Frobenius Manifolds and F-manifolds in Singularity Theory

其目的是研究自由因子上函数的奇异性。我们对所谓的线性自由因子特别感兴趣--那些自由因子，如正规交叉因子，其Der(-logD)是以权零生成的，特别是由有限维李子代数生成的。这些自由因子上的函数表现出丰富的现象学，与Arnold、Saito和Brieskorn所研究的经典情形既有显著的相似之处，也有显著的差异。我们希望在这些函数的形变理论中找到更多Frobenius流形的例子，并将它们与量子上同调中出现的Frobenius流形进行比较。

The aim is to study the singularities of functions on free divisors. We are particularly interested in the so-called linear free divisors - those free divisors, like the normal crossing divisor, for which Der(-log D) is generated in weight zero, and in particular is generated by a finite-dimensional Lie sub-algebra. The complement of such a divisor is a symmetric space which is the finite image of a complex Lie group.Functions on these free divisors exhibit a rich phenomenology which has striking resemblances, and also striking differences, with the classical case studied by Arnold, Saito and Brieskorn. We hope to find, in the deformation theory of these functions, further examples of Frobenius manifolds, and compare these with the Frobenius manifolds arising in quantum cohomology.