Quantum Teichmuller spaces of bordered Riemann surfaces and generalized Frobenius Manifolds

这是可积系统的一个项目，吸引杰出的科学家L.契诃夫教授在曼彻斯特呆一年。物理现象一般用微分方程来描述。这些问题通常很难或不可能解决。然而，有一类特殊的微分方程在某种意义上是可解的。它们被称为可积系统。当我们设法用一个可积系统来描述一个物理现象时，我们可以理解并经常预测它的行为。最近，可积系统的理论被重新表述为Frobenius流形的语言。Frobenius流形理论是纯数学、应用数学和理论物理学中许多学科的交叉点。这个理论的美妙之处在于它的普适性：对一类特殊的Frobenius流形所证明的结果，对其他种类的Frobenius流形也成立。例如，量子上同调和奇点理论中某些Frobenius流形的同构是镜像对称的一个版本。在这个项目中，我们计划探索Frobenius流形理论与另外两个令人着迷的数学分支之间的新联系：量子引力中已知的Teichmuller空间的量子化问题以及最近由Fomin和Zelevinsky引入的簇代数理论。这项研究将开辟突破性研究的新领域。事实上，当两个丰富的数学分支统一起来时，总是会产生许多有趣的新问题，并证明许多意想不到的结果。

This is a project in Integrable Systems, to attract an outstanding scientist, Prof. L. Chekhov, for one year in Manchester.Physical phenomena are generally described by differential equations. These are usually very difficult or impossible to solve. Nevertheless there is a special class of differential equations which are solvable in some sense. They are called integrable systems. When we manage to describe a physical phenomenon by an integrable system, we can understand and often predict its behavior. Recently the theory of integrable systems has been reformulated in the language of Frobenius manifolds. The theory of Frobenius manifolds lies at the crossroad of many disciplines in Pure, Applied Mathematics and Theoretical Physics. One of the beauties of this theory consists in its universality: results proved for a special class of Frobenius manifolds turn out to be true also for other classes of Frobenius manifolds. For example the isomorphy of certain Frobenius manifolds in quantum cohomology and in singularity theory is one version of mirror symmetry.In this project we plan to explore anew link between the theory of Frobenius manifolds and another two fascinating branch of mathematics: the problem of quantization of Teichmuller space known in quantum gravity and the theory of Cluster Algebras recently intruduced by Fomin and Zelevinsky.This research will open up new lines of ground breaking research. In fact, it is always the case that when two rich branches of mathematics are unified, many interesting new question will arise and many unexpected result will be proved.