Related papers: Mapping Class Groups and Moduli Spaces of Curves
We investigate topological properties of the moduli space of spin structures over genus two curves. In particular, we provide a combinatorial description of this space and give a presentation of the (rational) cohomology ring via generators…
We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the…
We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this…
These are notes for the Bootcamp volume for the 2015 AMS Summer Institute in Algebraic Geometry. They are based on earlier notes for the "Positive Characteristic Algebraic Geometry Workshop" held at University of Illinois at Chicago in…
In this paper, we explore the virtual technique that is very useful in studying moduli problem from differential geometric point of view. We introduce a class of new objects "virtual manifolds/orbifolds", on which we develop the integration…
We construct a smooth Deligne-Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m,…
Recently we suggested a new quantum algebra, the moduli algebra, which was conjectured to be a quantum algebra of observables of the Hamiltonian Chern Simons theory. This algebra provides the quantization of the algebra of functions on the…
In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth…
In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.
This paper agrees basically with the talk of the author at the workshop "Homological Mirror Symmetry and Applications", Institute for Advanced Study, Princeton, March 2007.
The main objective of this paper is to give a summary of our recent work on recursion formulae for intersection numbers on moduli spaces of curves and their applications. We also present a conjectural relation between tautological rings and…
We survey various constructions of finite dimensional projective representations of mapping class groups derived from stated skein algebras.
This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and…
This is a survey article to appear in the "Handbook on Teichmueller Theory".
The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and…
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification…
The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results…
We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of…
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to…