Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques

Part I. Traditional Methods for Computing Heat Kernels.- Introduction.- Stochastic Analysis Method.- A Brief Introduction to Calculus of Variations.- The Path Integral Approach.- The Geometric Method.- Commuting Operators.- Fourier Transform Method.- The Eigenfunctions Expansion Method.- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds.- Laplacians and Sub-Laplacians.- Heat Kernels for Laplacians and Step 2 Sub-Laplacians.- Heat Kernel for Sub-Laplacian on the Sphere S^3.- Part III. Laguerre Calculus and Fourier Method.- Finding Heat Kernels by Using Laguerre Calculus.- Constructing Heat Kernel for Degenerate Elliptic Operators.- Heat Kernel for the Kohn Laplacian on the Heisenberg Group.- Part IV. Pseudo-Differential Operators.- The Psuedo-Differential Operators Technique.- Bibliography.- Index.