Related papers: Loops in surfaces and star-fillings
We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main…
The action of the mapping class group of a surface on the collection of homotopy classes of disjointly embedded curves or arcs in the surface is discussed here as a tool for understanding Riemann's moduli space and its topological and…
We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution $\tau$,…
Boolean operations of geometric models is an essential issue in computational geometry. In this paper, we develop a simple and robust approach to perform Boolean operations on closed and open triangulated surfaces. Our method mainly has two…
In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic…
We construct a graph complex calculating the integral ho- mology of the bordered mapping class groups. We compute the ho- mology of the bordered mapping class groups of various surfaces. Using the circle action on this graph complex, we…
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…
Given two free homotopy classes $\alpha_1, \alpha_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(\alpha_1, \alpha_2)$ of loops in these two classes. We show that for…
In the context of quantum gravity for spacetimes of dimension 2+1, we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in R^2 (representing loops on the…
We construct a new grading on the Goldman Lie algebra of a closed oriented surface by the winding number. This grading induces a grading on the HOMFLY-PT skein algebra and related algebras. Our work supports the conjectures of B. Cooper and…
In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative…
We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete…
We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.
Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6)(1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie…
The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and…
In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the…
This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…
We develop a new, intrinsic, computationally friendly approach to Lie coalgebras through graph coalgebras, which are new and likely to be of independent interest. Our graph coalgebraic approach has advantages both in finding relations…