This definitive synthesis of mathematician Gregory Margulis’s research brings together leading experts to cover the breadth and diversity of disciplines Margulis’s work touches upon.
 
This edited collection highlights the foundations and evolution of research by widely influential Fields Medalist Gregory Margulis. Margulis is unusual in the degree to which his solutions to particular problems have opened new vistas of mathematics; his ideas were central, for example, to developments that led to the recent Fields Medals of Elon Lindenstrauss and Maryam Mirzhakhani. Dynamics, Geometry, Number Theory introduces these areas, their development, their use in current research, and the connections between them. Divided into four broad sections—“Arithmeticity, Superrigidity, Normal Subgroups”; “Discrete Subgroups”; “Expanders, Representations, Spectral Theory”; and “Homogeneous Dynamics”—the chapters have all been written by the foremost experts on each topic with a view to making them accessible both to graduate students and to experts in other parts of mathematics. This was no simple feat: Margulis’s work stands out in part because of its depth, but also because it brings together ideas from different areas of mathematics. Few can be experts in all of these fields, and this diversity of ideas can make it challenging to enter Margulis’s area of research. Dynamics, Geometry, Number Theory provides one remedy to that challenge.

Pioneer Texas Buildings opened people's eyes when it was first published in 1968. At a time when "progress" meant tearing down the weathered houses, barns, churches, and stores built by the original settlers of Central Texas, this book taught people to see the beauty, simplicity, and order expressed in the unadorned geometric forms of early Texas buildings. It inspired the preservation and restoration of many of the remaining pioneer buildings, as well as the design of modern buildings that employ the same simple geometries.

This revised edition of Pioneer Texas Buildings juxtaposes the historic structures with works by twenty contemporary architects who are inspired by the pioneer tradition to show how seamlessly the basic geometries translate from one era to another. As in the first edition, sketches and brief commentary by Clovis Heimsath explain how squares, triangles, and circles take shape in the cubic, triangular, and cylindrical forms that comprise houses and other buildings. Then black-and-white photographs, the heart of the book, illustrate these geometric forms in historic and modern buildings. The book also includes two essays in which Heimsath discusses the factors that led him and his wife Maryann to document early Texas buildings and the results in historic preservation and timeless architectural designs that have followed from their efforts.

In this profound and hopeful book, a mathematician and celebrated teacher shows how mathematics may help all of us—even the math-averse—to understand and cope with grief.
 
We all know the euphoria of intellectual epiphany—the thrill of sudden understanding. But coupled with that excitement is a sense of loss: a moment of epiphany can never be repeated. In Geometry of Grief, mathematician Michael Frame draws on a career’s worth of insight—including his work with a pioneer of fractal geometry Benoit Mandelbrot—and a gift for rendering the complex accessible as he delves into this twinning of understanding and loss. Grief, Frame reveals, can be a moment of possibility.

Frame investigates grief as a response to an irrevocable change in circumstance. This reframing allows us to see parallels between the loss of a loved one or a career and the loss of the elation of first understanding a tricky concept. From this foundation, Frame builds a geometric model of mental states. An object that is fractal, for example, has symmetry of magnification: magnify a picture of a mountain or a fern leaf—both fractal—and we see echoes of the original shape. Similarly, nested inside great loss are smaller losses. By manipulating this geometry, Frame shows us, we may be able to redirect our thinking in ways that help reduce our pain. Small‐scale losses, in essence, provide laboratories to learn how to meet large-scale losses.

Interweaving original illustrations, clear introductions to advanced topics in geometry, and wisdom gleaned from his own experience with illness and others’ remarkable responses to devastating loss, Frame’s poetic book is a journey through the beautiful complexities of mathematics and life. With both human sympathy and geometrical elegance, it helps us to see how a geometry of grief can open a pathway for bold action.

Addressing both the literature and the visual arts of Anglo-American modernism, The Geometry of Modernism recovers a crucial development of modernism's early years that until now has received little sustained critical attention: the distinctive idiom composed of geometric forms and metaphors generated within the early modernist movement of Vorticism, formed in London in 1914. Focusing on the work of Wyndham Lewis, leader of the Vorticist movement, as well as Ezra Pound, H.D., and William Butler Yeats, Hickman examines the complex of motives out of which Lewis initially forged the geometric lexicon of Vorticism—and then how Pound, H.D., and Yeats later responded to it and the values that it encoded, enlisting both the geometric vocabulary and its attendant assumptions and ideals, in transmuted form, in their later modernist work.

Placing the genesis and appropriation of the geometric idiom in historical context, Hickman explores how despite its brevity as a movement, Vorticism in fact exerted considerable impact on modernist work of the years between the wars, in that its geometric idiom enabled modernist writers to articulate their responses to both personal and political crises of the 1930s and 1940s. Informed by extensive archival research as well as treatment of several of the least-known texts of the modernist milieu, The Geometry of Modernism clarifies and enriches the legacy of this vital period.

Uyechi presents an extremely thorough and formal empirical description of the various features of ASL signs, of interest to any theoretician in developing a theory of sign phonology or in testing claims in the theory of the phonology of spoken languages against data from a signed language. The author also presents a formalism for representing signs and makes a number of theoretical proposals based on this formalism. The volume's analysis indicates that the properties of core constructs of the spoken-language phonology, namely the segment and the syllable, differ from the properties of the core constructs in a formal framework of visual phonology. The Geometry of Visual Phonology also differs from other analyses in concluding that such differences are not immediately reconcilable. This volume provides a framework for discussing crucial differences between signs and speech.

A Provocative Examination of the Origin of Imagination

Aristotle was the first philosopher to divide the imagination—what he called phantasia—from other parts of the psyche, placing it between perception and intellect. A mathematician and philosopher of mathematical sciences, Aristotle was puzzled by the problem of geometrical cognition—which depends on the ability to “produce” and “see” a multitude of immaterial objects—and so he introduced the category of internal appearances produced by a new part of the psyche, the imagination. As Justin Humphreys argues, Aristotle developed his theory of imagination in part to explain certain functions of reason with a psychological rather than metaphysical framework. Investigating the background of this conceptual development, The Invention of Imagination reveals how imagery was introduced into systematic psychology in fifth-century Athens and ultimately made mathematical science possible. It offers new insights about major philosophers in the Greek tradition and significant events in the emergence of ancient mathematics while offering space for a critical reflection on how we understand ourselves as thinking beings. 

The surprising history behind a ubiquitous facet of the United States: the gridded landscape.
 
Seen from an airplane, much of the United States appears to be a gridded land of startling uniformity. Perpendicular streets and rectangular fields, all precisely measured and perfectly aligned, turn both urban and rural America into a checkerboard landscape that stretches from horizon to horizon. In evidence throughout the country, but especially the West, the pattern is a hallmark of American life. One might consider it an administrative convenience—an easy way to divide land and lay down streets—but it is not. The colossal grid carved into the North American continent, argues historian and writer Amir Alexander, is a plan redolent with philosophical and political meaning.
 
In 1784 Thomas Jefferson presented Congress with an audacious scheme to reshape the territory of the young United States. All western lands, he proposed, would be inscribed with a single rectilinear grid, transforming the natural landscape into a mathematical one. Following Isaac Newton and John Locke, he viewed mathematical space as a blank slate on which anything is possible and where new Americans, acting freely, could find liberty. And if the real America, with its diverse landscapes and rich human history, did not match his vision, then it must be made to match it.
 
From the halls of Congress to the open prairies, and from the fight against George III to the Trail of Tears, Liberty’s Grid tells the story of the battle between grid makers and their opponents. When Congress endorsed Jefferson’s plan, it set off a struggle over American space that has not subsided. Transcendentalists, urban reformers, and conservationists saw the grid not as a place of possibility but as an artificial imposition that crushed the human spirit. Today, the ideas Jefferson associated with the grid still echo through political rhetoric about the country’s founding, and competing visions for the nation are visible from Manhattan avenues and Kansan pastures to Yosemite’s cliffs and suburbia’s cul-de-sacs. An engrossing read, Liberty’s Grid offers a powerful look at the ideological conflict written on the landscape.

Explore the surprising connections between math and maps—and the myriad ways they’ve shaped our world and us.

Why are coastlines and borders so difficult to measure? How does a UPS driver deliver hundreds of packages in a single day? And where do elusive serial killers hide? The answers lie in the crucial connection between math and maps.

In Mapmatics, mathematician Paulina Rowińska leads us on a journey around the globe to discover how math and maps are deeply entwined, and always have been. From a sixteenth-century map, an indispensable navigation tool that exaggerates the size of northern countries, to public transport maps that both guide and confound passengers, to congressional maps that can empower or silence whole communities, maps and math have shaped not only our sense of space but our worldview. Rowińska shows that by understanding the math behind maps, we can recognize their biases. And we can appreciate the ingenious tools mathematicians are developing to resolve them.

Written with authority and compassion, wit and unforgettable storytelling, Mapmatics is math exposition at its best. By unpacking the math underlying the maps we depend on, this book illuminates how our world works, and, ultimately, how we can better look after it.

For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.

In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.

Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

These reports, at the forefront of relativity theory when they were written, in particular the geometrical aspects of spacetime theory, were the result of the Alfred Schild Memorial Lecture Series presented at the University of Texas at Austin beginning in 1977. Each article is a self-contained summary of an important area of contemporary gravitational physics, while the book as a whole provides an overview of a wide variety of the problems of general relativity and gravitation.