Quantum Frobenius manifolds, Nelson-Regge algebra and Riemann-Hilbert problem.

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，我们就可以理解并经常预测它的行为。最近，可积系统的理论已经被重新表述为Frobenius流形的语言。Frobenius流形理论处于纯数学、应用数学和理论物理学中许多学科的交叉点。这个理论的一个优点在于它的普遍性：对于一类特殊的Frobenius流形证明的结果对于其他类的Frobenius流形也是正确的。例如，在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流形理论与另一个迷人的数学分支之间的又一个联系：量子引力中已知的Teichmuller空间的量子化问题。这项研究将开辟新的突破性研究领域。事实上，当两个丰富的数学分支被统一起来时，总是会出现许多有趣的新问题，并会证明许多意想不到的结果。

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 yet one more link between the theory of Frobenius manifolds and another fascinating branch of mathematics: the problem of quantization of Teichmuller space known in quantum gravity. 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.