Mathematical studies on the statistical properties of complex systems

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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.