Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections

We develop some cohomological tools for the study of the localgeometry of moduli and parameter spaces in complex AlgebraicGeometry. Notably, we develop canonical formulae for the differentialoperators of arbitrary order and their natural action on suitable‘natural’ modules (for example, functions); in particular,we obtain a formula, in terms of the moduli problem, for theLie bracket of vector fields on a moduli space. As an application,we obtain another construction and proof of flatness for thefamiliar KZW or Hitchin connection on moduli spaces of curves.2000 Mathematics Subject Classification 14D15, 32G05.     

Configuration Models For Moduli Spaces of Riemann surfaces with boundary

In this article we consider Riemann surfacesF of genus g ≥ 0 with n ≥ 1 incoming and m ≥ 1 outgoing boundary circles, where on each incoming circle a point is marked. For the moduli space m_g*(m, n) of all suchF of genusg ≥ 0 a configuration space model Rad_h(m, n) is described: it consists of configurations of h = 2g-2+m+n pairs of radial slits distributed over n annuli; certain combinatorial conditions must be satisfied to guarantee the genusg and exactly m outgoing circles. Our main result is a homeomorphism between Rad_h(m, n) and M_g*(m,n). The space Rad_h(m, n) is a non-compact manifold, and the complement of a subcomplex in a finite cell complex. This can be used for homological calculations. Furthermore, the family of spaces Rad_h(m, n ) form an operad, acting on various spaces connected to conformai field theories.     

二维井眼轨道设计模型及其精确解

讨论了典型的二维井眼轨道设计问题,建立了二维井眼轨道设计的一般数学模型,并求得了其全部精确解.这种方法避免了在设计中进行试算,设计计算简单、精确、快速.该模型具有普遍适用性,可广泛用于二维定向井、水平井和多目标井的井眼轨道设计.     

Tropicalization of Classical Moduli Spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.     

Duality Construction of Moduli Spaces

The duality of moduli spaces of coherent sheaves on curves and surfaces gives a GIT free construction of these spaces. It is a suitable generalization for topics as: strange duality, jumping lines of vector bundles on projective space, and Faltings' construction of the moduli space of semistable vector bundles on a complex curve given in his article Stable G-bundles and projective connections. Using this duality we propose a construction for the moduli space of semistable sheaves on a surface.     

Numerical Solutions of Matrix Differential Models Using Higher-Order Matrix Splines

This paper deals with the construction of approximate solution of first-order matrix linear differential equations using higher-order matrix splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included.     

Partition Functions of $$mathcal{N}=(2,2)$$N=(2,2) Supersymmetric Sigma Models and Special Geometry on the Moduli Spaces of Calabi-Yau Manifolds

Theoretical and Mathematical Physics - We study a new example of a mirror relation between the exact partition functions of $$mathcal{N}=(2,2)$$ super-symmetric gauged linear sigma models on the...     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

Moduli Spaces of Singular Yamabe Metrics

Complete, conformally flat metrics of constant positive scalar curvature on the complement of points in the -sphere, , , were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension . For a generic set of nearby conformal classes the moduli space is shown to be a -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

     

Moduli Spaces of Maslov Complex Germs

Maslov complex germs (complex vector bundles, satisfying certain additional conditions, over isotropic submanifolds of the phase space) are one of the central objects in the theory of semiclassical quantization. To these bundles one assigns spectral series (quasimodes) of partial differential operators. We describe the moduli spaces of Maslov complex germs over a point and a closed trajectory and find the moduli of complex germs generated by a given symplectic connection over an invariant torus.     

Periodic Besov Spaces and Generalized Moduli of Smoothness

Mathematical Notes -     

Rationality of Moduli Spaces of Parabolic Bundles

The moduli space of parabolic bundles with fixed determinantover a smooth curve of genus greater than one is proved to berational whenever one of the multiplicities of the quasi-parabolicstructure equals one. This gives a new proof that the modulispace of vector bundles of coprime rank and degree is stablyrational, a result originally due to Ballico, and the boundon the level is strong enough to deduce rationality in manycases, extending results of Newstead.     

Recursion Between Mumford Volumes of Moduli Spaces

We propose a new proof, as well as a generalization of Mirzakhani??s recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich??s integral, i.e., we relate them to a ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani??s recursions to measures containing all higher Mumford??s ?? classes, and not only ??₁ as in the Weil?CPetersson case.     

Beauville Surfaces,Moduli Spaces and Finite Groups

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.     

Eigenvalues and Entropy Moduli of Operators in Interpolation Spaces

The paper approaches in an abstract way the spectral theory of operators in abstract interpolation spaces. We introduce entropy numbers and spectral moduli of operators, and prove a relationship between them and eigenvalues of operators. We also investigate interpolation variants of the moduli, and offer a contribution to the theory of eigenvalues of operators. Specifically, we prove an interpolation version of the celebrated Carl–Triebel eigenvalue inequality. Based on these results we are able to prove interpolation estimates for single eigenvalues as well as for geometric means of absolute values of the first n eigenvalues of operators. In particular, some of these estimates may be regarded as generalizations of the classical spectral radius formula. We give applications of our results to the study of interpolation estimates of entropy numbers as well as of the essential spectral radius of operators in interpolation spaces.     

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV _r equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r–1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity A _r–1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov–Witten invariants and quantum cohomology.