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.

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.

“I love maps. I love math. And gosh, do I love this book, which so beautifully and clearly sounds the depths of both.” —Ben Orlin, author of Math with Bad Drawings

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 maps and math.

In Mapmatics, mathematician Paulina Rowińska leads us on a riveting journey around the globe to discover how maps and math 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, she reveals how maps and math have shaped not only our sense of space but our worldview. In her hands, we learn how to read maps like a mathematician—to extract richer information and, just as importantly, to question our conclusions by asking what we don’t see.

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

Paul Lockhart’s Mathematician’s Lament outlined how we introduce math to students in the wrong way. Measurement explains how math should be done. With plain English and pictures, Lockhart makes complex ideas about shape and motion intuitive and graspable, and offers a solution to math phobia by introducing us to math 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.

On the creation of the world, and the destruction of Atlantis.

Plato, the great philosopher of Athens, was born in 427 BC. In early manhood an admirer of Socrates, he later founded the famous school of philosophy in the grove Academus. Much else recorded of his life is uncertain; that he left Athens for a time after Socrates’ execution is probable; that later he went to Cyrene, Egypt, and Sicily is possible; that he was wealthy is likely; that he was critical of “advanced” democracy is obvious. He lived to be 80 years old. Linguistic tests including those of computer science still try to establish the order of his extant philosophical dialogues, written in splendid prose and revealing Socrates’ mind fused with Plato’s thought.

In Laches, Charmides, and Lysis, Socrates and others discuss separate ethical conceptions. Protagoras, Ion, and Meno discuss whether righteousness can be taught. In Gorgias, Socrates is estranged from his city’s thought, and his fate is impending. The Apology (not a dialogue), Crito, Euthyphro, and the unforgettable Phaedo relate the trial and death of Socrates and propound the immortality of the soul. In the famous Symposium and Phaedrus, written when Socrates was still alive, we find the origin and meaning of love. Cratylus discusses the nature of language. The great masterpiece in ten books, the Republic, concerns righteousness (and involves education, equality of the sexes, the structure of society, and abolition of slavery). Of the six so-called dialectical dialogues Euthydemus deals with philosophy; metaphysical Parmenides is about general concepts and absolute being; Theaetetus reasons about the theory of knowledge. Of its sequels, Sophist deals with not-being; Politicus with good and bad statesmanship and governments; Philebus with what is good. The Timaeus seeks the origin of the visible universe out of abstract geometrical elements. The unfinished Critias treats of lost Atlantis. Unfinished also is Plato’s last work, Laws, a critical discussion of principles of law which Plato thought the Greeks might accept.

The Loeb Classical Library edition of Plato is in twelve volumes.