°üÓÊ ÀèÂü-Ï£¶û²®ÌØÎÊÌâÓëÁ¿×Ó³¡ÂÛ¡ª¡ª¿É»ýÖØÕý»¯¡¢´÷É­-ʩθñ·½³Ì(Ó¢ÎÄ)

1 Introduction 2 Hopf Algebra Structures on Combinatorial Objects 2.1 Elementary properties 2.2 Combinatorial Hopf algebras 2.2.1 Connes-Kreimer Hopf algebra 2.2.2 Rooted trees and (quasi-)symmetric functions 2.2.3 Incidence Hopf algebras 3 Perturbative Renormalization 3.1 Insertion operator: From a pre-Lie algebra structure on Feynman diagrams to a Hopf algebra 3.2 Hopf algebraic formalism 4 Integrable Renormalization 4.1 Integrable systems: from finite dimension (geometric approach) to infinite dimen-sion (algebraic approach) 4.2 Rota-Baxter type algebras 4.3 Theory of quantum integrable systems 4.4 Integrable systems on the basis of the renormalization group 4.5 Baditoiu-Rosenberg framework: The continuation of the standard process 4.6 Fixed point equations 5 Connes-Marcolli Theory 5.1 Geometric nature of counterterms: Category of flat equi-singular connections 5.2 The construction of a universal Tannakian category 6 Universal Hopf Algebra of Renormalization 6.1 Shuffle type representation. 6.2 Rooted tree type representation 6.3 Universal counterterms 7 Combinatorial Dyson-Schwinger Equations and Connes-Marcolli Universal Treat-ment 7.1 Quantum motions in terms of the renormalization Hopf algebra 7.2 Universal Hopf algebra of renormalization and factorization problem 7.3 DSEs in a categorical framework 8 From Combinatorial Dyson-Schwinger Equations to the Category of Feynman Motivic Sheaves 8.1 DSEs in the context of the theory of motives 9 Conclusion ±à¼­ÊÖ¼Ç