Progress in Mathematics

Pattern formation in physical systems is one of the major research frontiers of mathematics. A central theme of this book is that many instances of pattern formation can be understood within a single framework: symmetry. The book applies symmetry methods to increasingly complex kinds of dynamic behavior: equilibria, period-doubling, time-periodic states, homoclinic and heteroclinic orbits, and chaos. Examples are drawn from both ODEs and PDEs. In each case the type of dynamical behavior being studied is motivated through applications, drawn from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary biology. An extensive bibliography is provided.

"The action of a compact Lie group, G, on a compact symplectic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytope, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope." "The moment polytope also encodes quantum information about the action of G. Using the methods of geometric quantization, one can frequently convert this action into a representation, p, of G on a Hilbert space, and in some sense the moment polytope is a diagramatic picture of the irreducible representations of G which occur as subrepresentations of p. Precise versions of this item of folklore are discussed in Chapters 3 and 4. Also, midway through Chapter 2 a more complicated object is discussed: the Duistermaat-Heckman measure, and the author explains in Chapter 4 how one can read off from this measure the approximate multiplicities with which the irreducible representations of G occur in p." "The last two chapters of this book are a self-contained and somewhat unorthodox treatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed. This book is addressed to researchers and can be used as a semester text."--BOOK JACKET.

The impetus for current research in pro-p groups comes from four main directions: from new applications in number theory, which continue to be a source of deep and challenging problems; from the traditional problem of classifying finite p-groups; from questions arising in infinite group theory; and finally, from the younger subject of ‘profinite group theory’. A correspondingly diverse range of mathematical techniques is being successfully applied, leading to new results and pointing to exciting new directions of research. In this work important theoretical developments are carefully presented by leading mathematicians in the field, bringing the reader to the cutting edge of current research. With a systematic emphasis on the construction and examination of many classes of examples, the book presents a clear picture of the rich universe of pro-p groups, in its unity and diversity. Thirty open problems are discussed in the appendix. For graduate students and researchers in group theory, number theory, and algebra, this work will be an indispensable reference text and a rich source of promising avenues for further exploration.