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Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of varifold surfaces and uses the methods of geometric measure theory to develop a mathematical description of the motion of a surface by its mean curvature. This mathematical description encompasses, among other subtleties, those of changing geometries and instantaneous mass losses. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston's geometrization conjecture.
This text is the result of an AMS Short Course on Knots and Physics that was held in San Francisco in January 1994. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds. The book features a basic introduction to knot polynomials in relation to statistical link invariants as well as concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity.
The first of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 1 begins with a problem list by S.T. Yau, successor to his 1980 list ( Sem
This book describes fruitful past collaborations between the mathematical and materials sciences and indicates future challenges. It seeks both to encourage mathematical sciences research that will complement vital research in materials science and to raise awareness of the value of quantitative methods. The volume encourages both communities to increase cross-disciplinary collaborations, emphasizing that each has much to gain from such an increase, and it presents recommendations for facilitating such work. This book is written for both mathematical and materials science researchers interested in advancing research at this interface; for federal and state agency representatives interested in encouraging such collaborations; and for anyone wanting information on how such cross-disciplinary, collaborative efforts can be accomplished successfully.
This book is the first volume of the SpringerBriefs in the Mathematics of Materials and provides a comprehensive guide to the interaction of mathematics with materials science. The anterior part of the book describes a selected history of materials science as well as the interaction between mathematics and materials in history. The emergence of materials science was itself a result of an interdisciplinary movement in the 1950s and 1960s. Materials science was formed by the integration of metallurgy, polymer science, ceramics, solid state physics, and related disciplines. We believe that such historical background helps readers to understand the importance of interdisciplinary interaction such as mathematics–materials science collaboration. The middle part of the book describes mathematical ideas and methods that can be applied to materials problems and introduces some examples of specific studies—for example, computational homology applied to structural analysis of glassy materials, stochastic models for the formation process of materials, new geometric measures for finite carbon nanotube molecules, mathematical technique predicting a molecular magnet, and network analysis of nanoporous materials. The details of these works will be shown in the subsequent volumes of this SpringerBriefs in the Mathematics of Materials series by the individual authors. The posterior section of the book presents how breakthroughs based on mathematics–materials science collaborations can emerge. The authors' argument is supported by the experiences at the Advanced Institute for Materials Research (AIMR), where many researchers from various fields gathered and tackled interdisciplinary research.
Acta Numerica is an annual volume presenting survey papers in numerical analysis. Each year the editorial board selects significant topics and invites papers from authors who have made notable contributions to the development of that topic. The articles are intended to summarize the field at a level accessible to graduate students and researchers. Acta Numerica has proved to be a valuable tool not only for researchers and professionals wishing to develop their understanding of the subject and follow developments, but also as an advanced teaching aid at colleges and universities. Articles in previous volumes have been expanded into both monographs and textbooks, and many of the original articles themselves have been used as the prime resource for graduate courses.
This volume offers a unique collection of some of the work of Frederick J. Almgren, Jr., the man most noted for defining the shape of geometric variational problems and for his role in founding The Geometry Center. Included in the volume are the following: a summary by Sheldon Chang of the famous 1700 page paper on singular sets of area-minimizing $m$-dimensional surfaces in $Rn$, a detailed summary by Brian White of Almgren's contributions to mathematics, his own announcements of several longer papers, important shorter papers, and memorable expository papers. Almgren's enthusiasm for the subject and his ability to locate mathematically beautiful problems that were "ready to be solved" attracted many students who further expanded the subject into new areas. Many of these former students are now known for the clarity of their expositions and for the beauty of the problems that they work on. As Almgren's former graduate student, wife, and colleague, Professor Taylor has compiled an important volume on an extraordinary mathematician. This collection presents a fine comprehensive view of the man's mathematical legacy