of complex structures, then Teichmuller rays are a distinguished family.  If Teichmuller space is viewed as a space of hyperbolic metrics, the Thurston stretch paths are a distinguished family.  Harmonic maps from a Riemann surface to a hyperbolic surface offer a curious hybrid: the domain is anchored in the conformal perspective, but the map depends on the metric structure on the target - but there are still rays.  We aim to connect these three ray structures.  (Joint with Huiping Pan.)

▷ Q. Li ( Lecture Note Download ↓ )
- Title: Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli
- Abstract: In this talk, we study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.

▷ Xu Wang ( Lecture Note Download ↓ )

- Title: Negative curvature property of a Poisson Kahler fibration

- Abstract: I would like to talk about a recent joint work with Xueyuan Wan. We obtain a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the Monge-Ampere foliation (which we call the Poisson-Kahler fibration). Furthermore, we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula for Weil--Petersson type metrics associated to general relative Kahler fibrations. An application to equivalent criterions of polystability of projective bundles is also given.  

▷ S. Wolpert ( Lecture Note Download ↓ )

- Title: The Hilbert area of inscribed polygons

- Abstract: Proper functions on moduli spaces of geometric structures provide a means of describing completions and compactifications. Hilbert area is an invariant of convex real projective plane structures. We review the Fock-Goncharov parameterization of inscribed polygons and apply the approach to study quadrilaterals. We describe completions of the parameter space and study the Hilbert area near the completions. A micro local condition is developed for bounded Hilbert area under degeneration. A sequence of strictly convex domains with bounded Hilbert area and divergent parameters is described.

▷ Yunhui Wu ( Lecture Note Download ↓ )
- Title: A new uniform lower bound on Weil-Petersson distance
- Abstract: In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $1.5537$-Lipschitz on Teichmuller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is Lipschitz on Teichmuller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be choosen to be $2.1792$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.

▷ X. Wan ( Lecture Note Download ↓ )

- Title:  Plurisubharmonicity and convexity of energy functions on Teichmuller space. 

- Abstract:  In this talk, we will consider the energy functions on Teichmuller space associated to the  harmonic maps between Riemann surfaces and Riemannian manifolds. We obtain the precise formulas of the second variation of energy functions, and we will show the plurisubharmonicity and convexity of energy functions by using these formulas.  This work is joint with Professors Inkang Kim and Genkai Zhang. 

▷ G. Schmacher
- Title: Analytic Application of Geometric Invariant Theory

- Abstract:

▷ C. Mese

- Title: Holomorphic Rigidity of Teichmuller space.
- Abstract: The holomorphic rigidity of Teichm¨uller space which can be loosely stated as follows: The action of the mapping class group uniquely determines the complex structure of Teichm¨uller space (as defined by Riemann and Teichm¨uller). We discuss its proof that applies the theory of finite and infinite energy harmonic maps into NPC (non-positively curved) metric spaces.

▷ Weixu Su ( Lecture Note Download ↓ )
 

▷ Jinsong Liu( Lecture Note Download ↓ )

- Title: Factoring Quasiconformal & quasisymmetric mappings

- Abstract: It follows from the Measurable Riemann Mapping Theorem that we can always present a 2-dimensional quasi-conformal mapping as a composition of quasi-conformal mappings with smaller dilatation. In this talk we will construct n(≥ 3)-dimensional quasi-conformal homeomorphisms between Eucilidean spaces which admit no minimal factorizations in linear, inner, or outer dilatations. If time permits, I will discuss the composition of quasi-symmetric mappings between metric spaces.