In the summer of 2014 leading experts in the theory of water waves gathered at the Newton Institute for Mathematical Sciences in Cambridge for four weeks of research interaction. A cross-section of those experts was invited to give introductory-level talks on active topics. This book is a compilation of those talks and illustrates the diversity, intensity, and progress of current research in this area. The key themes that emerge are numerical methods for analysis, stability and simulation of water waves, transform methods, rigorous analysis of model equations, three-dimensionality of water waves, variational principles, shallow water hydrodynamics, the role of deterministic and random bottom topography, and modulation equations. This book is an ideal introduction for PhD students and researchers looking for a research project. It may also be used as a supplementary text for advanced courses in mathematics or fluid dynamics.

The monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point.The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits ofa symplectic map and critical points of an action functional. New results onsingularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is that a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps. The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant bifurcation theory.

Synopsis: "In this dark, when we all talk at once, some of us must learn to whistle." In this comprehensive collection of his work, Craig Keen's voice emerges as that of a theologian who has indeed learned to whistle. In a day when much of what passes for academic "theology" is careful to maintain a safe distance from any determinate act of faith or work of praise, Keen evinces a single-minded determination to think and to speak, to write and to live doxologically. And whether writing or lecturing, teaching or conversing, Keen understands theology to be nothing less than an invitation to work out one's faith with fear and trembling. Throughout this volume Keen argues that the life, death, and resurrection of Jesus disrupt all metaphysical attempts to determine the reality of "God," and suggests instead that theology is to be done liturgically and eucharistically-as the work of a people whose labor is carried out with open hands, free from all attempts to grasp and control. Keen discusses doctrinal issues-the Trinity, incarnation, creation-as well as a number of critical theological concerns-church and culture, justice, holiness, Christian education-in this light. The result is a profound set of reflections on the ways in which the word of the cross simultaneously transgresses our constructions of "God" and gives us to live transgressively in love. Endorsements: "We've come to expect from Craig Keen that he will make things theological more difficult and complex than we thought they were. Then, after inviting us to accompany him in several unfamiliar paths, he makes us more trusting of the gospel without insisting that we eschew the complexity or arrive at a presumptuous conclusion. Masterful teacher that he is, he proffers only accompaniment, in all the richness of that term, knowing that each one will find the way only by being found by the Way. This is the method of these essays. What they also reveal is a writer whose humility and deference to God's grace is palpable holiness. Would that this holy way could spread among theologians." -M. Douglas Meeks Cal Turner Chancellor Professor of Theology and Wesleyan Studies Vanderbilt Divinity School "Talk of self-involving knowledge and performative speech has become so widespread in recent years that I almost hesitate to use it. But in relation to Craig Keen's work, there is really no alternative. Keen's writing is animated by a deep personal desire for an authentically kenotic existence, and a longing for the coming of a community of women and men who understand that they cannot live until they die. There is pain and anguish in these essays but there is also well-founded hope. I cannot read them without being reminded very forcibly of the crisis theology of the young Karl Barth. This is a book to be read and savored-and read again." -Bruce McCormack Charles Hodge Professor of Systematic Theology Princeton Theological Seminary Author Biography: Craig Keen is Professor of Systematic Theology at Azusa Pacific University, Azusa, California. He is the author of the forthcoming After Crucifixion (Cascade Books).

Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.