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The quest to build a quantum computer is arguably one of the major
scientific and technological challenges of the twenty-first century,
and quantum information theory (QIT) provides the mathematical
framework for that quest. Over the last dozen or so years, it has
become clear that quantum information theory is closely linked to
geometric functional analysis (Banach space theory, operator spaces,
high-dimensional probability), a field also known as asymptotic
geometric analysis (AGA). In a nutshell, asymptotic geometric analysis
investigates quantitative properties of convex sets, or other
geometric structures, and their approximate symmetries as the
dimension becomes large. This makes it especially relevant to quantum
theory, where systems consisting of just a few particles naturally
lead to models whose dimension is in the thousands, or even in the
billions.
This book is aimed at multiple audiences connected through their
interest in the interface of QIT and AGA: at quantum information
researchers who want to learn AGA or apply its tools; at
mathematicians interested in learning QIT, or at least the part of QIT
that is relevant to functional analysis/convex geometry/random matrix
theory and related areas; and at beginning researchers in either
field. We have tried to make the book as user-friendly as possible,
with numerous tables, explicit estimates, and reasonable constants
when possible, so as to make it a useful reference even for
established mathematicians generally familiar with the
subject.
Graduate students and researchers interested in mathematical aspects of quantum information theory and quantum computing.
This is the second part of a series of papers called “HAG”, devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category \(C\), and prove that this notion satisfies the expected properties.
This volume contains the proceedings of the CATS4 Conference on Higher
Categorical Structures and their Interactions with Algebraic Geometry,
Algebraic Topology and Algebra, held from July 2–7, 2012, at CIRM in
Luminy, France.
Over the past several years, the CATS conference series has brought
together top level researchers from around the world interested in
relative and higher category theory and its applications to classical
mathematical domains.
Included in this volume is a collection of articles covering the
applications of categories and stacks to geometry, topology and
algebra. Techniques such as localization, model categories, simplicial
objects, sheaves of categories, mapping stacks, dg structures,
hereditary categories, and derived stacks, are applied to give new
insight on cluster algebra, Lagrangians, trace theories, loop spaces,
structured surfaces, stability, ind-coherent complexes and
1-affineness showing up in geometric Langlands, branching out to many
related topics along the way.
Graduate students and research mathematicians interested in category theory and its applications in algebraic geometry, algebraic topology, representation theory, symplectic geometry, and mathematical physics.
Significantly revised and expanded, this new Second Edition
provides readers at all levels—from beginning students to practicing
analysts—with the basic concepts and standard tools necessary to solve
problems of analysis, and how to apply these concepts to research in
a variety of areas.
Authors Elliott Lieb and Michael Loss take you quickly from basic
topics to methods that work successfully in mathematics and its
applications. While omitting many usual typical textbook topics,
Analysis
includes all necessary definitions, proofs, explanations, examples,
and exercises to bring the reader to an advanced level of understanding with
a minimum of fuss, and, at the same time, doing so in a rigorous and
pedagogical way. Many topics that are useful and important, but
usually left to advanced monographs, are presented in Analysis, and
these give the beginner a sense that the subject is alive and growing.
This new Second Edition incorporates numerous changes since the publication
of the original 1997 edition and includes:
Features:
Introductory-level graduate students in mathematics; research mathematicians, natural scientists, and engineers interested in learning some of the important tools of modern analysis.
This is an excellent textbook on analysis and it has several unique features: Proofs of heat kernel estimates, the Nash inequality and the logarithmic Sobolev inequality are topics that are seldom treated on the level of a textbook. Best constants in several inequalities, such as Young's inequality and the logarithmic Sobolev inequality, are also included. A thorough treatment of rearrangement inequalities and competing symmetries appears in book form for the first time. There is an extensive treatment of potential theory and its applications to quantum mechanics, which, again, is unique at this level. Uniform convexity of \(L^p\) space is treated very carefully. The presentation of this important subject is highly unusual for a textbook. All the proofs provide deep insights into the theorems. This book sets a new standard for a graduate textbook in analysis.
-- Shing-Tung Yau
Begins with a down-to-earth intro … aims at a wide range of essential applications … The book should work equally well in a one-, or in a two-semester course … great for students to have … This choice of book is also especially agreeable to grad students in physics who need to read up on the tools of analysis.
-- Palle Jorgensen
I find the selection of the material covered in the book very attractive and I recommend the book to anybody who wants to learn about classical as well as modern mathematical analysis.
-- European Mathematical Society Newsletter
The essentials of modern analysis … are presented in a rigorous and pedagogical way … readers … are guided to a level where they can read the current literature with understanding … treatment of the subject is as direct as possible.
-- Zentralblatt MATH
Lieb and Loss offer a practical presentation of real and functional analysis at the beginning graduate level … could be used as a two-semester introduction to graduate analysis … not all of the topics covered are typical. The authors introduce the subject with a thorough presentation … [an] informative exposition.
-- CHOICE
This is definitely a beautiful book … useful reference even for specialists since the authors present basic tools in a very rigorous way … they show clever methods how to calculate, equally useful for beginners as well as advanced specialists … well known exercises.
-- Mathematica Bohemica
Interesting textbook ... brings the reader quickly to a level where a wide range of topics can be appreciated ... well-written textbook ... can be read by anyone with a good knowledge of calculus ... useful for graduate students in mathematics and physics.
-- ZAMM–Journal of Applied Mathematics and Mechanics
I liked the book very much. The topics chosen were suited toward concepts that I wanted students to master.
-- Gary Sampson, Auburn University
In the area of analysis / real analysis / functional analysis there are a very large number of books at all levels, many of them very well known: the one under review is an unusual addition to the list. The book by Lieb and Loss assumes little on the part of the reader beyond a good college calculus course and, as such, begins with the basics of Lebesgue integral and yet is able to go deep into quite a few topics usually treated in advanced or more specialised texts. This unorthodox development makes it possible for a reader to reach, in the space of less than three hundred pages, completely rigorous mathematical treatment of several interesting physical problems. The authors have exercised remarkable discipline in their choice of topics to reach such depths quickly, yet they have not made it a linear development with the sole aim of showing these applications.
To sum up, this is an excellent book and the present inexpensive edition is recommended for the libraries of all interested in analysis.
-- Resonance: Journal of Science Edition
We live in a highly connected world with multiple self-interested
agents interacting and myriad opportunities for conflict and
cooperation. The goal of game theory is to understand these
opportunities.
This book presents a rigorous introduction to the mathematics of
game theory without losing sight of the joy of the subject. This is
done by focusing on theoretical highlights (e.g., at least six Nobel
Prize winning results are developed from scratch) and by presenting
exciting connections of game theory to other fields such as computer
science (algorithmic game theory), economics (auctions and matching
markets), social choice (voting theory), biology (signaling and
evolutionary stability), and learning theory. Both classical topics,
such as zero-sum games, and modern topics, such as sponsored search
auctions, are covered. Along the way, beautiful mathematical tools
used in game theory are introduced, including convexity, fixed-point
theorems, and probabilistic arguments.
The book is appropriate for a first course in game theory at either
the undergraduate or graduate level, whether in mathematics,
economics, computer science, or statistics. The
importance of game-theoretic thinking transcends the academic
setting—for every action we take, we must consider not only its
direct effects, but also how it influences the incentives of
others.
Undergraduate and graduate students and professors interested in game theory and decision making.
This is an attractive candidate as a text for a mathematically rigorous course in game theory at the upper undergraduate or elementary graduate level. It offers an appealing and versatile selection of topics (including some modern ones), clear and well-motivated exposition, a good selection of examples, and a nicely-chosen selection of exercises (some but not all of which have solutions in the back of the book). People who teach, or simply have an interest in game theory, will certainly find this book worth a look.
-- Mark Hunacek, MAA Reviews
Game theory's influence is felt in a wide range of disciplines, and the authors deliver masterfully on the challenge of presenting both the breadth and coherence of its underlying world-view. The book achieves a remarkable synthesis, introducing the reader to the blend of economic insight, mathematical elegance, scientific impact, and counter-intuitive punch that characterizes game theory as a field.
-- Jon Kleinberg, Cornell University, 2006 Nevanlinna Prize winner
A game theory textbook by people who love … games! It covers many classic as well as recent topics of game theory. Its rigorous treatment, interspersed with illuminating examples, makes it a challenging pleasure to read.
-- Sergiu Hart, The Hebrew University of Jerusalem
This beautifully written book introduces the reader to the rich and flourishing subject of game theory. From an exhaustive account of basics of the classical theory to a wide range of modern themes—including auctions, matching markets, sequential decision making—the book presents a great variety of topics in a rigorous yet entertaining manner. The authors guide the reader through the intricacies of game theory with an exquisite mathematical elegance. For anyone interested in the topic, experts and beginners alike, this book is a must.
-- Gábor Lugosi, Pompeu Fabra University, Barcelona, Spain
It is great to have a comprehensive game theory textbook including so many modern topics.
-- Éva Tardos, Cornell University
This book serves as a very good resource and teaching material for anyone who wants to discover the beauty of Induction and its applications, from novice mathematicians to Olympiad-driven students and professors teaching undergraduate courses. The authors explore 10 different areas of mathematics, including topics that are not usually discussed in an Olympiad-oriented book on the subject. Induction is one of the most important techniques used in competitions and its applications permeate almost every area of mathematics.
A publication of XYZ Press. Distributed in North America by the American Mathematical Society.
Mathematicians and mathletes of all ages will benefit from this book, which is focused on the power and elegance of mathematical induction as a method of proof.
It's an eminently useful, well-written, and fun book, with huge pedagogical appeal: lots and lots of examples and problems to do. I expect to use my copy a lot.
-- Michael Berg, MAA Reviews
What's so funny about math? Lots! Especially if you're mathematically
bent. In the world of Colin Adams, differential equations bring on tears
of laughter. Hollywood producers hire algebraic geometers to punch up a
script. In this world, math and humor are synonymous.
A collection of humorous math stories, this book gives a window into
mathematics and the culture of mathematicians. Appropriate for
mathematicians, math students, math teachers, lay people with an
interest in mathematics, and indeed everyone else. This book is a romp
through the wild world of mathematics.
General mathematical audience interested in light mathematical reading.
A collection of humorous math stories, this book gives a window into mathematics and the culture of mathematicians. Appropriate for mathematicians, math students, math teachers, lay people with an interest in mathematics, and indeed everyone else. This book is a romp through the wild world of mathematics.
-- Zentralblatt MATH
Colin Adams writes some pretty funny mathematical tales, covering everything from the job interview to theorem proving to mathematical couture...had me laughing out loud. None of the stories require any mathematical sophistication to enjoy... any student would be able to enjoy this book.
-- Gregory P. Dresden, MAA Online
...a funny book. The author plays everything for laughs, and there's a lot to laugh at. ...I consider [the author] one of the funniest mathematicians I have ever met. This book bears that out.
-- Cap Khoury, Not About Apples (Blog)
Finally, a collection of hilarious mathematical stories by Colin Adams! ...I'm very pleased to see that this book is available because Colin Adams is the funniest mathematician I know...
-- Mathematical Fiction (Blog)
This book would make a great gift for that special person in your life who likes to read funny stories about math ...my Funny-o-Meter was definitely pointing somewhere between "amazing" and "hilarious".
-- The Math Less Traveled (Blog)
... rush for a copy...the reading is murderously funny, but includes some subtleties to think [about].
-- Jesus M. Ruiz, The European Mathematical Society
This book is written as a textbook on algebraic topology. The first
part covers the material for two introductory courses about homotopy and
homology. The second part presents more advanced applications and
concepts (duality, characteristic classes, homotopy groups of spheres,
bordism). The author recommends starting an introductory course with homotopy
theory. For this purpose, classical results are presented with new elementary
proofs. Alternatively, one could start more traditionally with singular and
axiomatic homology. Additional chapters are devoted to the geometry of
manifolds, cell complexes and fibre bundles. A special feature is the rich
supply of nearly 500 exercises and problems. Several sections include topics
which have not appeared before in textbooks as well as simplified proofs for
some important results.
Prerequisites are standard point set topology (as recalled in the first
chapter), elementary algebraic notions (modules, tensor product), and some
terminology from category theory. The aim of the book is to introduce advanced
undergraduate and graduate (master's) students to basic tools, concepts and
results of algebraic topology. Sufficient background material from geometry and
algebra is included.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Advanced undergraduates and graduate students interested in algebraic topology.
This book, updated and enlarged for the second edition, is
written as a text for a course aimed at third or fourth year graduate
students in discrete mathematics, computer science, or communication
engineering. Only some familiarity with elementary linear algebra and
probability is assumed.
The book is also a suitable introduction to coding theory for
researchers from related fields or for professionals who want to
supplement their theoretical basis. The book gives the coding basics
for working on projects in any of the above areas, but material
specific to one of these fields has not been included. The chapters
cover the codes and decoding methods that are currently of most
interest in research, development, and application. They give a
relatively brief presentation of the essential results, emphasizing
the interrelations between different methods and proofs of all
important results. A sequence of problems at the end of each chapter
serves to review the results and give the student an appreciation of
the concepts. In addition, some problems and suggestions for projects
indicate direction for further work.
The presentation encourages the use of programming tools for
studying codes, implementing decoding methods, and simulating
performance. Specific examples of programming exercises are provided
on the book's homepage.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Graduate students and research mathematicians interested in discrete mathematics, computer science, or communication engineering.
The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures.