Quanta of Maths


Author: Alain Connes,Institut Henri Poincaré,Institut des hautes études scientifiques (Paris, France),Institut de mathématiques de Jussieu
Publisher: American Mathematical Soc.
ISBN: 0821852035
Category: Mathematics
Page: 675
View: 2282

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The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.

Proofs from THE BOOK


Author: Martin Aigner,Günter M. Ziegler
Publisher: Springer
ISBN: 3662572656
Category: Mathematics
Page: 326
View: 2278

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This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters. From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition" “... It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [...] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.” From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011

Real Quanta

Simplifying Quantum Physics for Einstein and Bohr
Author: Martijn van Calmthout
Publisher: Dundurn
ISBN: 1459740513
Category: Science
Page: 192
View: 2627

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Albert Einstein and Niels Bohr walk into the famous Hotel Métropole and sit down at the author’s table to discuss the state of quantum mechanics today. Particles that exist in two places at once, consequences that occur without a cause, objects that exist only if you look at them — quantum mechanics proves that all of this is possible, and not just in dark science labs. Look no further than your smartphone or tablet for technology made conceivable by quantum theory. From quantum computers to “teleporting” data, medicine to photosynthesis and the quantum compass in some migratory birds, Martijn van Calmthout plainly explains — to his readers and to an astounded Einstein and Bohr — how Quantum 2.0 is increasingly part of everyone’s daily life. Rather than being the exceptional domain, Van Calmthout shows how quantum mechanics is actually part of our tangible world, and may even be the very crux of our existence.

The Book of Why

The New Science of Cause and Effect
Author: Judea Pearl,Dana Mackenzie
Publisher: Basic Books
ISBN: 0465097618
Category: Computers
Page: 432
View: 686

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A Turing Award-winning computer scientist and statistician shows how understanding causality has revolutionized science and will revolutionize artificial intelligence "Correlation is not causation." This mantra, chanted by scientists for more than a century, has led to a virtual prohibition on causal talk. Today, that taboo is dead. The causal revolution, instigated by Judea Pearl and his colleagues, has cut through a century of confusion and established causality--the study of cause and effect--on a firm scientific basis. His work explains how we can know easy things, like whether it was rain or a sprinkler that made a sidewalk wet; and how to answer hard questions, like whether a drug cured an illness. Pearl's work enables us to know not just whether one thing causes another: it lets us explore the world that is and the worlds that could have been. It shows us the essence of human thought and key to artificial intelligence. Anyone who wants to understand either needs The Book of Why.

Creating Symmetry

The Artful Mathematics of Wallpaper Patterns
Author: Frank A. Farris
Publisher: Princeton University Press
ISBN: 1400865670
Category: Art
Page: 248
View: 8444

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This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks—a sort of potato-stamp method—Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics. Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own. Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.

The Algebraic and Geometric Theory of Quadratic Forms


Author: Richard S. Elman,Nikita Karpenko,Alexander Merkurjev
Publisher: American Mathematical Soc.
ISBN: 9780821873229
Category: Mathematics
Page: 435
View: 7400

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This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given. Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.

Beyond Weird

Why Everything You Thought You Knew about Quantum Physics is Different
Author: Philip Ball
Publisher: N.A
ISBN: 022655838X
Category: Science
Page: 384
View: 4626

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No one can say what quantam mechanics means (and this is a book about it) -- Quantum mechanics is not really about the quantum -- Quantum objects are neither wave nor particle (but sometimes they might as well be) -- Quantum particles aren't in two states at once (but sometimes they might as well be) -- What "happens" depends on what we find out about it -- There are many ways of interpreting quantum theory (and none of them quite make sense) -- Whatever the question, the answer is "yes" (unless it's "no") -- Not everything is knowable at once -- The properties of quantum objects don't have to be contained within the objects -- There is no "spooky action at a distance"--The everyday world is what quantum becomes at human scales -- Everything you experience is a (partial) copy of what causes it -- Schrödinger's cat has had kittens -- Quantum mechanics can be harnessed for technology -- Quantum computers don't necessarily perform "many calculations at once" -- There is no other "quantum" you -- Things could be even more "quantum" than they are (so why aren't they)? -- The fundamental laws of quantum mechanics might be simpler than we imagine -- Can we ever get to the bottom of it?

Quantum Questions

Mystical Writings of the World's Great Physicists
Author: Ken Wilber
Publisher: Shambhala Publications
ISBN: 0834822830
Category: Philosophy
Page: 224
View: 7318

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Here is a collection of writings that bridges the gap between science and religion. Quantum Questions collects the mystical writings of each of the major physicists involved in the discovery of quantum physics and relativity, including Albert Einstein, Werner Heisenberg, and Max Planck. The selections are written in nontechnical language and will be of interest to scientists and nonscientists alike.

Mathematics Everywhere


Author: Martin Aigner,Ehrhard Behrends
Publisher: American Mathematical Soc.
ISBN: 0821843494
Category: Mathematics
Page: 330
View: 783

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Mathematics is all around us. Often we do not realize it, though. Mathematics Everywhere is a collection of presentations on the role of mathematics in everyday life, through science, technology, and culture. The common theme is the unique position of mathematics as the art of pure thought and at the same time as a universally applicable science. The authors are renowned mathematicians; their presentations cover a wide range of topics. From compact discs to the stock exchange, from computer tomography to traffic routing, from electronic money to climate change, they make the ``math inside'' understandable and enjoyable. An additional attractive feature is the leisurely treatment of some hot topics that have gained prominence in recent years, such as Fermat's Theorem, Kepler's packing problem, and the solution of the Poincare Conjecture. Or maybe you have heard about the Nash equilibrium (of ``A Beautiful Mind'' fame), or the strange future of quantum computers, and want to know what it is all about? Well, open the book and take an up-to-date trip into the fascinating world of the mathematics all around us.

The Scientist as Rebel


Author: Freeman Dyson
Publisher: New York Review of Books
ISBN: 1590178815
Category: Science
Page: 400
View: 5488

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From Galileo to today’s amateur astronomers, scientists have been rebels, writes Freeman Dyson. Like artists and poets, they are free spirits who resist the restrictions their cultures impose on them. In their pursuit of nature’s truths, they are guided as much by imagination as by reason, and their greatest theories have the uniqueness and beauty of great works of art.Dyson argues that the best way to understand science is by understanding those who practice it. He tells stories of scientists at work, ranging from Isaac Newton’s absorption in physics, alchemy, theology, and politics, to Ernest Rutherford’s discovery of the structure of the atom, to Albert Einstein’s stubborn hostility to the idea of black holes. His descriptions of brilliant physicists like Edward Teller and Richard Feynman are enlivened by his own reminiscences of them. He looks with a skeptical eye at fashionable scientific fads and fantasies, and speculates on the future of climate prediction, genetic engineering, the colonization of space, and the possibility that paranormal phenomena may exist yet not be scientifically verifiable. Dyson also looks beyond particular scientific questions to reflect on broader philosophical issues, such as the limits of reductionism, the morality of strategic bombing and nuclear weapons, the preservation of the environment, and the relationship between science and religion. These essays, by a distinguished physicist who is also a prolific writer, offer informed insights into the history of science and fresh perspectives on contentious current debates about science, ethics, and faith.

Math with Bad Drawings

Illuminating the Ideas That Shape Our Reality
Author: Ben Orlin
Publisher: Black Dog & Leventhal
ISBN: 0316509027
Category: Mathematics
Page: 376
View: 6404

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A hilarious reeducation in mathematics-full of joy, jokes, and stick figures-that sheds light on the countless practical and wonderful ways that math structures and shapes our world. In Math With Bad Drawings, Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician. Truth and knowledge come in multiple forms: colorful drawings, encouraging jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Orlin shows us how to think like a mathematician by teaching us a brand-new game of tic-tac-toe, how to understand an economic crises by rolling a pair of dice, and the mathematical headache that ensues when attempting to build a spherical Death Star. Every discussion in the book is illustrated with Orlin's trademark "bad drawings," which convey his message and insights with perfect pitch and clarity. With 24 chapters covering topics from the electoral college to human genetics to the reasons not to trust statistics, Math with Bad Drawings is a life-changing book for the math-estranged and math-enamored alike.

The Calculus of Friendship

What a Teacher and a Student Learned about Life while Corresponding about Math
Author: Steven Strogatz
Publisher: Princeton University Press
ISBN: 9781400830886
Category: Mathematics
Page: 184
View: 3513

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The Calculus of Friendship is the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough. Like calculus itself, The Calculus of Friendship is an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics, The Calculus of Friendship is warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters. For anyone whose life has been changed by a mentor, The Calculus of Friendship will be an unforgettable journey.

Arithmetic Noncommutative Geometry


Author: Matilde Marcolli
Publisher: American Mathematical Soc.
ISBN: 0821838334
Category: Mathematics
Page: 136
View: 8985

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Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable ``fibers at infinity''), by adding boundaries that are invisible to algebraic geometry, such as degenerations of elliptic curves to noncommutative tori. The text of the book is organized around series of invited lectures delivered by the author at various universities, and the results presented are based on work of the author in collaboration with Alain Connes, Katia Consani, Yuri Manin, and Niranjan Ramachandran.

Not Even Wrong

The Failure of String Theory and the Search for Unity in Physical Law for Unity in Physical Law
Author: Peter Woit
Publisher: Basic Books
ISBN: 046500363X
Category: Science
Page: 320
View: 3834

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When does physics depart the realm of testable hypothesis and come to resemble theology? Peter Woit argues that string theory isn't just going in the wrong direction, it's not even science. Not Even Wrong shows that what many physicists call superstring “theory” is not a theory at all. It makes no predictions, not even wrong ones, and this very lack of falsifiability is what has allowed the subject to survive and flourish. Peter Woit explains why the mathematical conditions for progress in physics are entirely absent from superstring theory today, offering the other side of the story.

Lost in Math

How Beauty Leads Physics Astray
Author: Sabine Hossenfelder
Publisher: Basic Books
ISBN: 0465094260
Category: Science
Page: 304
View: 384

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A contrarian argues that modern physicists' obsession with beauty has given us wonderful math but bad science Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.

Vortices in the Magnetic Ginzburg-Landau Model


Author: Etienne Sandier,Sylvia Serfaty
Publisher: Springer Science & Business Media
ISBN: 9780817645502
Category: Mathematics
Page: 322
View: 6119

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This book presents the mathematical study of vortices of the two-dimensional Ginzburg-Landau model, an important phenomenological model used to describe superconductivity. The vortices, identified as quantized amounts of vorticity of the superconducting current localized near points, are the objects of many observational and experimental studies, both past and present. The Ginzburg-Landau functionals considered include both the model cases with and without a magnetic field. The book acts a guide to the various branches of Ginzburg-Landau studies, provides context for the study of vortices, and presents a list of open problems in the field.

Wonders Beyond Numbers

A Brief History of All Things Mathematical
Author: Johnny Ball
Publisher: Bloomsbury Publishing
ISBN: 1472939964
Category: Mathematics
Page: 336
View: 8492

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In this book, Johnny Ball tells one of the most important stories in world history – the story of mathematics. By introducing us to the major characters and leading us through many historical twists and turns, Johnny slowly unravels the tale of how humanity built up a knowledge and understanding of shapes, numbers and patterns from ancient times, a story that leads directly to the technological wonderland we live in today. As Galileo said, 'Everything in the universe is written in the language of mathematics', and Wonders Beyond Numbers is your guide to this language. Mathematics is only one part of this rich and varied tale; we meet many fascinating personalities along the way, such as a mathematician who everyone has heard of but who may not have existed; a Greek philosopher who made so many mistakes that many wanted his books destroyed; a mathematical artist who built the largest masonry dome on earth, which builders had previously declared impossible; a world-renowned painter who discovered mathematics and decided he could no longer stand the sight of a brush; and a philosopher who lost his head, but only after he had died. Enriched with tales of colourful personalities and remarkable discoveries, there is also plenty of mathematics for keen readers to get stuck into. Written in Johnny Ball's characteristically light-hearted and engaging style, this book is packed with historical insight and mathematical marvels; join Johnny and uncover the wonders found beyond the numbers.

The Joy of X

A Guided Tour of Math, from One to Infinity
Author: Steven Henry Strogatz
Publisher: Houghton Mifflin Harcourt
ISBN: 0547517653
Category: MATHEMATICS
Page: 316
View: 4027

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A comprehensive tour of leading mathematical ideas by an award-winning professor and columnist for the New York Times Opinionator series demonstrates how math intersects with philosophy, science and other aspects of everyday life. By the author of The Calculus of Friendship. 50,000 first printing.

Shadows of the Mind

A Search for the Missing Science of Consciousness
Author: Roger Penrose
Publisher: Oxford University Press, USA
ISBN: 9780195106466
Category: Computers
Page: 457
View: 2189

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Presenting a look at the human mind's capacity while criticizing artificial intelligence, the author makes suggestions about classical and quantum physics and the role of microtubules

Cyclic Cohomology and Noncommutative Geometry


Author: Joachim J. R. Cuntz,Masoud Khalkhali
Publisher: American Mathematical Soc.
ISBN: 9780821871249
Category: Mathematics
Page: 189
View: 949

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Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras - such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory - on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics. This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications.