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One is a science, the other an art; one useful, the other seemingly decorative, but mathematics and music share common origins in cult and mystery and have been linked throughout history. Emblems of Mind is Edward Rothstein’s classic exploration of their profound similarities, a journey into their “inner life.” Along the way, Rothstein explains how mathematics makes sense of space, how music tells a story, how theories are constructed, how melody is shaped. He invokes the poetry of Wordsworth, the anthropology of Lévi-Strauss, the imagery of Plato, and the philosophy of Kant. Math and music, Rothstein shows, apply comparable methods as they create their abstractions, display similar concerns with ratio and proportion, and depend on metaphors and analogies to create their meanings. Ultimately, Rothstein argues, they reveal the ways in which we come to understand the world. They are images of the mind at work and play; indeed, they are emblems of Mind itself. 

Jacques Barzun called this book “splendid.” Martin Gardner said it was “beautifully written, marvelous and entertaining.” It will provoke all serious readers to think in new ways about the grand patterns in art and life. 

“Lovely, wistful. . . . Rothstein is a wonderful guide to the architecture of musical space, its tensions and relations, its resonances and proportions. . . . His account of what is going on in the music is unfailingly felicitous.”—New Yorker

“Provocative and exciting. . . . Rothstein writes this book as a foreign correspondent, sending dispatches from a remote and mysterious locale as a guide for the intellectually adventurous. The remarkable fact about his work is not that it is profound, as much of the writing is, but that it is so accessible.”—Christian Science Monitor

We may learn from our mistakes, but Deborah Mayo argues that, where experimental knowledge is concerned, we haven't begun to learn enough. Error and the Growth of Experimental Knowledge launches a vigorous critique of the subjective Bayesian view of statistical inference, and proposes Mayo's own error-statistical approach as a more robust framework for the epistemology of experiment. Mayo genuinely addresses the needs of researchers who work with statistical analysis, and simultaneously engages the basic philosophical problems of objectivity and rationality.

Mayo has long argued for an account of learning from error that goes far beyond detecting logical inconsistencies. In this book, she presents her complete program for how we learn about the world by being "shrewd inquisitors of error, white gloves off." Her tough, practical approach will be important to philosophers, historians, and sociologists of science, and will be welcomed by researchers in the physical, biological, and social sciences whose work depends upon statistical analysis.

Functional analysis is a broad mathematical area with strong connections to many domains within mathematics and physics. This book, based on a first-year graduate course taught by Robert J. Zimmer at the University of Chicago, is a complete, concise presentation of fundamental ideas and theorems of functional analysis. It introduces essential notions and results from many areas of mathematics to which functional analysis makes important contributions, and it demonstrates the unity of perspective and technique made possible by the functional analytic approach.

Zimmer provides an introductory chapter summarizing measure theory and the elementary theory of Banach and Hilbert spaces, followed by a discussion of various examples of topological vector spaces, seminorms defining them, and natural classes of linear operators. He then presents basic results for a wide range of topics: convexity and fixed point theorems, compact operators, compact groups and their representations, spectral theory of bounded operators, ergodic theory, commutative C*-algebras, Fourier transforms, Sobolev embedding theorems, distributions, and elliptic differential operators. In treating all of these topics, Zimmer's emphasis is not on the development of all related machinery or on encyclopedic coverage but rather on the direct, complete presentation of central theorems and the structural framework and examples needed to understand them. Sets of exercises are included at the end of each chapter.

For graduate students and researchers in mathematics who have mastered elementary analysis, this book is an entrée and reference to the full range of theory and applications in which functional analysis plays a part. For physics students and researchers interested in these topics, the lectures supply a thorough mathematical grounding.

The Everyday Mathematics (EM) program was developed by the University of Chicago School Mathematics Project (UCSMP) and is now used in more than 185,000 classrooms by almost three million students. Its research-based learning delivers the kinds of results that all school districts aspire to. Yet despite that tremendous success, EMoften leaves parents perplexed. Learning is accomplished not through rote memorization, but by actually engaging in real-life math tasks. The curriculum isn’t linear, but rather spirals back and forth, weaving concepts in and out of lessons that build overall understanding and long-term retention. It’s no wonder that many parents have difficulty navigating this innovative mathematical and pedagogic terrain.

Now help is here. Inspired by UCSMP’s firsthand experiences with parents and teachers, Everyday Mathematics for Parents will equip parents with an understanding of EM and enable them to help their children with homework—the heart of the great parental adventure of ensuring that children become mathematically proficient.

Featuring accessible explanations of the research-based philosophy and design of the program, and insights into the strengths of EM, this little book provides the big-picture information that parents need. Clear descriptions of how and why this approach is different are paired with illustrative tables that underscore the unique attributes of EM. Detailed guidance for assisting students with homework includes explanations of the key EM concepts that underlie each assignment. Resources for helping students practice math more at home also provide an understanding of the long-term utility of EM. Easy to use, yet jam-packed with knowledge and helpful tips, Everyday Mathematics for Parents will become a pocket mentor to parents and teachers new to EM who are ready to step up and help children succeed. With this book in hand, you’ll finally understand that while this may not be the way that you learned math, it’s actually much better.

In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws.

This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.