Mathematical studies on the statistical properties of complex systems

随机矩阵在50年代末被S用来模拟核光谱，现在是量子混沌、介观现象和无序系统理论的标准。这些都是物理复杂系统的例子，由于对称性，未知的相互作用导致可预测的行为。大量的数学结构源于对称性--可积系统、Painleve方程、麦克唐纳多项式理论等等。这些结构将进一步发展，导致分布函数的解析形式量化复杂系统的类别。类似的统计量化是最近提出的分析人工复杂系统(如股市)的方法的本质。

Introduced in the late `50's to model nuclear spectra, random matrices are now standard in the theory of quantum chaos, mesoscopic phenomena and disordered systems. These are all examples of physical complex systems, characterized by unknown interactions leading to predictable behaviour due to symmetries. Vast mathematical structures result from the symmetries - integrable systems, Painleve equations, Macdonald polynomial theory to name a few. These structures will be further developed, leading to the analytic form of distribution functions quantifying classes of complex systems. Analogous statistical quantification is the essence of recently proposed methods to analyze artificial complex systems such as the stock market.