Related papers: Teichmuller geometry of moduli space, II: M(S) see…

We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichm\"uller space of the torus, inspired by Thurston's Lipschitz metric…

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S. We…

As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique…

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichm\"uller…

Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving…

We construct the smooth, compact moduli space of similarity classes of labeled, oriented triangles. The space, denoted $\mathfrak D$, is a connected sum of three projective planes, and projects via blowdown to two shape spaces that have…

We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…

Let $X$ be any compact connected Riemann surface of genus $g \geq 3$. For any $r\geq 2$, let $M_X$ denote the moduli space of holomorphic $SL(r,C)$-connections over $X$. It is known that the biholomorphism class of the complex variety $M_X$…

We classify the Teichm\"uller curves in the moduli space of genus three Riemann surfaces $\mathcal M_3$ that are obtained by a covering construction from a primitive Teichm\"uller curve in $\mathcal M_2$. We describe the action on homology…

We present an outline of the theory of universal Teichmuller space, viewed as part of the theory of QS, the space of quasisymmetric homeomorphisms of a circle. Although elements of QS act in one dimension, most results depend on a…

Veech groups uniformize Teichm\"uller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite…

The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts…

Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. A cone spherical metric is called irreducible if each developing map of the metric does not have…

For any analytically finite Riemann surface, the Teichm\"uller modular group is countable, but it is not easy to find an analytically infinite Riemann surface for which the Teichm\"uller modular group is countable. In this paper, we show…

The paper exhibits a product-to-sum formula for the observables of a certain quantization of the moduli space of flat SU(2)-connections on the torus. This quantization was defined using the topological quantum field theory that was…

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact K\"{a}hler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact,…

Applying the authors' preceding work, we construct a version of the moduli space of $G$-torsors over the formal punctured disk for a finite group $G$. To do so, we introduce two Grothendieck topologies, the sur (surjective) and luin…

We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the…

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we…

We show the existence of a symplectic structure on the moduli space of the Seiberg-Witten equations on $\Sigma \times \Sigma$ where $\Sigma$ is a compact oriented Riemann surface. To prequantize the moduli space, we construct a Quillen-type…