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The field of random matrix theory has seen an explosion of activity
in recent years, with connections to many areas of mathematics and
physics. However, this makes the current state of the field almost
too large to survey in a single book. In this graduate text, we
focus on one specific sector of the field, namely the spectral
distribution of random Wigner matrix ensembles (such as the
Gaussian Unitary Ensemble), as well as iid matrix ensembles. The
text is largely self-contained and starts with a review of relevant
aspects of probability theory and linear algebra. With over 200
exercises, the book is suitable as an introductory text for
beginning graduate students seeking to enter the field.
This is part one of a two-volume introduction to real analysis and
is intended for honours undergraduates who have already been
exposed to calculus. The emphasis is on rigour and on foundations.
The material starts at the very beginning--the construction of the
number systems and set theory--then goes on to the basics of
analysis (limits, series, continuity, differentiation, Riemann
integration), through to power series, several variable calculus
and Fourier analysis, and finally to the Lebesgue integral. These
are almost entirely set in the concrete setting of the real line
and Euclidean spaces, although there is some material on abstract
metric and topological spaces. There are also appendices on
mathematical logic and the decimal system. The entire text
(omitting some less central topics) can be taught in two quarters
of twenty-five to thirty lectures each. The course material is
deeply intertwined with the exercises, as it is intended that the
student actively learn the material (and practice thinking and
writing rigorously) by proving several of the key results in the
theory. The fourth edition incorporates a large number of
additional corrections reported since the release of the third
edition, as well as some additional new exercises.
MATRIX is Australia's international and residential mathematical
research institute. It facilitates new collaborations and
mathematical advances through intensive residential research
programs, each 1-4 weeks in duration. This book is a scientific
record of the eight programs held at MATRIX in 2018: -
Non-Equilibrium Systems and Special Functions - Algebraic Geometry,
Approximation and Optimisation - On the Frontiers of High
Dimensional Computation - Month of Mathematical Biology - Dynamics,
Foliations, and Geometry In Dimension 3 - Recent Trends on
Nonlinear PDEs of Elliptic and Parabolic Type - Functional Data
Analysis and Beyond - Geometric and Categorical Representation
Theory The articles are grouped into peer-reviewed contributions
and other contributions. The peer-reviewed articles present
original results or reviews on a topic related to the MATRIX
program; the remaining contributions are predominantly lecture
notes or short articles based on talks or activities at MATRIX.
MATRIX is Australia's international and residential mathematical
research institute. It facilitates new collaborations and
mathematical advances through intensive residential research
programs, each 1-4 weeks in duration. This book is a scientific
record of the eight programs held at MATRIX in 2018: -
Non-Equilibrium Systems and Special Functions - Algebraic Geometry,
Approximation and Optimisation - On the Frontiers of High
Dimensional Computation - Month of Mathematical Biology - Dynamics,
Foliations, and Geometry In Dimension 3 - Recent Trends on
Nonlinear PDEs of Elliptic and Parabolic Type - Functional Data
Analysis and Beyond - Geometric and Categorical Representation
Theory The articles are grouped into peer-reviewed contributions
and other contributions. The peer-reviewed articles present
original results or reviews on a topic related to the MATRIX
program; the remaining contributions are predominantly lecture
notes or short articles based on talks or activities at MATRIX.
MATRIX is Australia's international and residential mathematical
research institute. It facilitates new collaborations and
mathematical advances through intensive residential research
programs, each 1-4 weeks in duration. This book is a scientific
record of the eight programs held at MATRIX in its second year,
2017: - Hypergeometric Motives and Calabi-Yau Differential
Equations - Computational Inverse Problems - Integrability in
Low-Dimensional Quantum Systems - Elliptic Partial Differential
Equations of Second Order: Celebrating 40 Years of Gilbarg and
Trudinger's Book - Combinatorics, Statistical Mechanics, and
Conformal Field Theory - Mathematics of Risk - Tutte Centenary
Retreat - Geometric R-Matrices: from Geometry to Probability The
articles are grouped into peer-reviewed contributions and other
contributions. The peer-reviewed articles present original results
or reviews on a topic related to the MATRIX program; the remaining
contributions are predominantly lecture notes or short articles
based on talks or activities at MATRIX.
This is part two of a two-volume introduction to real analysis and
is intended for honours undergraduates who have already been
exposed to calculus. The emphasis is on rigour and on foundations.
The material starts at the very beginning--the construction of the
number systems and set theory--then goes on to the basics of
analysis (limits, series, continuity, differentiation, Riemann
integration), through to power series, several variable calculus
and Fourier analysis, and finally to the Lebesgue integral. These
are almost entirely set in the concrete setting of the real line
and Euclidean spaces, although there is some material on abstract
metric and topological spaces. There are also appendices on
mathematical logic and the decimal system. The entire text
(omitting some less central topics) can be taught in two quarters
of twenty-five to thirty lectures each. The course material is
deeply intertwined with the exercises, as it is intended that the
student actively learn the material (and practice thinking and
writing rigorously) by proving several of the key results in the
theory. The fourth edition incorporates a large number of
additional corrections reported since the release of the third
edition, as well as some additional new exercises.
MATRIX is Australia's international, residential mathematical
research institute. It facilitates new collaborations and
mathematical advances through intensive residential research
programs, each lasting 1-4 weeks. This book is a scientific record
of the five programs held at MATRIX in its first year, 2016: -
Higher Structures in Geometry and Physics - Winter of
Disconnectedness - Approximation and Optimisation - Refining
C*-Algebraic Invariants for Dynamics using KK-theory - Interactions
between Topological Recursion, Modularity, Quantum Invariants and
Low- dimensional Topology The MATRIX Scientific Committee selected
these programs based on their scientific excellence and the
participation rate of high-profile international participants. Each
program included ample unstructured time to encourage collaborative
research; some of the longer programs also included an embedded
conference or lecture series. The articles are grouped into
peer-reviewed contributions and other contributions. The
peer-reviewed articles present original results or reviews on
selected topics related to the MATRIX program; the remaining
contributions are predominantly lecture notes based on talks or
activities at MATRIX.
This is a graduate text introducing the fundamentals of measure
theory and integration theory, which is the foundation of modern
real analysis. The text focuses first on the concrete setting of
Lebesgue measure and the Lebesgue integral (which in turn is
motivated by the more classical concepts of Jordan measure and the
Riemann integral), before moving on to abstract measure and
integration theory, including the standard convergence theorems,
Fubini's theorem, and the Caratheodory extension theorem. Classical
differentiation theorems, such as the Lebesgue and Rademacher
differentiation theorems, are also covered, as are connections with
probability theory. The material is intended to cover a quarter or
semester's worth of material for a first graduate course in real
analysis. There is an emphasis in the text on tying together the
abstract and the concrete sides of the subject, using the latter to
illustrate and motivate the former. The central role of key
principles (such as Littlewood's three principles) as providing
guiding intuition to the subject is also emphasized. There are a
large number of exercises throughout that develop key aspects of
the theory, and are thus an integral component of the text. As a
supplementary section, a discussion of general problem-solving
strategies in analysis is also given. The last three sections
discuss optional topics related to the main matter of the book.
Authored by a leading name in mathematics, this engaging and
clearly presented text leads the reader through the various tactics
involved in solving mathematical problems at the Mathematical
Olympiad level. Covering number theory, algebra, analysis,
Euclidean geometry, and analytic geometry, Solving Mathematical
Problems includes numerous exercises and model solutions
throughout. Assuming only a basic level of mathematics, the text is
ideal for students of 14 years and above in pure mathematics.
There are many bits and pieces of folklore in mathematics that are
passed down from advisor to student, or from collaborator to
collaborator, but which are too fuzzy and nonrigorous to be
discussed in the formal literature. Traditionally, it was a matter
of luck and location as to who learned such "folklore mathematics".
But today, such bits and pieces can be communicated effectively and
efficiently via the semiformal medium of research blogging. This
book grew from such a blog. In 2007 Terry Tao began a mathematical
blog to cover a variety of topics, ranging from his own research
and other recent developments in mathematics, to lecture notes for
his classes, to nontechnical puzzles and expository articles. The
first two years of the blog have already been published by the
American Mathematical Society. The posts from the third year are
being published in two volumes. This second volume contains a broad
selection of mathematical expositions and self-contained technical
notes in many areas of mathematics, such as logic, mathematical
physics, combinatorics, number theory, statistics, theoretical
computer science, and group theory. Tao has an extraordinary
ability to explain deep results to his audience, which has made his
blog quite popular. Some examples of this facility in the present
book are the tale of two students and a multiple-choice exam being
used to explain the $P = NP$ conjecture and a discussion of "no
self-defeating object" arguments that starts from a schoolyard
number game and ends with results in logic, game theory, and
theoretical physics. The first volume consists of a second course
in real analysis, together with related material from the blog, and
it can be read independently.
There are many bits and pieces of folklore in mathematics that are
passed down from advisor to student, or from collaborator to
collaborator, but which are too fuzzy and non-rigorous to be
discussed in the formal literature. Traditionally, it was a matter
of luck and location as to who learned such folklore mathematics.
But today, such bits and pieces can be communicated effectively and
efficiently via the semiformal medium of research blogging. This
book grew from such a blog.In 2007, Terry Tao began a mathematical
blog, as an outgrowth of his own website at UCLA. This book is
based on a selection of articles from the first year of that blog.
These articles discuss a wide range of mathematics and its
applications, ranging from expository articles on quantum
mechanics, Einstein's equation E = mc2, or compressed sensing, to
open problems in analysis, combinatorics, geometry, number theory,
and algebra, to lecture series on random matrices, Fourier
analysis, or the dichotomy between structure and randomness that is
present in many subfields of mathematics, to more philosophical
discussions on such topics as the interplay between finitary and
infinitary in analysis.Some selected commentary from readers of the
blog has also been included at the end of each article. While the
articles vary widely in subject matter and level, they should be
broadly accessible to readers with a general graduate mathematics
background; the focus in many articles is on the 'big picture' and
on informal discussion, with technical details largely being left
to the referenced literature.
Among nonlinear PDEs, dispersive and wave equations form an
important class of equations. These include the nonlinear
Schrodinger equation, the nonlinear wave equation, the Korteweg de
Vries equation, and the wave maps equation. This book is an
introduction to the methods and results used in the modern analysis
(both locally and globally in time) of the Cauchy problem for such
equations. Starting only with a basic knowledge of graduate real
analysis and Fourier analysis, the text first presents basic
nonlinear tools such as the bootstrap method and perturbation
theory in the simpler context of nonlinear ODE, then introduces the
harmonic analysis and geometric tools used to control linear
dispersive PDE.These methods are then combined to study four model
nonlinear dispersive equations. Through extensive exercises,
diagrams, and informal discussion, the book gives a rigorous
theoretical treatment of the material, the real-world intuition and
heuristics that underlie the subject, as well as mentioning
connections with other areas of PDE, harmonic analysis, and
dynamical systems. As the subject is vast, the book does not
attempt to give a comprehensive survey of the field, but instead
concentrates on a representative sample of results for a selected
set of equations, ranging from the fundamental local and global
existence theorems to very recent results, particularly focusing on
the recent progress in understanding the evolution of
energy-critical dispersive equations from large data. The book is
suitable for a graduate course on nonlinear PDE.
Fourier analysis is a subject that was born in physics but grew up
in mathematics. Now it is part of the standard repertoire for
mathematicians, physicists and engineers. This diversity of
interest is often overlooked, but in this much-loved book, Tom
Koerner provides a shop window for some of the ideas, techniques
and elegant results of Fourier analysis, and for their
applications. These range from number theory, numerical analysis,
control theory and statistics, to earth science, astronomy and
electrical engineering. The prerequisites are few (a reader with
knowledge of second- or third-year undergraduate mathematics should
have no difficulty following the text), and the style is lively and
entertaining. This edition of Koerner's 1989 text includes a
foreword written by Professor Terence Tao introducing it to a new
generation of fans.
Expander graphs are an important tool in theoretical computer
science, geometric group theory, probability, and number theory.
Furthermore, the techniques used to rigorously establish the
expansion property of a graph draw from such diverse areas of
mathematics as representation theory, algebraic geometry, and
arithmetic combinatorics. This text focuses on the latter topic in
the important case of Cayley graphs on finite groups of Lie type,
developing tools such as Kazhdan's property (T), quasirandomness,
product estimates, escape from subvarieties, and the
Balog-Szemeredi-Gowers lemma. Applications to the affine sieve of
Bourgain, Gamburd, and Sarnak are also given. The material is
largely self-contained, with additional sections on the general
theory of expanders, spectral theory, Lie theory, and the Lang-Weil
bound, as well as numerous exercises and other optional material.
Winner of the 2015 Prose Award for Best Mathematics Book! In the
fifth of his famous list of 23 problems, Hilbert asked if every
topological group which was locally Euclidean was in fact a Lie
group. Through the work of Gleason, Montgomery-Zippin, Yamabe, and
others, this question was solved affirmatively; more generally, a
satisfactory description of the (mesoscopic) structure of locally
compact groups was established. Subsequently, this structure theory
was used to prove Gromov's theorem on groups of polynomial growth,
and more recently in the work of Hrushovski, Breuillard, Green, and
the author on the structure of approximate groups. In this graduate
text, all of this material is presented in a unified manner,
starting with the analytic structural theory of real Lie groups and
Lie algebras (emphasising the role of one-parameter groups and the
Baker-Campbell-Hausdorff formula), then presenting a proof of the
Gleason-Yamabe structure theorem for locally compact groups
(emphasising the role of Gleason metrics), from which the solution
to Hilbert's fifth problem follows as a corollary. After reviewing
some model-theoretic preliminaries (most notably the theory of
ultraproducts), the combinatorial applications of the
Gleason-Yamabe theorem to approximate groups and groups of
polynomial growth are then given. A large number of relevant
exercises and other supplementary material are also provided.
There are many bits and pieces of folklore in mathematics that are
passed down from advisor to student, or from collaborator to
collaborator, but which are too fuzzy and non-rigorous to be
discussed in the formal literature. Traditionally, it was a matter
of luck and location as to who learned such ""folklore
mathematics"". But today, such bits and pieces can be communicated
effectively and efficiently via the semiformal medium of research
blogging. This book grew from such a blog. The articles, essays,
and notes in this book are derived from the author's mathematical
blog in 2010. It contains a broad selection of mathematical
expositions, commentary, and self-contained technical notes in many
areas of mathematics, such as logic, group theory, analysis, and
partial differential equations. The topics range from the
foundations of mathematics to discussions of recent mathematical
breakthroughs. Lecture notes from the author's courses that
appeared on the blog have been published separately in the Graduate
Studies in Mathematics series
Traditional Fourier analysis, which has been remarkably effective
in many contexts, uses linear phase functions to study functions.
Some questions, such as problems involving arithmetic progressions,
naturally lead to the use of quadratic or higher order phases.
Higher order Fourier analysis is a subject that has become very
active only recently. Gowers, in groundbreaking work, developed
many of the basic concepts of this theory in order to give a new,
quantitative proof of Szemerédi's theorem on arithmetic
progressions. However, there are also precursors to this theory in
Weyl's classical theory of equidistribution, as well as in
Furstenberg's structural theory of dynamical systems. This book,
which is the first monograph in this area, aims to cover all of
these topics in a unified manner, as well as to survey some of the
most recent developments, such as the application of the theory to
count linear patterns in primes. The book serves as an introduction
to the field, giving the beginning graduate student in the subject
a high-level overview of the field. The text focuses on the
simplest illustrative examples of key results, serving as a
companion to the existing literature on the subject. There are
numerous exercises with which to test one's knowledge.
There are many bits and pieces of folklore in mathematics that are
passed down from advisor to student, or from collaborator to
collaborator, but which are too fuzzy and non-rigorous to be
discussed in the formal literature. Traditionally, it was a matter
of luck and location as to who learned such folklore mathematics.
But today, such bits and pieces can be communicated effectively and
efficiently via the semiformal medium of research blogging. This
book grew from such a blog. In 2007, Terry Tao began a mathematical
blog to cover a variety of topics, ranging from his own research
and other recent developments in mathematics, to lecture notes for
his classes, to non-technical puzzles and expository articles. The
articles from the first year of that blog have already been
published by the AMS. The posts from 2008 are being published in
two volumes. This book is Part II of the second-year posts,
focusing on geometry, topology, and partial differential equations.
The major part of the book consists of lecture notes from Tao's
course on the Poincare conjecture and its recent spectacular
solution by Perelman. The course incorporates a review of many of
the basic concepts and results needed from Riemannian geometry and,
to a lesser extent, from parabolic PDE. The aim is to cover in
detail the high-level features of the argument, along with selected
specific components of that argument, while sketching the remaining
elements, with ample references to more complete treatments. The
lectures are as self-contained as possible, focusing more on the
'big picture' than on technical details. In addition to these
lectures, a variety of other topics are discussed, including
expository articles on topics such as gauge theory, the Kakeya
needle problem, and the Black-Scholes equation. Some selected
comments and feedback from blog readers have also been incorporated
into the articles. The book is suitable for graduate students and
research mathematicians interested in broad exposure to
mathematical topics.
In 2007 Terry Tao began a mathematical blog to cover a variety of
topics, ranging from his own research and other recent developments
in mathematics, to lecture notes for his classes, to nontechnical
puzzles and expository articles. The first two years of the blog
have already been published by the American Mathematical Society.
The posts from the third year are being published in two volumes.
The present volume consists of a second course in real analysis,
together with related material from the blog. The real analysis
course assumes some familiarity with general measure theory, as
well as fundamental notions from undergraduate analysis. The text
then covers more advanced topics in measure theory, notably the
Lebesgue-Radon-Nikodym theorem and the Riesz representation
theorem, topics in functional analysis, such as Hilbert spaces and
Banach spaces, and the study of spaces of distributions and key
function spaces, including Lebesgue's $L^p$ spaces and Sobolev
spaces. There is also a discussion of the general theory of the
Fourier transform. The second part of the book addresses a number
of auxiliary topics, such as Zorn's lemma, the Caratheodory
extension theorem, and the Banach-Tarski paradox. Tao also
discusses the epsilon regularisation argument--a fundamental trick
from soft analysis, from which the book gets its title. Taken
together, the book presents more than enough material for a second
graduate course in real analysis. The second volume consists of
technical and expository articles on a variety of topics and can be
read independently.
There are many bits and pieces of folklore in mathematics that are
passed down from advisor to student, or from collaborator to
collaborator, but which are too fuzzy and non-rigorous to be
discussed in the formal literature. Traditionally, it was a matter
of luck and location as to who learned such folklore mathematics.
But today, such bits and pieces can be communicated effectively and
efficiently via the semiformal medium of research blogging. This
book grew from such a blog. In 2007, Terry Tao began a mathematical
blog to cover a variety of topics, ranging from his own research
and other recent developments in mathematics, to lecture notes for
his classes, to non-technical puzzles and expository articles. The
articles from the first year of that blog have already been
published by the AMS. The posts from 2008 are being published in
two volumes. This book is Part I of the second-year posts, focusing
on ergodic theory, combinatorics, and number theory. Chapter 2
consists of lecture notes from Tao's course on topological dynamics
and ergodic theory. By means of various correspondence principles,
recurrence theorems about dynamical systems are used to prove some
deep theorems in combinatorics and other areas of mathematics. The
lectures are as self-contained as possible, focusing more on the
'big picture' than on technical details. In addition to these
lectures, a variety of other topics are discussed, ranging from
recent developments in additive prime number theory to expository
articles on individual mathematical topics such as the law of large
numbers and the Lucas-Lehmer test for Mersenne primes. Some
selected comments and feedback from blog readers have also been
incorporated into the articles. The book is suitable for graduate
students and research mathematicians interested in broad exposure
to mathematical topics.
Additive combinatorics is the theory of counting additive
structures in sets. This theory has seen exciting developments and
dramatic changes in direction in recent years thanks to its
connections with areas such as number theory, ergodic theory and
graph theory. This graduate-level 2006 text will allow students and
researchers easy entry into this fascinating field. Here, the
authors bring together in a self-contained and systematic manner
the many different tools and ideas that are used in the modern
theory, presenting them in an accessible, coherent, and intuitively
clear manner, and providing immediate applications to problems in
additive combinatorics. The power of these tools is well
demonstrated in the presentation of recent advances such as
Szemeredi's theorem on arithmetic progressions, the Kakeya
conjecture and Erdos distance problems, and the developing field of
sum-product estimates. The text is supplemented by a large number
of exercises and new results.
Authored by a leading name in mathematics, this engaging and
clearly presented text leads the reader through the various tactics
involved in solving mathematical problems at the Mathematical
Olympiad level. Covering number theory, algebra, analysis,
Euclidean geometry, and analytic geometry,
Solving Mathematical Problems includes numerous exercises and model
solutions throughout. Assuming only a basic level of mathematics,
the text is ideal for students of 14 years and above in pure
mathematics.
Additive combinatorics is the theory of counting additive
structures in sets. This theory has seen exciting developments and
dramatic changes in direction in recent years thanks to its
connections with areas such as number theory, ergodic theory and
graph theory. This graduate-level 2006 text will allow students and
researchers easy entry into this fascinating field. Here, the
authors bring together in a self-contained and systematic manner
the many different tools and ideas that are used in the modern
theory, presenting them in an accessible, coherent, and intuitively
clear manner, and providing immediate applications to problems in
additive combinatorics. The power of these tools is well
demonstrated in the presentation of recent advances such as
Szemeredi's theorem on arithmetic progressions, the Kakeya
conjecture and Erdos distance problems, and the developing field of
sum-product estimates. The text is supplemented by a large number
of exercises and new results.
This book contains five theses in analysis, by A C Gilbert, N
Saito, W Schlag, T Tao and C M Thiele. It covers a broad spectrum
of modern harmonic analysis, from Littlewood-Paley theory
(wavelets) to subtle interactions of geometry and Fourier
oscillations. The common theme of the theses involves intricate
local Fourier (or multiscale) decompositions of functions and
operators to account for cumulative properties involving size or
structure.
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