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"Circles Disturbed" brings together important thinkers in
mathematics, history, and philosophy to explore the relationship
between mathematics and narrative. The book's title recalls the
last words of the great Greek mathematician Archimedes before he
was slain by a Roman soldier--"Don't disturb my circles"--words
that seem to refer to two radically different concerns: that of the
practical person living in the concrete world of reality, and that
of the theoretician lost in a world of abstraction. Stories and
theorems are, in a sense, the natural languages of these two
worlds--stories representing the way we act and interact, and
theorems giving us pure thought, distilled from the hustle and
bustle of reality. Yet, though the voices of stories and theorems
seem totally different, they share profound connections and
similarities.
A book unlike any other, "Circles Disturbed" delves into topics
such as the way in which historical and biographical narratives
shape our understanding of mathematics and mathematicians, the
development of "myths of origins" in mathematics, the structure and
importance of mathematical dreams, the role of storytelling in the
formation of mathematical intuitions, the ways mathematics helps us
organize the way we think about narrative structure, and much
more.
In addition to the editors, the contributors are Amir Alexander,
David Corfield, Peter Galison, Timothy Gowers, Michael Harris,
David Herman, Federica La Nave, G.E.R. Lloyd, Uri Margolin, Colin
McLarty, Jan Christoph Meister, Arkady Plotnitsky, and Bernard
Teissier.
The intention of the authors is to examine the relationship between
piecewise linear structure and differential structure: a
relationship, they assert, that can be understood as a homotopy
obstruction theory, and, hence, can be studied by using the
traditional techniques of algebraic topology. Thus the book attacks
the problem of existence and classification (up to isotopy) of
differential structures compatible with a given combinatorial
structure on a manifold. The problem is completely "solved" in the
sense that it is reduced to standard problems of algebraic
topology. The first part of the book is purely geometrical; it
proves that every smoothing of the product of a manifold M and an
interval is derived from an essentially unique smoothing of M. In
the second part this result is used to translate the classification
of smoothings into the problem of putting a linear structure on the
tangent microbundle of M. This in turn is converted to the homotopy
problem of classifying maps from M into a certain space PL/O. The
set of equivalence classes of smoothings on M is given a natural
abelian group structure.
This work is a comprehensive treatment of recent developments in
the study of elliptic curves and their moduli spaces. The
arithmetic study of the moduli spaces began with Jacobi's
"Fundamenta Nova" in 1829, and the modern theory was erected by
Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade
mathematicians have made further substantial progress in the field.
This book gives a complete account of that progress, including not
only the work of the authors, but also that of Deligne and
Drinfeld.
Barry Mazur invites lovers of poetry to make a leap into mathematics. Through discussions of the role of the imagination and imagery in both poetry and mathematics, Mazur reviews the writings of the early mathematical explorers and reveals the early bafflement of these Renaissance thinkers faced with imaginary numbers. Then he shows us, step-by-step, how to begin imagining these strange mathematical objects ourselves.
Prime numbers are beautiful, mysterious, and beguiling mathematical
objects. The mathematician Bernhard Riemann made a celebrated
conjecture about primes in 1859, the so-called Riemann hypothesis,
which remains one of the most important unsolved problems in
mathematics. Through the deep insights of the authors, this book
introduces primes and explains the Riemann hypothesis. Students
with a minimal mathematical background and scholars alike will
enjoy this comprehensive discussion of primes. The first part of
the book will inspire the curiosity of a general reader with an
accessible explanation of the key ideas. The exposition of these
ideas is generously illuminated by computational graphics that
exhibit the key concepts and phenomena in enticing detail. Readers
with more mathematical experience will then go deeper into the
structure of primes and see how the Riemann hypothesis relates to
Fourier analysis using the vocabulary of spectra. Readers with a
strong mathematical background will be able to connect these ideas
to historical formulations of the Riemann hypothesis.
In these volumes, a reader will find all of John Tate's published
mathematical papers-spanning more than six decades-enriched by new
comments made by the author. Included also is a selection of his
letters. His letters give us a close view of how he works and of
his ideas in process of formation.
Prime numbers are beautiful, mysterious, and beguiling mathematical
objects. The mathematician Bernhard Riemann made a celebrated
conjecture about primes in 1859, the so-called Riemann hypothesis,
which remains one of the most important unsolved problems in
mathematics. Through the deep insights of the authors, this book
introduces primes and explains the Riemann hypothesis. Students
with a minimal mathematical background and scholars alike will
enjoy this comprehensive discussion of primes. The first part of
the book will inspire the curiosity of a general reader with an
accessible explanation of the key ideas. The exposition of these
ideas is generously illuminated by computational graphics that
exhibit the key concepts and phenomena in enticing detail. Readers
with more mathematical experience will then go deeper into the
structure of primes and see how the Riemann hypothesis relates to
Fourier analysis using the vocabulary of spectra. Readers with a
strong mathematical background will be able to connect these ideas
to historical formulations of the Riemann hypothesis.
Papers based on selected lectures given at the Current Development
Mathematics Conference, held in November 2011 at Harvard
University. Contents: The Laplacian on planar graphs and graphs on
surfaces (Richard Kenyon); Hyperbolicity and stable polynomials in
combinatorics and probability (Robin Pemantle); Introduction to KPZ
(Jeremy Quastel)
Papers based on selected lectures given at the Current Development
Mathematics Conference, held in November 2010 at Harvard
University.
Papers based on selected lectures given at the Current Development
Mathematics Conference, held in November 2009 at Harvard
University.
These are the proceedings of the joint seminar by M.I.T. and
Harvard on the current Developments in mathematics for the year
2001. Established in 1995, this seminar has been continued on the
third weekend of November every year. The organizing committee for
the seminar consisted of distinguished mathematicians from the
mathematics departments of both institutions: Barry Mazur, Wilfried
Schmid, and Shing-Tung Yau from Harvard, and David Jerison, A. J.
de Jong and George Lustig from M.I.T.. We trust that these
proceedings will be of interest to many mathematicians, and will
inspire future developments and research pursuits in mathematics.
This book offers an entertaining, yet thorough, explanation of the
concept of, yes, infinity. Accessible to non-mathematicians, this
book also cleverly connects mathematical reasoning to larger issues
in society.
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