- 3 hours talk

Speaker: Masa-Hiko Saito (Kobe University) 
Title: Geometry of the moduli spaces of Parabolic Higgs bundles and Connections and applications to Painleve equations.
Abstract: In a series of three lectures, I will explain about the geometry of moduli spaces of stable parabolic Higgs bundles and parabolic connections on a smooth curves and their relations to differential equations of Painleve type.

In the first lecture, we will start by reviewing algebraic constructions of moduli spaces of stable parabolic Higgs bundles and parabolic connections on a smooth curve via GIT. Due to the works of Maruyama and Yokogawa, and Inaba, Iwasaki and Saito, one can show that the moduli spaces are smooth quasi-projective algebraic schemes with natural holomorphic symplectic structures.

In the second lecture, we will review Riemann-Hilbert correspondence from moduli spaces of parabolic connections to moduli spaces of monodromy representations. By analysing the RH correspondence, we can give a rigorous proof of the fact that isomonodromic eformations of parabolic connections induces differential equations with geometric Painleve property.

In the third lecture, we will give an explicit description of moduli spaces of parabolic Higgs bundles and parabolic connections by apparent singularities and their duals(a joint work of S. Szabo). We will explain the relation between the geometry of moduli spaces and spectral curves.

Speaker: Jan Swoboda(University of Heidelberg)
Title: Ends of Higgs bundle moduli spaces
Abstract: Moduli spaces of Higgs bundles arise naturally in several quite different contexts: as generalizations of the concept of stable bundles in algebraic geometry, as solution spaces of certain elliptic equations over a Riemann surface, and as varieties of conjugacy classes of representations of surface groups. At the same time, they admit a rich geometric structure as they are complete noncompact hyperk?hler manifolds.

In the first part of this series of talks, I shall give an introduction to the basic theory of Higgs bundles over a Riemann surface. I will focus on the construction of its moduli space as the set of gauge equivalence classes of solutions to Hitchin's self-duality equation, discuss concepts such as stability and the Hitchin fibration, and describe its realization as a hyperk?hler quotient.

Analytic aspects of the moduli space are in the focus of the second part of the course. Here I plan to report about joint work with Mazzeo, Weiß and Witt on the asymptotic behaviour of solutions in the limit of large Higgs fields. I start by giving a very explicit description of special solutions on the unit disc. A partial compactification of the moduli space is then obtained using a gluing construction. Along the way, differential operators with singular coefficients naturally show up. It is therefore planned to include a discussion of some aspects of the relevant solution theory.

In the final part of the lecture series I intend to discuss some aspects of the large scale structure of the underlying $L^2$-hyperk?hler metric. According to a conjecture due to Gaiotto, Moore and Neitzke this metric is asymptotically close (at least on a large compact sector) to the so-called semiflat hyperk?hler metric, an incomplete metric constructed from the Hitchin integrable system. I will present the recent proof of this conjecture which also has been obtained in joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt.

Speaker: Michael Tuite (National University of Ireland, Galway)
Title: Vertex operator algebras and Riemann surfaces
Abstract: A Vertex Operator Algebra (VOA) is an algebraic system closely related to conformal field theory in physics. VOAs have a rich and well established connection with elliptic and modular functions. In recent work connections with higher genus Riemann surfaces have also emerged.

In my first lecture I will give a gentle introduction to VOA theory. My second lecture will be concerned with Zhu theory which relates VOAs to elliptic and modular forms. My last lecture will concentrate on a particular VOA, the Heisenberg Model (the bosonic string) on a general genus Riemann surface constructed by a Schottky sewing procedure where the partition function is related to the Zograf McIntyre Takhtajan formula for the Laplacian determinant line bundle.

Speaker: Richard A. Wentworth (University of Maryland)
Title: Higgs bundles at the Fuchsian locus
Abstract: In this series of talks I will present joint work with Francois Labourie which analyzes the variational behavior of dynamical quantities at certain distinguished points in the character variety of representations of surface groups into complex reductive Lie groups. I will begin with a review of the nonabelian Hodge correspondence which relates Higgs bundles to flat connections for a complex reductive Lie group G.  The deformation theory of the two moduli spaces has a direct relationship in the case of representations into G that come from uniformizing representions into PSL(2,R) followed a principal embedding into G. These are called "Fuchsian representations." The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations along this Fuchsian locus. The expression is in terms holomorphic differentials, and it gives a precise relationship with the Petersson pairing. Along the way, variational formulas are established that generalize results from classical Teichmueller theory, such as Gardiner's formula, the relationship between length functions and Fenchel-Nielsen deformations, and variations of cross ratios.

Speaker: Graeme Wilkin (National University of Singapore)
Title: Algebraic classification of Yang-Mills-Higgs flow lines
Abstract: The Yang-Mills-Higgs flow was originally studied by Simpson in the context of constructing Hermitian-Einstein metrics on Higgs bundles. Methods of Donaldson and Simpson show that the Yang-Mills-Higgs flow resembles a nonlinear heat equation on the space of Hermitian metrics on the bundle and therefore the downwards flow is well-behaved with respect to existence and uniqueness, and it also has nice smoothing properties. On the other hand, the upwards flow is ill-posed and therefore even existence is problematic.

In these lectures I will describe how to construct long-time solutions to the reverse Yang-Mills-Higgs flow that converge to a given critical point. On the space of Hermitian metrics, these solutions are asymptotic to a solution of the linear heat equation with initial condition at an eigenfunction of the Laplacian. This gives an algebro-geometric criterion for two critical points to be connected by a flow line, which leads naturally to a Morse-theoretic construction of the Hecke correspondence for Higgs bundles.

Lecture 1. Review of basic properties of the Yang-Mills-Higgs flow.
Lecture 2. Constructing solutions to the reverse heat flow.
Lecture 3. An algebro-geometric interpretation of flow lines.

- 1 hour talk

Speaker: Thilo Kuessner (KIAS)
Title: Fundamental class of representations
Abstract: Let $Gamma$ be the fundamental group of a compact manifold, possibly with boundary, and $rhocolonGammato G$ a boundary-unipotent representation into a Lie group $G$. Many deformation-invariant invariants like Euler number, hyperbolic volume, Chern-Simons invariant, Borel class and Cheeger-Chern-Simons classes can be computed by applying some cocycle to the fundamental class $(Brho)_*left[M,partial Mright]in H_*(BG^delta;Z)$ in the group homology of $G$. The fundamental class is constant on components of the character variety of boundary-unipotent characters, and it is related to some similarly constructed invariants in the Bloch group or algebraic K-theory.
We will present two results. First, we show that this fundamental class behaves nicely under certain cut and paste operations ("generalized mutation"). This implies that invariants like the hyperbolic volume are preserved, in particular giving a topological proof of Ruberman's theorem on mutation of knots.
In another direction we will discuss the conjecture that $(rho_notimesoverline{rho}_m)_*h=noverline{h}+mh$ should hold for the irreducible representations $rho_ncolon SL(2,C)to SL(n,C)$ and for $hin H_3(SL(2,C))$. We will prove this conjecture by explicit computation (using the recently developed approach of Garoufalidis-D.Thurston-Zickert) for $n=m=2$. In the general case, the conjecture would be implied by a conjecture of Neumann's that all elements of $H_3(SL(2,C))$ can be represented by representations of hyperbolic 3-manifold groups.

Speaker: Xin Nie (KIAS)
Title: Geometry of SL(2,R) and SL(3,R) Higgs bundles on punctured surfaces.
Abstract: SL(n,R)-Higgs bundles are constructed from holomorphic differentials on a Riemann surface and, when the surface is closed, parametrize a connected component in the moduli space of fundamental group representations in SL(n,R). The representations come from solving Hitchin's self-duality equation. Given holomorphic differentials on a punctured Riemann surface with poles on the punctures, unique solvability of the corresponding Hitchin's equation is no longer guaranteed, but we will discuss geometric significance of solutions satisfying certain asymptotic conditions, in the SL(2,R) and SL(3,R) case.
On the complex plane, when n=2 these solutions correspond to constant mean curvature surfaces in the Minkowski space asymptotic to polytopes will null plane faces; when n=3 they correspond to hyperbolic affine spheres in R^3 asymptotic to a convex polygonal cone.

Speaker: Binbin Xu (KIAS)
Title: Pressure metric on Teichm?ller space of surface with boundary
Abstract: Let S be an oriented compact surface with negative Euler characteristic. Teichm?ller space of S is the space of isotopy classes of the space of marked hyperbolic structures on S. There is a well-known Riemannian metric on the Teichm?ller space, called Weil-Petersson metric. It has many interesting properties, for example: K?hler, negatively curved, incomplete, geodesically convex, etc. When S is closed, this metric can be interpreted as the pressure metric by using the thermordynamic formalism. When S has boundary, both of these two metrics are still well-defined but the relation between them are not known. By studying the incompleteness of the pressure metric in the second case, we are capable to answer that they are not equivalent to each other.

Speaker: Sangbum Yoo (POSTECH)
Title: Fibers of Hitchin map and elementary modifications of stable rank 2 twisted Higgs bundles on a curve
Abstract: Let X be a smooth complex projective curve and let L be a line bundle on X. Let M be the moduli space of semistable rank 2 L-twisted Higgs bundles on X with trivial determinant. In this talk we construct a cycle in the product space of a Hilbert scheme of curves and Pic(X) by using an elementary modification of stable rank 2 L-twisted Higgs bundles. Then we explain how the locus of these cycles parametrizes fibers of Hitchin map restricted to the stable locus of M.