Geometry and Topology of Orbifolds

DMS-4724Ernesto Lupercio该提案包括一些关于orbifolds和相关主题的上同调理论的研究项目。该方法将联合收割机代数拓扑学与代数几何中的栈理论和辛几何中的工具集相结合。它提出了使用理论的群胚作为一个统一的工具，在解决这些问题的方法，已被证明是成功的，在我们的研究ofbifolds。以下是该项目某些方面的简要摘要。第一个项目的目标是研究上同调不变量，这些不变量可以从与一个轨道相关的几个自然空间中自然地获得。第二个项目是从同伦理论的角度研究椭圆形亏格的模性和刚性。第三个项目描述了一个程序来定义和研究Orbifold Deligne cohomologies和他们的关系gerbes定义的orbifold。第四个项目是研究一个orbifold的奇性分解的Hodge-Deligne数与原orbifold的关系。Orbifolds是几何空间，在其中跟踪特定情况的局部对称性是非常重要的。所以一般来说，轨道褶皱的点是根据局部对称性的大小来分类的。在平凡空间中，所有的点都是相等的，而在一个轨道中，对应于局部对称性的量，点具有不同的权。Orbifolds已被用于结构晶体学的研究中，通过减少晶体群空间群图中的冗余信息，使其成为一种方便的簿记设备。在理论物理学领域，轨道折叠也变得极其重要，称为超弦理论。在弦论中，经典高能物理学中的粒子种类繁多，取而代之的是一个基本的组成部分，即“弦”。当弦在时间中运动时，它会根据它是封闭的还是开放的，描绘出一个管或一个片。此外，弦可以振动，弦的不同振动模式代表不同的粒子类型，因为不同的模式被视为不同的质量或自旋。为了使这些理论描述物理宇宙，物理学家们被激励着让弦在一个轨道空间上运动。在这些情况下，理论获得了非常理想的属性，以模拟现实。在这个项目中提出的工作涉及到严谨的数学方法来研究这些空间的几何发展。

DMS-4724Ernesto LupercioThis proposal consists of a number of research projects on the topicof cohomology theories for orbifolds and related subjects. The suggestedmethods combine algebraic topology with techniquescoming from stack theory in algebraic geometry and tools arisingin symplectic geometry. It proposes the use of the theory ofgroupoids as a unifying tool in the solution these questions; amethodology that has proved successful before in our study oforbifolds. The following is a brief summary of some aspects of the project. The first project is program whose objective isthe study of the cohomological invariants that can be naturallyobtained from several natural spaces associated to an orbifold. Asecond project consist of a study the orbifold elliptic genus, its modularity and rigidity properties from the homotopy theory point of view. The third project describes a program to define and study Orbifold Deligne cohomologies and their relation to gerbes defined over the orbifold. The fourth is a project to study the relation between the Hodge-Deligne numbers of an orbifold resolution of singularities of an orbifold and the original orbifold. Orbifolds are geometric spaces in which it is very important to keep trackof the local symmetries of the particular situation. So in general thepointsof an orbifold are classified by the amount of local symmetry. In ordinaryspaces all points are equal, while in an orbifold points carry differentweights corresponding to the amount of local symmetry. Orbifolds have beenused in the study of structural crystallography allowing for a convenientbook-keeping device by reducing the redundant information in the space group diagrams of the crystallographic groups. Orbifolds have also becomeextremely important in the field of theoretical physics referred to asSuperstringtheory. In String Theory, the multitude of particle types of classicalhigh energy physics is replaced by a single fundamental building block, a`string'.As the string moves through time it traces out a tube or a sheet,according to whether it is closed or open. Moreover, the string can vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins. In an attempt to make these theories describe the physical universe, physicistshavebeen motivated to let the string move on an orbifold space. In these casesthetheory acquires properties that are very desirable for it to modelreality. The work proposed in this project involves the development of rigorousmathematical methods to study the geometry of these spaces.