Nonlinear Sociophysics, Quantum Sociology, and Multiply Connected Topological Economics

First, nonlinear sociophysics and the nonlinear whole sociology are researched, which have chaos, fractal, etc. Second, quantum sociology is discussed. We propose that its bases are the extensive quantum theory and the social individual-wave duality. Third, we research the double solutions of some nonlinear equations with soliton and chaos, and the possible social meaning of chaos. On the one hand, chaos may correspond to the spread of ideas, the popularization of information, etc. On the other hand, it corresponds to the economic crisis and various social crises, etc. Fourth, chaos in corruption is discussed by mathematics. Fifth, we propose the multiply connected topological economics, in which the confidence relations and the influence functions represent various interacting strengths of different families, cliques, and systems of organization. This has a fractal structure. We propose the binary periods of the political economy by the complex function and the elliptic functions. Various applications of the mathematical and physical method are the important developing direction of modern social sciences.