This volume is a pedagogical research introduction to several recently discovered and actively developing mathematical and mathematical physics subjects. The main benefiting audience consists of graduate students and young postdocs interested in interaction between quantum field theory and mathematics related to automorphic forms, representation theory, number theory and geometry, and mirror symmetry. The main subjects of the book are: 1) Feynman integrals and modular functions, 2) hyperbolic and Lorentzian Kac-Moody algebras, related automorphic forms and applications to quantum gravity, 3) superconformal indices and elliptic hypergeometric integrals, related instanton partition functions, 4) moonshine, its arithmetic aspects, Jacobi forms, elliptic genus, and string theory, 5) theory and applications of the elliptic Painleve equation, aspects of Painleve equations in quantum field theories. All these topics are unified in the sense that all of them are related to various partition functions emerging in different supersymmetric and ordinary quantum field theories in curved space-times of different (d=2,3,…,6) dimensions. The reader should be interested to see different multidisciplinary methods (localization, Borcherds products, theory of special functions, Cremona maps, etc) targeted at treating partition functions of interest. weiterlesen