Geometry and Topology of Orbifolds

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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.