On 2D Inverse Problems/Overview

The inverse problems, that are the main objects of this book, have their roots in the study of the relationships between local and global properties of the embedded graphs and surfaces. The two-dimensional setting has an important structure, due to the duality b/w the harmonic conjugate functions and the connection to classical special matrices, that can be viewed as two-dimensional objects. Also, the dimension of the unknown parameter fits the dimension of the measurement data in several important instances of the inverse problems. thumbnail|Discretisation technique The solutions of the inverse problems are presented in a way bridging the discrete and continuous settings. The techniques of solutions are motivated by the following commutative diagram: thumbnail|center|Calculation scheme The context of the inverse problems provides a unified point of view on the work of many great mathematicians, and even though there are references to many areas of mathematics in this book, it is practically self-contained, and is intended for a wide audience of readers interested in the subject. There are exercises, ranked by difficulty, through most of the chapters of the book, and it can be used as a textbook for an applied mathematics class. In the later volumes we'll discuss the connection and relationship between the spectrum of the interior operator and its boundary data Dirichlet-to-Neumann map (eigenvalues of the monodromy with singularity at the origin) and eta and zeta zeros (trivial and non-trivial kernels of the operator and the function .