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
Undergraduate and graduate students interested in group theory and geometry.
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
Undergraduate and graduate students interested in broadening their view of geometry and topology.
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
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.
Graduate students and research mathematicians interested in ergodic theory.
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
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.
Graduate students and research mathematicians interested in ergodic theory and its applications.