In this talk, I will firstly introduce some background on the problem of joint moments of the characteristic polynomial of a random unitary matrix. Secondly, I will discuss the connections between joint moments and integrable systems, specifically, connections to solutions of the sigma-Painleve V and the sigma-Painleve III' equations, respectively. Thirdly, I will talk about the applications of these results, including explicit formulae for joint moments of finite matrix size, an efficient way to compute the expectation of some random variables arise from a determinantal process.
