This monograph studies the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It introduces many original ideas, methods, and tools developed by the author and provides a list of all known exact results in explicit form and derives them for the heat kernel on spheres and hyperbolic spaces. Part I considers the geometry of simple Lie groups and maximally symmetric spaces in detail, and Part II discusses the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. This text will be a valuable resource for researchers and graduate students working in various areas of mathematics such as global analysis, spectral geometry, stochastic processes, and financial mathematics as well in areas of mathematical and theoretical physics including quantum field theory, quantum gravity, string theory, and statistical physics.

Autorentext

Dr. Ivan Avramidi is an accomplished researcher with more than 30 years of research experience in mathematical physics. He has a cutting-edge expertise in asymptotic analysis of partial differential equations on manifolds and the applications of these methods to quantum physics, differential geometry and financial mathematics. His research program spans such areas as global analysis, geometric analysis, mathematical physics, spectral geometry, differential geometry, quantum field theory, quantum gravity and financial mathematics. Dr. Avramidi has an extensive publication record including two books and more than 60 refereed papers. Dr. Ivan Avramidi got his Ph.D. in Theoretical and Mathematical Physics at the Lomonosov Moscow State University (Russia) in 1987. He held several research scientist positions at Rostov State University (1987-1990), University of Karlsruhe (1989-1990), University of Naples (1995), University of Greifswald (1993-1998) and a visiting faculty position at the University of Iowa (1997-1999); since 1999 he is a Professor of Mathematics at the New Mexico Institute of Mining and Technology. Dr. Avramidi was awarded Research Fellowships by Alexander von Humboldt Foundation (1993-1996 and 2018) and by the DAAD (German Academic Exchange Service) (1989-1990). His research was supported by the Deutsche Forschungsgemeinschaft (German Science Foundation) and the Department of Science and Higher Education (Russia).

Inhalt

Part I. Manifolds.- Chapter. 1. Introduction.- Chapter. 2. Geometry of Simple Groups.- Chapter. 3. Geometry of SU(2).- Chapter. 4. Maximally Symmetric Spaces.- Chapter. 5. Three-dimensional Maximally Symmetric Spaces.- Part II: Heat Kernel.- Chapter. 6. Scalar Heat Kernel.- Chapter. 7. Spinor Heat Kernel.- Chapter. 8. Heat Kernel in Two Dimensions.- Chapter. 9. Heat Kernel on S3 and H3.- Chapter. 10. Algebraic Method for the Heat Kernel.- Appendix A.- References.- Index.