Operads and the Topology of Possibly Singular Spaces

PI的研究将集中在继续研究和理解几何范畴中的空间、流形和奥布里流形以及代数范畴中的簇和堆栈的拓扑。这种分析的一般工具是对与这些空间相关的代数结构的运算，例如上同调运算或K-理论运算。这类运算的著名例子是弦拓扑学和Gromov-Witten理论。后者产生关于簇的上同调的运算，而前者提供关于紧流形的环空间的同调的运算。此外，PI还将在分析中使用其他方法和技巧，如算子、(拟)Hopf代数、表示论、范畴理论、模空间、顶点算子代数和余单纯集。具体地说，PI期望从一个检测循环空间的算子定义和构造一个新的谱，并为弦拓扑中的模空间作用建立一个余单纯集。此外，他还期望定义量子K理论、Gromov-Witten不变量和全局商的特征类。将这些奥布伦德结构推广到其他类型的整体商，将产生具有对称性的奇点的形变空间和手征德勒姆复形的奥布朗德版本。这些结构通过猜想对称联系在一起，如Landau-Ginzburg/Calabi-Yau对应和镜像对称，它们起源于物理。这一提议对数学的几个领域做出了贡献，由于这些结构通常是出于对弦理论和量子场论的考虑，所以它在物理中也得到了应用。特别是，上述数学研究有望带来重要的新结果，一方面交叉培养拓扑学、代数、几何和数论的学科，另一方面增加理论物理和数学之间的知识转移。其结果将有助于理解一系列广泛的问题，从纯数学问题，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.