Mirror Symmetry and Frobenius Manifolds

该项目将在Frobenius流形的框架下研究镜像对称猜想。 他的猜想有许多层次，从纯粹的拓扑陈述霍奇数的最先进的配方方面的等价性的A-无穷大范畴。 有一个由弗罗贝纽斯流形描述的中间层次，在那里，被认为与这种对称性相关的结构比拓扑图像更丰富，但也不具有完整的不易解释的范畴结构。 在物理框架中，有一种方法可以通过使用基本构建块以及张量积和轨道折叠这两种运算来构造镜像对称的伙伴。 张量积是两种运算中的一种，也是R.考夫曼和完全理解内的理论弗罗贝纽斯流形。 等价的基本积木将是一个版本的minimitudes开折的奇异性的功能与孤立的临界点。 然而，最后一步仍有待完成，即轨道折叠的定义。 这相当于研究有限群作用并在范畴中定义正确的商。在过去的几年里，数学和物理界之间的富有成效的互动在弦理论的推动下发展起来。 例如，这使代数几何学家和物理学家在研究镜像对称问题时联合起来。 这是一种基于物理论证的几何对称，它对代数几何和枚举几何有着惊人的影响。

The project will study the mirror-symmetry conjecture in the framework of Frobenius manifolds. his conjecture has many levels, from a purely topological statement about Hodge numbers to the most advanced formulation in terms of equivalences of A-infinity categories. There is an intermediate level described by Frobenius manifolds, where the structures which are supposed to be related by this symmetry are richer than the topological picture, but also do not have the full not easily handleable categorical structure. In the physical framework there is a way to construct mirror-symmetric partners by using elementary building blocks and the two operations of tensor product and orbifolding. One of the two operations, the tensor product has been the object of previous research of R. Kaufmann and is fully understood within the theory Frobenius manifolds. The equivalent of the elementary building blocks will be a version of the miniversal unfoldings of singularities of functions with isolated critical points. The last step, however, remains to be completed; i.e. the definition of orbifolding. This amounts to studying finite group actions and defining the correct quotient in the category.In the past few years a fruitful interaction between the mathematics and physics communities has developed driven by the subject of string theory. This has for instance united algebraic geometers and physicists in looking into questions about mirror symmetry. This is a conjectural symmetry based on physical arguments, which has striking implications for algebraic and enumerative geometry.