Orbifold concepts in equivariant singularity theory

拓扑空间的不变量和具有群（有限群）作用的复解析变异体在拓扑学、代数几何和数学物理中起着重要的作用。申请人及其合著者S. M. Gusein-Zade和A. Takahashi也对本研究做出了贡献。特别是，他们描述了一些所谓的可逆多项式与有限群对称之间的对称性（轨道朗道-金兹堡模型）。他们定义并研究了不变或等变向量场和1-型及其集合的一些指标。该项目的主要目标是进一步发展有限群作用下的不变量理论。特别的重点将放在轨道设置和轨道类型不变量上。更准确地说，计划继续寻找berglund - h<s:1> bsch对偶可逆多项式（也可能具有非阿贝尔对偶群的作用）的不变量之间的对称性，研究1-形式和向量场的等变指标的代数公式，并研究Milnor晶格的轨道类似物。提出的研究还包括麦凯对应的推广研究，陈氏特征数的轨道，和奥利克-兰德尔猜想。

Invariants of topological spaces and complex analytic varieties with actions of groups (say, finite ones) play an important role, in particular in topology, algebraic geometry and mathematical physics. The applicant and his coauthors S. M. Gusein-Zade and A. Takahashi also contributed to this research. In particular, they have described a number of symmetries between so called invertible polynomials with finite group symmetries (orbifold Landau-Ginzburg models). They have defined and studied some indices of invariant or equivariant vector fields and 1-forms and of their collections. The main objective of the project is a further development of the theory of invariants in the presence of a finite group action. The special focus will be on the orbifold setting and on orbifold type invariants. More precisely, it is planned to continue the search for symmetries between invariants of Berglund-Hübsch dual invertible polynomials (also with actions of possibly non-abelian dual groups), to study algebraic formulae for equivariant indices of 1-forms and vector fields, and to investigate orbifold analogues of the Milnor lattice. The proposed research also includes a study of generalizations of the McKay correspondence, of Chern characteristic numbers of orbifolds, and of the Orlik-Randell conjecture.