Compact And Non Compact Spaces

This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices.

Authors: Terras, Audrey
New edition extensively revised and updated
Includes many new figures and examples
New topics include random matrix theory and quantum chaos
Includes recent work on modular forms and their corresponding L-functions in higher rank, the heat equation on Pn solution, the central limit theorem for Pn, densest lattice packing of spheres in Euclidean space, and much more

by Tim Kirschner (Author)
Presents sheaves with a clear connection to the set-theoretic foundations
Strives for a maximum of rigor (concerning proofs, statements, definitions, and notation)
Overcomes the “canonical isomorphism” paradigm; all morphisms are given/constructed explicitly
Introduces a Gauß-Manin connection for families of possibly non-compact manifolds

Since the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, and it has become a valuable tool in nonlinear analysis. Presenting a survey of advances made in generalizations of degree theory during the past decade, this book focuses on topological degree theory in normed spaces and its applications.

The book deals with the representation in series form of compact linear operators acting between Banach spaces, and provides an analogue of the classical Hilbert space results of this nature that have their roots in the work of D. Hilbert, F. Riesz and E. Schmidt.