Resurgence theory and its applications

This paper gives an introduction to the resurgence theory developed by French mathematician Jean écalle and its important applications in mathematics and physics. The theory of simple resurgent functions and mould calculus are systematically presented with briefly mentioning general singularities. Euler series, Stirling series, Airy function and quantum dilogarithm are taken as examples to explain the subtlety of the theory. Applications in quantum mechanics (exact WKB theory, semi-classical trace formula), gauge theory (especially complex Chern-Simons theory) and topological string theory are also mentioned.