Hoermander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hoermander and is intimately connected to the geometry of vector fields.Motivations for the study of Hoermander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hoermander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.

Over the course of the last century, the systematic exploration of the relationship between Fourier analysis and other branches of mathematics has lead to important advances in geometry, number theory, and analysis, stimulated in part by Hurwitzs proof of the isoperimetric inequality using Fourier series. This unified, self-contained volume is dedicated to Fourier analysis, convex geometry, and related topics. Specific topics covered include: the geometric properties of convex bodies the study of Radon transforms the geometry of numbers the study of translational tilings using Fourier analysis irregularities in distributions Lattice point problems examined in the context of number theory, probability theory, and Fourier analysis restriction problems for the Fourier transform The book presents both a broad overview of Fourier analysis and convexity as well as an intricate look at applications in some specific settings; it will be useful to graduate students and researchers in harmonic analysis, convex geometry, functional analysis, number theory, computer science, and combinatorial analysis. A wide audience will benefit from the careful demonstration of how Fourier analysis is used

Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians