Applications of Renormalization Group Methods in Mathematical Sciences

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We applied the renormalization group methods to the longstanding problems (the so-called Millennium problems) in mathematical sciences which go back to the last century. We apply this method to solve the 2D O(N) symmetric spin model which is a prototype model of the gauge theory model. In this model, one needs to integrate over infinite variables which form a functional space. We find a set of subspaces (called flow) which dominate the integral. This enables us to solve the recursive integrals and obtain the final result.