Probability theory related to analysis and geometry

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The following symposiums were held, where talks and discussions were given with fruitful results.1."Probability Theory and Computor Mathematics" : With several researchers in the field of physics and engineering, we approached to the topics both from theoritical and applicative point of view. Among all, Sugita proposed a new method of generating pseudo-random numbers by means of dynamics, Fukuyama discussed on the convergence of the Monte Carlo method with this random numbers, and Takashima reported on the test in which the pseudo-random numbers are regarded as a stochastic proccess.2."Topics on Stochastic Control" : In this research, it was revealed that the condition for avoiding break down in the risk sensitive stochastic control problem can be much relaxd than was known before in the case when the stochastic control processes are stable.3."Topics on Probability Theory for Quantum Chaos 2" : Talks and discussions on mathematical aspect of quantum chaos such as quantum chaos in numbe … More r theory or quantum chaos in stochastic analysis were given.4."Stochastic analysis and geometry" : Masahiko Kanai introduced a new approach to the rigidity of discrete group action, K.D.Elworthy discussed on the homology and homotopy vanishing theorems from the point of view of stochastic differential equations and Hajime Urakawa introduced his spectral geometry of graphs. Hironori Kumura discussed the structure of esential spectre for Riemannian manifold, and Atsushi Kasue geometric analysis of Riemannian submersions.5."Stochastic Processes in Population Genetics" : In the discussion of this symposium, in which S.N.Ethier and Tachida participate, the mathematical formulation of the model and research subject for House-of-Cards model due to Ohta-Tanchida was given.6."Non-equilibrium Statistical Mechanics and Probability Theory" : With the intensive lectures of H.Spohn (Ludwig-Maximilians-Univ.) and Kyoji Kawasaki (Chubu Univ.) and talks of some other probabilists, the present situations and prospect in the furture for therelation of non-equilibrium statistical mechanics and probability theory was analyzed.7."Some topics on Diffusion Processes and Random Walks" : Talks and discussion mainly on new results on the local time of Brownian motions and survey on diffusion processes were given.8."Stochastic Analysis and Asymptotics Theory" : Many talks and discussions on stochastic asymptotics theory were given. They covered not only the already established asymptotic expansions and large deviation priciple (Laplace method), but also the aymptotics theory on stochastic oscilatory integral, whose future development is strongly expected. Less