Related papers: Moduli Spaces and Formal Operads

If $X$ is a variety with an additional structure $\xi$, such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There…

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb].…

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…

We prove the existence of minimal models a la Sullivan for operads with non-trivial arity zero. So, up to homotopy algebras with strict units are just operad algebras over these minimal models. As an application we give another proof of the…

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn…

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to…

Recent algebraic structures of string theory, including homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras, are deduced from the topology of the moduli spaces of punctured Riemann spheres. The principal reason for these…

We study moduli spaces of flat metrics on closed Riemannian orbifolds admitting such metrics. We show that for such orbifolds $\mathcal{O}$, the Teichm\"uller space of flat metrics $\mathcal{T}_{\text{flat}}(\mathcal{O})$ serves as a…

A $\mathrm{PGL}_n^{(N)}$-oper is a specific type of flat $\mathrm{PGL}_n$-bundle on an algebraic curve in prime characteristic $p$ enhanced by an action of the sheaf of differential operators of level $N-1$. In this paper, we introduce and…

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space…

We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we…

This paper is dedicated to the memory of Moshe Flato, and will appear in Lett. Math. Phys. 48 (1) It became clear during last 5-6 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has…

In this article we give a survey of homology computations for moduli spaces $\mathfrak{M}_{g,1}^m$ of Riemann surfaces with genus $g\geqslant 0$, one boundary curve, and $m\geqslant 0$ punctures. While rationally and stably this question…

We prove that the space of holomorphic $p$-forms on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points vanishes for $p=14, 16, 18$ unconditionally and also for $p=20$ under a natural…

We establish that $E_n$-operads satisfy a rational intrinsic formality theorem for $n\geq 3$. We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in…

For every genus g, we construct a smooth, complete, rational polarized algebraic variety DM_g together with a normal crossing divisor D = sum D_i, such that for every moduli space M_C(2,0) of semistable topologically trivial vector bundles…

We compute the structure of the homology of an operad built from the spaces TH_{d,n} of configurations of points in C^d, modulo translation and homothety. We find that it is a mild generalization of Getzler's gravity operad, which occurs in…

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical…

We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to…

In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad $\{\bar{M}_{0,n+1}\}$ can be endowed with natural descent data that allow it to be considered as the lift to ${\rm Spec}\,\Z$ of an…