Operads and the Topology of Possibly Singular Spaces

PI的调查将集中在继续研究和理解空间，流形和orbifolds的拓扑结构的几何类别和品种和堆栈的代数类别。这种分析的一般工具是与这些空间相关的代数结构上的操作，例如上同调或K-理论上的操作。这类运算的著名例子是弦拓扑和Gromov-Witten理论。另外，PI还将使用其他的方法和技巧，如运算、（拟）Hopf代数、表示论、范畴论、模空间、顶点算子代数和余单集等。具体地说，基于他以前的工作，PI希望从检测循环空间的操作数中定义和构建一个新的谱，并为弦拓扑中的模空间动作建立一个余单设置。此外，他希望定义量子K理论，Gromov-Witten不变量和全局不变量的特征类。将这些轨道结构扩展到其他类型的全局对称性，将产生具有对称性的奇点的变形空间和手征deRham复形的轨道版本。这些构造是由物理学中的几何对称性如朗道-金兹伯格/卡拉比-丘对应和镜像对称性联系在一起的。这个提议对数学的几个领域都有贡献，并且由于这些构造通常是由弦理论和量子场论的考虑所激发的，它也在物理学中得到应用。特别是，上述数学调查预计将带来重要的新成果，交叉施肥学科的拓扑，代数，几何和数论一方面，另一方面增加了理论物理和数学之间的知识转移。 结果将有助于理解广泛的问题，从纯粹的数学问题，如解的基本结构，表现出对称性或不光滑的方程组，到弦理论预测的宇宙的更好的数学描述。

The PI's investigations will focus on the continuing study and understanding of the topology of spaces, manifolds and orbifolds in the geometric category and varieties and stacks in the algebraic category. The general tools for this analysis are operations on algebraic structures associated to these spaces such as operations on cohomology or on K-theory. Famous examples of this type of operations are String Topology and Gromov-Witten theory. The latter yields operations on the cohomology of a variety while the former provides operations on the homology of the loop space of a compact manifold.Additionally, the PI will use other methods and techniques such as operads, (quasi-)Hopf algebras, representation theory, category theory, moduli spaces, vertex operator algebras and cosimplicial sets in his analysis. Concretely based on his previous work, the PI expects to define and construct a new spectrum from an operad that detects loop spaces as well as to establish a cosimplicial setup for moduli space actions in string topology. Furthermore he expects to define quantum K-theory, Gromov-Witten invariants and characteristic classes for global quotients. Extending these orbifold constructions to other types of global quotients will yield deformation spaces for singularities with symmetries and an orbifold version of the chiral deRham complex. These constructions are tied together by conjectural symmetries such as the Landau-Ginzburg/Calabi-Yau correspondence and mirror symmetry which have their origin in physics.The proposal contributes to several fields of mathematics and as the constructions are often motivated by considerations of string theory and quantum field theory it also finds applications in physics. In particular, the mathematical investigation of the above are expected to bring about important new results that cross-fertilize the subjects of topology, algebra, geometry, and number theory on one hand and on the other hand add to the transfer of knowledge between theoretical physics and mathematics. The outcome will be helpful in understanding a wide spectrum of problems ranging from the purely mathematical such as the basic structure of solutions sets of equations which exhibit symmetries or are not smooth to a better mathematical description of the universe predicted by string theory.