The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151)

Princeton University Press
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This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory.

The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary.

Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

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About the author

Michael Harris is Professor of Mathematics at Université Paris 7--Denis Diderot. His research focuses on automorphic forms and representation theory, with a view to applications in number theory and arithmetical algebraic geometry. Richard Taylor is Professor of Mathematics at Harvard University. His main research interest is number theory, particularly the theory of modular forms, automorphic forms, and Galois representations.
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Princeton University Press
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Published on
Oct 15, 2001
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Mathematics / Number Theory
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Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.

Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.

The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe.

To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Catch-22 with radiation.

Area 51 meets Dr. Strangelove.

Except it really happened.

The Atomic Times is the absolutely insane, incredibly f*cked-up, but true, eyewitness story of what happened in 1956 on a tiny island in the South Pacific when over 1600 young soldiers (including me) were turned into atomic guinea pigs by the US Army.

We were sent there to “observe” this nuclear test series, called Operation Redwing. Wearing only T-shirts and shorts and without any other protective gear while Army brass and nuclear scientists wore Hazmat suits, we were exposed to radiation and fallout.

Operation Redwing, the biggest and baddest of America's atmospheric nuclear weapons test series, mixed saber rattling with mad science, while overlooking the cataclysmic human, geopolitical and ecological effects. But mostly, it just messed with guys' heads. 

Major Maxwell, who put Safety First, Second and Third. Except when he didn't. 

Berko, the wise-cracking Brooklyn Dodgers fan forced to cope with the H-bomb and his mother's cookies.

Tony, who thought military spit and polish plus uncompromising willpower made him an exception.

Carl Duncan, who clung to his girlfriend's photos and a dangerous secret.

Major Vanish, who did just that.

In THE ATOMIC TIMES, Michael Harris welcomes readers into the U.S. Army's nuclear family where the F-words were Fallout and Fireball. In a distinctive narrative voice, Harris describes his H-bomb year with unforgettable imagery and insight into the ways isolation and isotopes change men for better—and for worse. 

A New York Times bestseller and a Pulitzer Prize nominee, THE ATOMIC TIMES was originally published in hardcover by Random House.

"A gripping memoir leavened by humor, loyalty and pride of accomplishment. A tribute to the resilience, courage and patriotism of the American soldier." —Henry Kissinger

"Harris' frank and disturbing descriptions of the criminally irresponsible proceedings on Eniwetok, and the physical and mental pain he and others endured, constitute shocking additions to atomic history. Amazingly enough, given his ordeal, Harris remains healthy." --Booklist

"An entertaining read in the bloodline of Catch-22, Harris achieves the oddest of victories: a funny, optimistic story about the H-bomb. Harris uses a chatty, dead-pan voice that highlights the horrifying absurdity of life on the island: the use of Geiger counters to monitor scrambled eggs' radiation level, three-eyed fish swimming in the lagoon, corroded, permanently open windows that fail to keep out the radioactive fall-out and men whose toenails glow in the dark." --Publisher's Weekly

From the author:

Three-eyed fish swimming in the lagoon. Men whose toenails glow in the dark. Operation Redwing where the F words were Fallout and Fireball. In 1956, I was an army draftee sent to the Marshall Islands to watch 17 H-bomb tests. An "observer," the Army called it. In plain English: a human guinea pig.

I knew at the time that the experience could make a fascinating book, and I wrote a novel based on it while I was still there. The problem was that Eniwetok was a security post. There were signs everywhere impressing on us that the work going on (I mopped floors, typed, filed requisitions and wrote movie reviews for the island newspaper “All the news that fits we print”) was Top Secret. “What you do here, what you see here, what you hear here, when you leave here leave it here.”

I was afraid they would confiscate the manuscript if they found it but a buddy who left Eniwetok before I did concealed the pages in his luggage. When he got back to the States, he mailed those pages to my father so I had what turned out to be a very rough draft.

What was wrong with the book? Let me count the ways. I didn’t know how to write action, plot and character. I did know how to leave out everything interesting that was happening around me. Back in the States after my discharge, I thought about writing Version #2 but for ten years, I had nightmares about the H-bomb almost every night. I survived the radiation (unlike some of my friends), but the memories were also a formidable foe. I tried to forget and more or less succeeded.

My perspective gradually changed over the years and I began to remember what I had tried to forget: 

We were told we had to wear high density goggles during the tests to avoid losing our sight but the shipment of goggles never arrived—the requisition was cancelled to make room for new furniture for the colonel's house.

We were told we had to stand with our backs to the blast—again to prevent blindness. But the first H-bomb ever dropped from a plane missed its target, and the detonation took place in front of us and our unprotected eyes.

Servicemen were sent to Ground Zero wearing only shorts and sneakers and worked side by side with scientists dressed in RadSafe suits. The exposed military men developed severe radiation burns and many died.

The big breakthrough came when enough years had passed and I had overcome the anger and the self-pity resulting from the knowledge that I and the men who served with me had been used as guinea pigs in a recklessly dangerous and potentially deadly experiment. At last I had the perspective to understand my nuclear year in its many dimensions and capture the tragedy and the black humor that came along with 17 H-bomb explosions. In addition, certain significant external realities had changed.

Top Secret documents about Operation Redwing had been declassified. I learned new details about the test known as Tewa: the fallout lasted for three days and the radiation levels exceeded 3.9 Roentgens, the MPE (Maximum Permissible Exposure). Three ships were rushed to Eniwetok to evacuate personnel but were ordered back after the military raised the MPE to 7. That, they reasoned, ensured everyone's safety. 

I made contact with other atomic veterans who told me about their own experiences and in some cases sent me copies of letters written to their families during the tests. As we talked, we also laughed: about officers who claimed Eniwetok was a one year paid vacation; about the officer who guarded the political purity of the daily island newspaper by deleting "pinko propaganda," including a speech by President Eisenhower.

By now, Ruth knew the material almost as well as I did and provided crucial perspective and detailed editing expertise.

At last, I was able to pull all the strands together. After 50 years, I was able write the book I had wanted to in the beginning.

Having struggled to write a memoir for so long and having been asked for advice by others contemplating writing a memoir, I can pass along a bit of what I learned along the way.

Make sure you have enough distance from the experience to have perspective on what happened. Exposure to radiation and the resulting reactions—anger, terror, incredulity—produce powerful emotions that take time to process.

Figure out how to use (or keep away) from your own intense feelings. In the case of the H-Bomb tests, anger and self-pity were emotions to stay away from. So was the hope of somehow getting “revenge.”

Sometimes the unexpected works. For me, finding humor in a tragic situation— the abject military incompetence in planning and executing the H-Bomb tests—freed my memory and allowed me to write about horrific experiences.

Figure out (most likely by trial and error) how much or how little of yourself you want to reveal.

Keywords: memoir, veterans, H-bomb, US Army, black humor, dark humor, military memoir, nuclear bombs, radiation, fission, fusion, fallout, danger, suspense, atomic bombs, hydrogen bombs, H-bomb, South Pacific, Eniwetok, Marshall Islands

An awesome, globe-spanning, and New York Times bestselling journey through the beauty and power of mathematics
What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren't even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.
In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we've never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.
Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man's journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century's leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat's last theorem, that had seemed intractable before.
At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.
Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program.

The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported.

The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.

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