This book, aimed at readers with some grounding in mathematics, describes the nascent evolution of mathematical analysis, its development as a subject in its own right, and its wide-ranging applications in mathematics and science, modelling reality from acoustics to fluid dynamics, from biological systems to quantum theory.

The analytic geometry part of the book links equations to many of these intuitive concepts, which we explore through in-depth explanations of manipulative models of conic sections. This book is aimed at high school students who might be in Algebra II or Pre-Calculus. It shows the geometrical and practical sides of these topics that otherwise can drown in their own algebra.

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space Hd. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum.