The method of the Riemann-Hilbert problem for singular phenomenas in mathematical physics

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Singular phenomenas are investigated by use of the method of the Riemann-Hilbert problems. The following three topics are treated :(1) The problems of divergence in quantum field theory are discussed. The method of the Riemann-Hilbert problems supply the canonical forms of the divergences in local. Hence we can divide the divergence by the canonical one and we can get the vector bundle associated to the divergences. The anomaly can be described in terms of the characteristic classes of the bundle, which can be imeadiatlly obtained through the Fuchs relations.(2) The quarkconfinement can be discussed by use of the Riemann-Hilbert problems. After the formulation of the A belianization method, we can obtain the singular fieilds and the singular gauge trasformations. Then we can formulate the Riemann-Hilbert problems for the singularities and we can get the non-trivial anomaly which gives the mass of the gluons. Following the discussions in the Meissner effects, we can gent the quarkconfinement.(3) Toward the Riemann-Hilbert problem for non pertubative field theory. It is expected that non-pertabative method will be necessary for the real field theory. Here we prepare fundamental materials for this purpose. We treat the fractal geometry and discuss the sigularities by use of the algebraic method. Namely we consider the representations of the Cuntz algebras and discuss the infinite dimensional (Clifford algebras and the periodicity theorems in them.