Geometric methods in discrete integrablesystems

我们主要研究离散painlevel方程。构造了包括E_8型椭圆差分painlevel方程在内的各种情形的Lax形式，并阐明了其几何性质。构造了相应的Pade插值问题，导出了painlevel方程、Lax形式和特解的简单表达式。研究了规范理论、共形场理论和量子痛级方程之间的关系。

We studied mainly the discrete Painleve equations. The Lax formalism for various cases including the elliptic difference Painleve equation of type E_8 was constructed, and its geometric characterization was clarified. We constructed the corresponding Pade interpolation problem and derive simple expressions for the Painleve equation, Lax form and special solutions. Relations among the Gauge theory, conformal field theory and quantum Painleve equations were also studied.