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In the first purpose, we concentrate on the theory of quantum integrablesystems underlying the Connes-Kreimer approach. We introduce a new family ofHamiltonian systems depended on the perturbative renormalization process inrenormalizable theories. It is observed that the renormalization group candetermine an infinite dimensional integrable system such that this factprovides a link between this proposed class of motion integrals andrenormalization flow. Moreover, with help of the integral renormalizationtheorems, we study motion integrals underlying Bogoliubv character and BCHseries to obtain a new family of fixed point equations. In the second goal, weconsider the combinatorics of Connes-Marcolli approach to provide a Hall rootedtree type reformulation from one particular object in this theory namely,universal Hopf algebra of renormalization $H_{\mathbb{U}}$. As theconsequences, interesting relations between this Hopf algebra and somewell-known combinatorial Hopf algebras are obtained and also, one can make anew Hall polynomial representation from universal singular frame such thatbased on the universal nature of this special loop, one can expect a Hall treetype scattering formula for physical information such as counterterms. In thethird aim, with attention to the given rooted tree version of $H_{\mathbb{U}}$and by applying the Connes-Marcolli's universal investigation, we discover anew geometric explanation from Dyson-Schwinger equations.

Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations