Universalities in random matrix theory and quantum chaos

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The Universalities in random matrix theory appears in the correlation functions, and there are several universal classes depending upon the symmetries. In this research project, we have studied the random matrix theory (RMT) with an external source. We study RMT by a parameter β, which corresponds to GOE, GUE and GSE for β21,2 and 4, respectively. We have extended the integral formula of Harish Chandra-Itzykson-Zuber to a general value of β. The relation to Jack polynomial has been also studied. From the correlation function at the edge point, we obtain the intersection numbers on Riemann surfaces and applications to quantum chaos and random laser have been extensively studied.