Frobenius Manifolds and F-manifolds in Singularity Theory

目的是研究函数在自由因子上的奇异性。我们对所谓的线性自由除数特别感兴趣——这些自由除数，比如正常交叉除数，其中Der（-log D）是在权值为零的情况下生成的，特别是由有限维李子代数生成的。这样一个除数的补是一个对称空间，它是复李群的有限像。这些自由因子上的函数表现出丰富的现象学，与Arnold、Saito和Brieskorn研究的经典案例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变形理论中找到更多的Frobenius流形的例子，并将它们与量子上同调中的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.