Böcker i Graduate Studies in Mathematics-serien

'Metric geometry' is an approach to geometry based on the notion of length on a topological space. This title offers an exposition of basic notions and techniques used in the theory of length spaces. It offers an elementary introduction into a broad variety of geometrical topics related to the notion of distance.

Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students.

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauss curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.

'Semiclassical Analysis' provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation.

Provides graduate students with exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces.

Presents the basic mathematical foundations of stochastic analysis (probability theory and stochastic processes) as well as some important practical tools and applications. The book strikes a nice balance between mathematical formalism and intuitive arguments, a style that is most suited for applied mathematicians.

This textbook is addressed to graduate students in mathematics or other disciplines who wish to understand the essential concepts of functional analysis and their applications to partial differential equations. The book is intentionally concise, presenting all the fundamental concepts and results but omitting the more specialized topics.

An extensive survey of many important topics in the theory of Hamilton-Jacobi equations with particular emphasis on modern approaches and viewpoints.

Develops a mathematical theory for finance, based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability, starting with initial capital arbitrarily near zero.

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. This book treats the spectral theory of automorphic forms as the study of the space $L^2 (H\Gamma)$ and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$.

Presents the theory of reproducing kernel Hilbert spaces. This work discusses the Pick problem of finding the function of smallest $H^\infty$ norm that has specified values at a finite number of points in the disk. It considers $H^\infty$ as the multiplier algebra of the Hardy space and uses Hilbert space techniques to solve the problem.

Nonlinear algebra provides modern mathematical tools to address challenges arising in the sciences and engineering. It is useful everywhere, where polynomials appear: in particular, data and computational sciences, statistics, physics, optimization. This book offers an invitation to this broad and fast-developing area.

Uses linear phase functions to study functions. Aims to cover all topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes."

Within applied topology, discrete Morse theory came into light as one of the main tools to understand cell complexes arising in different contexts, as well as to reduce the complexity of homology calculations. This book provides a gentle introduction into this beautiful theory.

Prime numbers have fascinated mathematicians since the time of Euclid. This book presents some of our best tools to capture the properties of these fundamental objects, beginning with the most basic notions of asymptotic estimates and arriving at the forefront of mathematical research.

Offers a gentle introduction to the enumerative part of combinatorics. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature such as the use of quotient posets to study the Moebius function and characteristic polynomial of a partially ordered set.

Focuses on the simplest case of scattering by compactly supported potentials, and provides pointers to modern literature where more general cases are studied. The book also presents an approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.

Provides a lucid yet thorough account of convection-dominated convection-diffusion problems and how to solve them numerically for beginning graduate students. The book includes a large number of exercises. An up-to-date bibliography provides the reader with further reading.

Many laws of physics are formulated in terms of differential equations. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view.

Using the combinatorial reciprocity theorems as a leitmotif, this book unfolds central ideas and techniques in enumerative and geometric combinatorics. Written in a friendly writing style, this is an accessible graduate textbook with almost 300 exercises, numerous illustrations, and pointers to the research literature.

A graduate text on the incompressible Navier-Stokes system, which is of fundamental importance in mathematical fluid mechanics as well as in engineering applications. The goal is to give a rapid exposition on the existence, uniqueness, and regularity of its solutions, with a focus on the regularity problem.

The theory of finite fields encompasses algebra, combinatorics, and number theory and has furnished widespread applications in other areas of mathematics and computer science. This book covers basic facts about finite fields, polynomials over finite fields, Gauss sums, algebraic number theory and cyclotomic fields, zeros of polynomials over finite fields, and classical groups over finite fields.