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The author studies self-similar sets and measures on
\(\mathbb{R}^{d}\). Assuming that the defining iterated
function system \(\Phi\) does not preserve a proper affine
subspace, he shows that one of the following holds: (1) the dimension
is equal to the trivial bound (the minimum of \(d\) and the
similarity dimension \(s\)); (2) for all large \(n\)
there are \(n\)-fold compositions of maps from \(\Phi\)
which are super-exponentially close in \(n\); (3) there is a
non-trivial linear subspace of \(\mathbb{R}^{d}\) that is
preserved by the linearization of \(\Phi\) and whose translates
typically meet the set or measure in full dimension. In particular,
when the linearization of \(\Phi\) acts irreducibly on
\(\mathbb{R}^{d}\), either the dimension is equal to
\(\min\{s,d\}\) or there are super-exponentially close
\(n\)-fold compositions. The author gives a number of
applications to algebraic systems, parametrized systems, and to some
classical examples.
The main ingredient in the proof is an inverse theorem for the entropy
growth of convolutions of measures on \(\mathbb{R}^{d}\), and the growth
of entropy for the convolution of a measure on the orthogonal group
with a measure on \(\mathbb{R}^{d}\). More generally, this part of
the paper applies to smooth actions of Lie groups on manifolds.
Topology Through Inquiry is a comprehensive introduction to
point-set, algebraic, and geometric topology, designed to support
inquiry-based learning (IBL) courses for upper-division undergraduate
or beginning graduate students. The book presents an enormous amount
of topology, allowing an instructor to choose which topics to
treat. The point-set material contains many interesting topics well
beyond the basic core, including continua and metrizability. Geometric
and algebraic topology topics include the classification of
2-manifolds, the fundamental group, covering spaces, and homology
(simplicial and singular). A unique feature of the introduction to
homology is to convey a clear geometric motivation by starting with
mod 2 coefficients.
The authors are acknowledged masters of IBL-style teaching. This
book gives students joy-filled, manageable challenges that
incrementally develop their knowledge and skills. The exposition
includes insightful framing of fruitful points of view as well as
advice on effective thinking and learning. The text presumes only a
modest level of mathematical maturity to begin, but students who work
their way through this text will grow from mathematics students into
mathematicians.
Michael Starbird is a University of Texas Distinguished Teaching
Professor of Mathematics. Among his works are two other co-authored
books in the Mathematical Association of America's (MAA) Textbook
series. Francis Su is the Benediktsson-Karwa Professor of Mathematics
at Harvey Mudd College and a past president of the MAA. Both authors
are award-winning teachers, including each having received the MAA's
Haimo Award for distinguished teaching. Starbird and Su are, jointly
and individually, on lifelong missions to make learning—of mathematics
and beyond—joyful, effective, and available to everyone. This book
invites topology students and teachers to join in the adventure.
Undergraduate and graduate students interested in topology and Inquiry Based Learning (IBL).
Recipient of the Mathematical Association of America's
Beckenbach Book Prize in 2007!
Where did math come from? Who thought up all those algebra symbols,
and why? What is the story behind \(\pi\)? … negative
numbers? … the metric system? … quadratic equations?
… sine and cosine? … logs? The 30 independent historical
sketches in Math through the Ages answer these questions and
many others in an informal, easygoing style that is accessible to
teachers, students, and anyone who is curious about the history of
mathematical ideas. Each sketch includes Questions and Projects to
help you learn more about its topic and to see how the main ideas fit
into the bigger picture of history.
The 30 short stories are preceded by a 58-page bird's-eye overview
of the entire panorama of mathematical history, a whirlwind tour of
the most important people, events, and trends that shaped the
mathematics we know today. “What to Read Next” and
reading suggestions after each sketch provide starting points for
readers who want to learn more.
This book is ideal for a broad spectrum of audiences, including
students in history of mathematics courses at the late high school or
early college level, pre-service and in-service teachers, and anyone
who just wants to know a little more about the origins of
mathematics.
An instructor's manual for this title is available to those
instructors who have adopted the textbook for classroom use. Please
send email to textbooks@ams.org
for more information.
This is a beautiful and important book, a pleasure to read, in which the history recounted fully illuminates the mathematical ideas, and the ideas themselves are superbly explained: a wonderful accomplishment.
-- Barry Mazur, Harvard University
“Math through the Ages” is a treasure, one of the best history of math books at its level ever written. Somehow, it manages to stay true to a surprisingly sophisticated story, while respecting the needs of its audience. Its overview of the subject captures most of what one needs to know, and the 30 sketches are small gems of exposition that stimulate further exploration.
-- Glen van Brummelen, Quest University, President (2012–14) of the Canadian Society for History and Philosophy of Mathematics
Derived algebraic geometry is a far-reaching generalization of
algebraic geometry. It has found numerous applications in various
parts of mathematics, most prominently in representation theory. This
two-volume monograph develops generalization of various topics in
algebraic geometry in the context of derived algebraic geometry.
Volume I presents the theory of ind-coherent sheaves, which are a
“renormalization” of quasi-coherent sheaves and provide a natural
setting for Grothendieck-Serre duality as well as geometric
incarnations of numerous categories of interest in representation
theory.
Volume II develops deformation theory, Lie theory and the theory of
algebroids in the context of derived algebraic geometry. To that end,
it introduces the notion of inf-scheme, which is an infinitesimal
deformation of a scheme and studies ind-coherent sheaves on
inf-schemes. As an application of the general theory, the six-functor
formalism for D-modules in derived geometry is obtained.
The books are carefully written...and they are not as difficult to read as one might expect from the content. This is mainly due to the many introductions scattered throughout the books, which explain the main ideas of each volume, part or chapter.
-- Adrian Langer, Mathematical Reviews
Quantum field theory has been a great success for physics, but it is
difficult for mathematicians to learn because it is mathematically
incomplete. Folland, who is a mathematician, has spent considerable
time digesting the physical theory and sorting out the mathematical
issues in it. Fortunately for mathematicians, Folland is a gifted
expositor.
The purpose of this book is to present the elements of quantum field
theory, with the goal of understanding the behavior of elementary
particles rather than building formal mathematical structures, in a form
that will be comprehensible to mathematicians. Rigorous definitions and
arguments are presented as far as they are available, but the text
proceeds on a more informal level when necessary, with due care in
identifying the difficulties.
The book begins with a review of classical physics and quantum
mechanics, then proceeds through the construction of free quantum fields
to the perturbation-theoretic development of interacting field theory
and renormalization theory, with emphasis on quantum electrodynamics.
The final two chapters present the functional integral approach and the
elements of gauge field theory, including the Salam–Weinberg model of
electromagnetic and weak interactions.
Graduate students and research mathematicans interested in mathematical physics, specifically, quantum field theory.
Folland's book is valuable for the mathematician who wants to understand how quantum field theory describes nature. ... [This book] is a great introduction to these issues. A mathematician who is serious about learning quantum field theory as a physical theory could do no better than to start with it. Physicists could also benefit from his careful and succinct survey.
-- SIAM Review
The style of the present monograph is clear and the author is honest about possible mathematical shortcomings of quantum field theory.
-- Mathematical Reviews
The goal of this book is to develop a theory of join and slices for strict \(\infty\)-categories. To any pair of strict \(\infty\)-categories, the authors associate a third one that they call their join. This operation is compatible with the usual join of categories up to truncation. The authors show that the join defines a monoidal category structure on the category of strict \(\infty\)-categories and that it respects connected inductive limits in each variable. In particular, the authors obtain the existence of some right adjoints; these adjoints define \(\infty\)-categorical slices, in a generalized sense. They state some conjectures about the functoriality of the join and the slices with respect to higher lax and oplax transformations and they prove some first results in this direction. These results are used in another paper to establish a Quillen Theorem A for strict \(\infty\)-categories. Finally, in an appendix, the authors revisit the Gray tensor product of strict \(\infty\)categories. One of the main tools used in this paper is Steiner's theory of augmented directed complexes.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Computational Complexity Theory is the study of how much of a given resource is required to perform the computations that interest us the most. Four decades of fruitful research have produced a rich and subtle theory of the relationship between different resource measures and problems. At the core of the theory are some of the most alluring open problems in mathematics.
This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity. The first week gives a general introduction to the field, including descriptions of the basic models, techniques, results and open problems. The second week focuses on lower bounds in concrete models. The final week looks at randomness in computation, with discussions of different notions of pseudorandomness, interactive proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). It is recommended for independent study by graduate students or researchers interested in computational complexity.
The volume is recommended for independent study and is suitable for graduate
students and researchers interested in computational
complexity.
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.
This book contains well written accounts of many branches of complexity theory. ... In particular, any CS grad student doing theory, or any math grad student, should be able to read this book. ... This book would be a good way to learn a lot of complexity theory quickly.
-- Mathematical Reviews
A vast literature has grown up around the value distribution theory of
meromorphic functions, synthesized by Rolf Nevanlinna in the 1920s and
singled out by Hermann Weyl as one of the greatest mathematical
achievements of this century. The multidimensional aspect, involving
the distribution of inverse images of analytic sets under holomorphic
mappings of complex manifolds, has not been fully treated in the
literature. This volume thus provides a valuable introduction to
multivariate value distribution theory and a survey of some of its
results, rich in relations to both algebraic and differential geometry
and surely one of the most important branches of the modern geometric
theory of functions of a complex variable.
Since the book begins with preparatory material from the contemporary
geometric theory of functions, only a familiarity with the elements of
multidimensional complex analysis is necessary background to understand
the topic. After proving the two main theorems of value distribution
theory, the author goes on to investigate further the theory of
holomorphic curves and to provide generalizations and applications of
the main theorems, focusing chiefly on the work of Soviet
mathematicians.
The main goals of this paper are:
(i) To develop an abstract differential calculus on metric measure
spaces by investigating the duality relations between differentials
and gradients of Sobolev functions. This will be achieved without
calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is
Radon and non-negative.
(ii) To employ these notions of calculus to provide, via
integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like \(\Delta
g=\mu\), where \(g\) is a function and \(\mu\) is a
measure.
(iii) To show that on spaces with Ricci curvature bounded from below and
dimension bounded from above, the Laplacian of the distance function is always a
measure and that this measure has the standard sharp comparison properties. This
result requires an additional assumption on the space, which reduces to strict
convexity of the norm in the case of smooth Finsler structures and is always
satisfied on spaces with linear Laplacian, a situation which is analyzed in
detail.