Publication list for Katok, Anatole

Latest Titles

  • From Groups to Geometry and Back
    Base Product Code Keyword List: stml; STML; stml/81; STML/81; stml-81; STML-81
    Print Product Code: STML/81
    Online Product Code: STML/81.E
    Title (HTML): From Groups to Geometry and Back
    Author(s) (Product display): Vaughn Climenhaga; Anatole Katok
    Affiliation(s) (HTML): University of Houston, Houston, TX; Pennsylvania State University, University Park, PA
    Abstract:

    Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory.

    The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras.

    The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth.

    The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness.

    This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.

    This book is published in cooperation with Mathematics Advanced Study Semesters

    Book Series Name: Student Mathematical Library
    Volume: 81
    Publication Month and Year: 2017-04-07
    Copyright Year: 2017
    Page Count: 420
    Cover Type: Softcover
    Print ISBN-13: 978-1-4704-3479-3
    Online ISBN 13: 978-1-4704-3753-4
    Print ISSN: 1520-9121
    Online ISSN: 1520-9121
    Primary MSC: 20; 51
    Secondary MSC: 22; 54; 57
    Textbook?: False
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    Home Page?: false
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    Reference?: false
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    Publisher (non-AMS): Published in cooperation with Mathematics Advanced Study Semesters
    SXG Subject: AA; GT
    Supplemental Materials (URL at AMS): http://www.ams.org/bookpages/stml-81
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    Review Copy: http://www.ams.org/exam-desk-review-request?&eisbn=978-1-4704-3753-4&pisbn=978-1-4704-3479-3&epc=STML/81.E&ppc=STML/81&title=From%20Groups%20to%20Geometry%20and%20Back&author=Vaughn%20Climenhaga%3B%20Anatole%20Katok&type=R
    Readership:

    Undergraduate and graduate students interested in group theory and geometry.

    Related Products: Geometries Lectures on Surfaces
    Cover Image URL: ~~FreeAttachments/stml-81-cov.jpg
  • Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov
    Base Product Code Keyword List: conm; CONM; conm/692; CONM/692; conm-692; CONM-692
    Print Product Code: CONM/692
    Online Product Code: CONM/692.E
    Title (HTML): Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov
    Author/Editor Label (optional): Edited by
    Author(s) (Product display): Anatole Katok; Yakov Pesin; Federico Rodriguez Hertz
    Affiliation(s) (HTML): Pennsylvania State University, University Park, PA; Pennsylvania State University, University Park, PA; Pennsylvania State University, University Park, PA
    Abstract:

    This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov.

    It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov's work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov's original work.

    Also included is an article by A. Katok that presents Anosov's scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform.

    Book Series Name: Contemporary Mathematics
    Volume: 692
    Copyright Year: 2017
    Page Count: 320
    Cover Type: Softcover
    Print ISBN-13: 978-1-4704-2560-9
    Online ISBN 13: 978-1-4704-4119-7
    Print ISSN: 0271-4132
    Online ISSN: 0271-4132
    Primary MSC: 37
    Textbook?: False
    Applied Math?: False
    Electronic Media?: False
    Apparel or Gift: False
    SXG Subject: AN
    Online Price 1 Label: List
    Online Price 1: 111.00
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    Print Price 1: 111.00
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    Online Price 2: 88.80
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    Print Outstock Reason: NYP
    Print Outstock Reason Avail Date: Expected publication date July 26, 2017
    Online Outstock Reason: NYP
    Online Outstock Reason Avail Date: Expected publication date July 26, 2017
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    Readership:

    Graduate students and research mathematicians interested in dynamical systems and their applications.

    Cover Image URL: ~~FreeAttachments/conm-692-cov.jpg
  • Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them
    Base Product Code Keyword List: stml; STML; stml/46; STML/46; stml-46; STML-46
    Print Product Code: STML/46
    Online Product Code: STML/46.E
    Title (HTML): Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them
    Author(s) (Product display): Anatole Katok; Vaughn Climenhaga
    Affiliation(s) (HTML): Pennsylvania State University, University Park, PA; Pennsylvania State University, University Park, PA
    Abstract:

    Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. At the same time, many of those notions appear in a technically simpler and more graphic form than in their general “natural” settings.

    The first, primarily expository, chapter introduces many of the principal actors—the round sphere, flat torus, Möbius strip, Klein bottle, elliptic plane, etc.—as well as various methods of describing surfaces, beginning with the traditional representation by equations in three-dimensional space, proceeding to parametric representation, and also introducing the less intuitive, but central for our purposes, representation as factor spaces. It concludes with a preliminary discussion of the metric geometry of surfaces, and the associated isometry groups. Subsequent chapters introduce fundamental mathematical structures—topological, combinatorial (piecewise linear), smooth, Riemannian (metric), and complex—in the specific context of surfaces.

    The focal point of the book is the Euler characteristic, which appears in many different guises and ties together concepts from combinatorics, algebraic topology, Morse theory, ordinary differential equations, and Riemannian geometry. The repeated appearance of the Euler characteristic provides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories.

    The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background.

    This book is a result of the MASS course in geometry in the fall semester of 2007.

    This book is published in cooperation with Mathematics Advanced Study Semesters

    Book Series Name: Student Mathematical Library
    Volume: 46
    Publication Month and Year: 2008-10-28
    Copyright Year: 2008
    Page Count: 286
    Cover Type: Softcover
    Print ISBN-13: 978-0-8218-4679-7
    Online ISBN 13: 978-1-4704-1817-5
    Print ISSN: 1520-9121
    Online ISSN: 1520-9121
    Primary MSC: 51; 53; 57
    Textbook?: False
    Applied Math?: False
    Electronic Media?: False
    Apparel or Gift: False
    Publisher (non-AMS): Published in cooperation with Mathematics Advanced Study Semesters
    SXG Subject: GT
    Supplemental Materials (URL at AMS): http://www.ams.org/bookpages/stml-46
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    Readership:

    Undergraduate and graduate students interested in broadening their view of geometry and topology.

    Reviews:

    This book will be a welcome addition to college and university libraries and an excellent source for supplementary reading.

    -- Mathematical Reviews

    (This book) does a masterful job of introducing the study of surfaces to advanced undergraduates. ... The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading.

    -- MAA Reviews

    Related Products: Mostly Surfaces Low-Dimensional Geometry
  • Combinatorial Constructions in Ergodic Theory and Dynamics
    Base Product Code Keyword List: ulect; ULECT; ulect/30; ULECT/30; ulect-30; ULECT-30
    Print Product Code: ULECT/30
    Online Product Code: ULECT/30.E
    Title (HTML): Combinatorial Constructions in Ergodic Theory and Dynamics
    Author(s) (Product display): Anatole Katok
    Affiliation(s) (HTML): Pennsylvania State University, University Park, PA
    Abstract:

    Ergodic theory studies measure-preserving transformations of measure spaces. These objects are intrinsically infinite, and the notion of an individual point or of an orbit makes no sense. Still there are a variety of situations when a measure-preserving transformation (and its asymptotic behavior) can be well described as a limit of certain finite objects (periodic processes).

    The first part of this book develops this idea systematically. Genericity of approximation in various categories is explored, and numerous applications are presented, including spectral multiplicity and properties of the maximal spectral type. The second part of the book contains a treatment of various constructions of cohomological nature with an emphasis on obtaining interesting asymptotic behavior from approximate pictures at different time scales.

    The book presents a view of ergodic theory not found in other expository sources. It is suitable for graduate students familiar with measure theory and basic functional analysis.

    Book Series Name: University Lecture Series
    Volume: 30
    Publication Month and Year: 2003-10-15
    Copyright Year: 2003
    Page Count: 121
    Cover Type: Softcover
    Print ISBN-13: 978-0-8218-3496-1
    Online ISBN 13: 978-1-4704-2175-5
    Print ISSN: 1047-3998
    Online ISSN: 1047-3998
    Primary MSC: 37
    Textbook?: False
    Applied Math?: False
    Electronic Media?: False
    Apparel or Gift: False
    SXG Subject: DE
    Online Price 1 Label: List
    Online Price 1: 34.00
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    Readership:

    Graduate students and research mathematicians interested in ergodic theory.

    Reviews:

    For more advanced readers, however, this volume will be highly rewarding: they will be learning from a master of the subject, presenting some of his tools.

    -- Mathematical Reviews

  • Smooth Ergodic Theory and Its Applications
    Base Product Code Keyword List: pspum; PSPUM; pspum/69; PSPUM/69; pspum-69; PSPUM-69
    Print Product Code: PSPUM/69
    Online Product Code: PSPUM/69.E
    Title (HTML): Smooth Ergodic Theory and Its Applications
    Author/Editor Label (optional): Edited by
    Author(s) (Product display): Anatole Katok; Rafael de la Llave; Yakov Pesin; Howard Weiss
    Affiliation(s) (HTML): Pennsylvania State University, University Park, PA; University of Texas at Austin, Austin, TX; Pennsylvania State University, University Park, PA; Pennsylvania State University, University Park, PA
    Abstract:

    During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student—or even an established mathematician who is not an expert in the area—to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools.

    Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference.

    Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincaré and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of correlations, and measure-theoretic entropy). Smooth ergodic theory also provides a foundation for numerous applications throughout mathematics (e.g., Riemannian geometry, number theory, Lie groups, and partial differential equations), as well as other sciences.

    This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

    Book Series Name: Proceedings of Symposia in Pure Mathematics
    Volume: 69
    Publication Month and Year: 2001-10-03
    Copyright Year: 2001
    Page Count: 881
    Cover Type: Hardcover
    Print ISBN-13: 978-0-8218-2682-9
    Online ISBN 13: 978-0-8218-9374-6
    Print ISSN: 0082-0717
    Online ISSN: 0082-0717
    Primary MSC: 11; 28; 34; 37; 53; 70
    Textbook?: False
    Applied Math?: False
    Electronic Media?: False
    Apparel or Gift: False
    SXG Subject: AN; GT
    Online Price 1 Label: List
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    Print Add to Cart URL: /some/url/at/AMS/PSPUM-69
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    Review Copy: http://www.ams.org/exam-desk-review-request?&eisbn=978-0-8218-9374-6&pisbn=978-0-8218-2682-9&epc=PSPUM/69.E&ppc=PSPUM/69&title=Smooth%20Ergodic%20Theory%20and%20Its%20Applications&author=Anatole%20Katok%3B%20Rafael%20de%20la%20Llave%3B%20Yakov%20Pesin%3B%20Howard%20Weiss&type=R
    Readership:

    Graduate students and research mathematicians interested in ergodic theory and its applications.