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Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 0821894927 Category : Mathematics Languages : en Pages : 256
Book Description
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
Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 0821894927 Category : Mathematics Languages : en Pages : 256
Book Description
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
Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 147041564X Category : Mathematics Languages : en Pages : 338
Book Description
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.
Author: Stephanie Alexander Publisher: Springer ISBN: 3030053121 Category : Mathematics Languages : en Pages : 88
Book Description
Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Author: David Eric Edmunds Publisher: Oxford University Press ISBN: 0198812051 Category : Mathematics Languages : en Pages : 610
Book Description
This revised edition corrects various errors, and adds extensive notes to the end of each chapter which describe the considerable progress that has been made on the topic in the last 30 years.--
Author: Julia Lossau Publisher: Springer Nature ISBN: 3658259906 Category : Philosophy Languages : en Pages : 254
Book Description
This volume focusses on contradiction as a key concept in the Humanities and Social Sciences. By bringing together theoretical and empirical contributions from a broad disciplinary spectrum, the volume advances research in contradiction and on contradictory phenomena, laying the foundations for a new interdisciplinary field of research: Contradiction Studies. Dealing with linguistic phenomena, urban geographies, business economy, literary writing practices, theory of the social sciences, and language education, the contributions show that contradiction, rather than being a logical exemption in the Aristotelian sense, provides a valuable approach to many fields of socially, culturally, and historically relevant fields of research.
Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 9780821886281 Category : Mathematics Languages : en Pages : 316
Book Description
"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 = mc[superscript 2], 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.
Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 1470466406 Category : Education Languages : en Pages : 206
Book Description
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 Carathéodory 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.
Author: César Ernesto Silva Publisher: American Mathematical Soc. ISBN: 1470449285 Category : Education Languages : en Pages : 318
Book Description
Provides a careful introduction to the real numbers with an emphasis on developing proof-writing skills. The book continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions.
Author: Marcel de Jeu Publisher: Birkhäuser ISBN: 3319278428 Category : Mathematics Languages : en Pages : 507
Book Description
This book presents the proceedings of Positivity VII, held from 22-26 July 2013, in Leiden, the Netherlands. Positivity is the mathematical field concerned with ordered structures and their applications in the broadest sense of the word. A biyearly series of conferences is devoted to presenting the latest developments in this lively and growing discipline. The lectures at the conference covered a broad spectrum of topics, ranging from order-theoretic approaches to stochastic processes, positive solutions of evolution equations and positive operators on vector lattices, to order structures in the context of algebras of operators on Hilbert spaces. The contributions in the book reflect this variety and appeal to university researchers in functional analysis, operator theory, measure and integration theory and operator algebras. Positivity VII was also the Zaanen Centennial Conference to mark the 100th birth year of Adriaan Cornelis Zaanen, who held the chair of Analysis in Leiden for more than 25 years and was one of the leaders in the field during his lifetime.
Author: Terence Tao Publisher: American Mathematical Soc. ISBN: 0821848836 Category : Differential equations, Partial Languages : en Pages : 306
Book Description
Focuses on ergodic theory, combinatorics, and number theory. This book discusses a variety of topics, ranging from 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.
Author: Terence Tao
Author: Terence Tao
Author: Terence Tao
Author: Stephanie Alexander
Author: David Eric Edmunds
Author: Julia Lossau
Author: Terence Tao
Author: Terence Tao
Author: César Ernesto Silva
Author: Marcel de Jeu
Author: Terence Tao