RUI: Versal Deformations, Deformation Quantization, Moduli Spaces and Graph Complexes

该项目包括三个部分：（1）研究了李代数和结合代数的推广--A_infty，L_infty代数的形变，（2）研究了多项式Poisson代数的形变量子化问题;（三）讨论了在带标记点的黎曼曲面模空间研究中，由于该空间与度量带轨道的组合等价而引起的一些问题图表。 第一部分继续与Alice Fialowski联合开展的项目。 第二部分是关于推广PI和Pol Vanhaecke的一个纯上同调构造的唯一的一般的三阶形变量子化，它扩展到四阶形变。 第三部分是与Motohico Mulase的联合工作，比较开放。 合理的目标可能是：建立了Riemann曲面模空间与带状图复形之间的标准轨道折叠同构，给出了Strebel微分在某些有趣情形下的具体刻画，研究了由带状图与Riemann曲面在有理数代数闭包上的对应关系所决定的KP系统的解，给出了Riemann曲面模空间与带状图复形之间的标准轨道折叠同构的具体刻画，给出了Riemann曲面模空间与带状图复形之间的标准轨道折叠同构的具体刻画。详细地描述了带状图复形的同调与带标记点的黎曼曲面的模空间之间的对应关系。这个项目包括三个与物理有关或受物理启发的数学问题。 第一个问题是研究当一种非常普遍的代数结构，一种在数学和物理中出现的代数结构，被“变形”时会发生什么，例如通过将一些特殊的小扭曲过程纳入通常的代数乘法运算中。 这项调查继续与匈牙利的一位同事合作进行。 第二个问题涉及应用于某类多项式代数的不同类型的变形过程。 第三，也是最开放的问题，是关于调查一个显着的和意想不到的对应关系之间的代数定义的几何空间和差分（如在微积分）定义的几何对象。 这第三部分也提供了大量的机会，本科生参与研究。

This project has three parts: (1) To study the versal deformations of A_\infty , L_\infty algebras, which are generalizations of Lie and associative algebras; (2) To study the problem of deformation quantization of polynomial Poisson algebras; (3) To address some problems in the study of the moduli space of Riemann surfaces with marked points that arise from the combinatorial equivalence of this space and the orbifold of metric Ribbon graphs. The first part continues a program being carried out jointly with Alice Fialowski. The second part is about generalizing a purely cohomological construction of the PI and Pol Vanhaecke of the unique generic deformation quantizationof order three which extends to a fourth order deformation. The third part, which is joint work with Motohico Mulase, is more open-ended. Reasonable goals might be: to establish a canonical orbifold diffeomorphism between the moduli space of Riemann surfaces and the Ribbon graph complex; to give concrete characterizations of the Strebel differential in some interesting cases; to study solutions to the KP system determined by the correspondence between ribbon graphs and Riemann surfaces over the algebraic closure of the rationals; to describe in detail the correspondence between the homology of the Ribbon graph comples, and the moduli space of Riemann surfaces with marked points.This project consists of three problems in mathematics related to or motivated by physics. The first problem is to study what happens when a very general kind of algebraic structure, one that arises in both mathematics and physics, is "deformed," for example by incorporating some special little twisting process into the usual algebraic operation of multiplication. This investigation continues collaborative work with a colleague in Hungary. The second problem involves a different kind of deformation process applied to a certain class of polynomial algebras. The third, and most open-ended problem, is about investigating a remarkable and unexpected correspondence between an algebraically defined geometric space and a differentially (as in calculus) defined geometric object. This third part also provides numerous opportunities for undergraduate involvement in research.