Deep Learning is more than a pack of techniques showing improving performance updating on benchmarks. I will focus on mathematics related to the comprehensive developments of the deep neural network (DNN). I plan to cover as many more topics on AI explanation, together with at least Generative AIs (GAN, VAE, and diffusion models), Generalization error analysis (PAC, Rademacher, and VC-dimension), Ranking by Sharpley values (Game), and but not the least Kenel machines (NTKs).