Preface Liao Heng; Part I. Computing: Introduction to Part I; 1. Mathematics, models and architectures Bill McColl; 2. Mathematics and software verification Chen Haibo and Gao Xin; 3. Mathematics for quantum computing Kong Yunchuan; 4. Mathematics for AI: categories, toposes, types Daniel Bennequin and Jean-Claude Belfiore; Part II. Communications: Introduction to Part II; 5. Mathematics and compressed sensing Zhang Rui and Long Zichao; 6. Mathematics, information theory, and statistical physics Merouane Debbah; 7. Mathematics of data networking Li Zongpeng, Miao Lihua and Tang Siyu; 8. Mathematics and network science Sun Jie; Part III. Artificial Intelligence: Introduction to Part III; 9. Mathematics, information and learning Tong Wen and Ge Yiqun; 10. Mathematics and Bayesian inference Guo Kaiyang, Lv Wenlong and Zhang Jianfeng; 11. Mathematics, optimization and machine learning Jiu Shangling; 12. Mathematics of reinforcement learning Wu Shuang and Wang Jun; Part IV. Future: 13. Mathematics and prospects for future breakthroughs Dang Wenshuan; Editors and contributing authors.