This volume includes expositions of key developments over the past four decades in commutative and non-commutative algebra, algebraic $K$-theory, infinite group theory, and applications of algebra to topology. Many of the articles are based on lectures given at a conference at Columbia University honoring the 65th birthday of Hyman Bass. Important topics related to Bass's mathematical interests are surveyed by leading experts in the field. Of particular note is a professional autobiography of Professor Bass, and an article by Deborah Ball on mathematical education. The range of subjects covered in the book offers a convenient single source for topics in the field.
The Mathematical Neighborhoods of School Mathematics visits regions beyond, but proximal to and accessible from school mathematics. Its aim is to give readers a glimpse of not just the rich diversity and adaptability of mathematics, but, most importantly, its interconnections and overall coherence, a perspective not easily available from the school curriculum. This aim entailed assembling a variety of substantial mathematical domains that do not typically cohabit the same volume. The book begins with an in-depth treatment of topics in the school curriculum, often with novel approaches and connections. A unifying thread is the group theoretic study of addition and multiplication in the various number systems of school mathematics. The exposition is mathematically rigorous, including proofs of many fundamental theorems not otherwise easily available in mathematically accessible form. The Mathematical Neighborhoods of School Mathematics is intended to be a conceptual contribution to mathematics education. It will be a valuable resource in professional development of mathematics teachers, and in mathematical enrichment programs, for both students and teachers. In this regard, many of the chapters are relatively self-contained. It could also serve as a text for undergraduate mathematics majors with an interest in teaching. The exceptional Chapter 11 presents some novel instructional designs for problem-solving activities meant to cultivate “connection-oriented mathematical thinking.” Hyman Bass is the Samuel Eilenberg Distinguished University Professor of Mathematics and Mathematics Education at the University of Michigan. He is a member of the National Academy of Sciences and of the National Academy of Education. Jason Brasel, a former high school mathematics teacher, is a mathematics educator and researcher in secondary mathematics, who works at TeachingWorks, University of Michigan.
This fifth volume of collected papers constitutes not only an important historical archive, but also a valuable resource for work in the fields treated. The papers are complemented by detailed surveys on the current state of the field in two areas: the congruence subgroup problem and the algebraic $K$-theory and quadratic forms.
This monograph extends this approach to the more general investigation of X-lattices, and these "tree lattices" are the main object of study. The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Tree Lattices should be a helpful resource to researchers in the field, and may also be used for a graduate course on geometric methods in group theory.
The work of Joseph Fels Ritt and Ellis Kolchin in differential algebra paved the way for exciting new applications in constructive symbolic computation, differential Galois theory, the model theory of fields, and Diophantine geometry. This volume assembles Kolchin's mathematical papers, contributing solidly to the archive on construction of modern differential algebra. This collection of Kolchin's clear and comprehensive papers--in themselves constituting a history of the subject--is an invaluable aid to the student of differential algebra. In 1910, Ritt created a theory of algebraic differential equations modeled not on the existing transcendental methods of Lie, but rather on the new algebra being developed by E. Noether and B. van der Waerden. Building on Ritt's foundation, and deeply influenced by Weil and Chevalley, Kolchin opened up Ritt theory to modern algebraic geometry. In so doing, he led differential geometry in a new direction. By creating differential algebraic geometry and the theory of differential algebraic groups, Kolchin provided the foundation for a "new geometry" that has led to both a striking and an original approach to arithmetic algebraic geometry. Intriguing possibilities were introduced for a new language for nonlinear differential equations theory. The volume includes commentary by A. Borel, M. Singer, and B. Poizat. Also Buium and Cassidy trace the development of Kolchin's ideas, from his important early work on the differential Galois theory to his later groundbreaking results on the theory of differential algebraic geometry and differential algebraic groups. Commentaries are self-contained with numerous examples of various aspects of differential algebra and its applications. Central topics of Kolchin's work are discussed, presenting the history of differential algebra and exploring how his work grew from and transformed the work of Ritt. New directions of differential algebra are illustrated, outlining important current advances. Prerequisite to understanding the text is a background at the beginning graduate level in algebra, specifically commutative algebra, the theory of field extensions, and Galois theory.
The Mathematical Sciences Education Board (MSEB) and the U.S. National Commission on Mathematics Instruction (USNCMI) took advantage of a unique opportunity to bring educators together. In August 2000, following the Ninth International Congress on Mathematics Education (ICME-9) in Makuhari, Japan, MSEB and USNCMI capitalized on the presence of mathematics educators in attendance from the United States and Japan by holding a two and a half-day workshop on the professional development of mathematics teachers. This workshop used the expertise of the participants from the two countries to develop a better, more flexible, and more useful understanding of the knowledge that is needed to teach well and how to help teachers to obtain this knowledge. A major focus of the workshop was to discuss teachers' opportunities in both societies-using teaching practice as a medium for professional development. Another focus of the workshop addressed practice by considering the records of teaching, including videos of classroom lessons and cases describing teachers and their work. These proceedings reflect the activities and discussion of the workshop using both print and video to enable others to share in their experience
To achieve national goals for education, we must measure the things that really count. Measuring What Counts establishes crucial research- based connections between standards and assessment. Arguing for a better balance between educational and measurement concerns in the development and use of mathematics assessment, this book sets forth three principlesâ€"related to content, learning, and equityâ€"that can form the basis for new assessments that support emerging national standards in mathematics education.
The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study "cyclic renormalization", a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of such renormalizations is naturally represented by a rooted tree, together with a "spherically transitive" automorphism; the infinite case corresponds to maps with an invariant Cantor set, a class of particular interest for its relevance to the description of the transition to chaos and of the Mandelbrot set. The normal subgroup structure of the automorphism group of such "spherically homogeneous" rooted trees is investigated in some detail. This work will be of interest to researchers in both dynamical systems and group theory.
Are you among the 22 million students now enrolled in college? Or a high school student thinking of joining them shortly? Or perhaps a parent of a college-bound junior or senior? Then this book is just for you. Written by college professors and successfully used by tens of thousands of students, The Secrets of College Success combines easy-to-use tips, techniques, and strategies with insider information that few professors are willing to reveal. The over 800 tips in this book will show you how to: pick courses and choose a major manage your time and develop college-level study skills get good grades and manage the “core” requirements get motivated and avoid stress interact effectively with the professor or TA prepare for a productive and lucrative career New to this third edition are high-value tips about: undergraduate and collaborative research summer internships staying safer on campus diversity and inclusion disabilities and accommodations …with special tips for international students at US colleges. Winner of the 2010 USA Book News Award for best book in the college category, The Secrets of College Success makes a wonderful back-to-college or high-school-graduation gift –or a smart investment in your own college success.
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