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.
Both fractal geometry and dynamical systems have a long history of
development and have provided fertile ground for many great
mathematicians and much deep and important mathematics. These two
areas interact with each other and with the theory of chaos in a
fundamental way: many dynamical systems (even some very simple ones)
produce fractal sets, which are in turn a source of irregular
“chaotic” motions in the system. This book is an introduction to
these two fields, with an emphasis on the relationship between them.
The first half of the book introduces some of the key ideas in
fractal geometry and dimension theory—Cantor sets, Hausdorff
dimension, box dimension—using dynamical notions whenever
possible, particularly one-dimensional Markov maps and symbolic
dynamics. Various techniques for computing Hausdorff dimension are
shown, leading to a discussion of Bernoulli and Markov measures and of
the relationship between dimension, entropy, and Lyapunov
exponents.
In the second half of the book some examples of dynamical systems
are considered and various phenomena of chaotic behaviour are
discussed, including bifurcations, hyperbolicity, attractors,
horseshoes, and intermittent and persistent chaos. These phenomena
are naturally revealed in the course of our study of two real models
from science—the FitzHugh–Nagumo model and the Lorenz
system of differential equations.
This book is accessible to undergraduate students and requires only
standard knowledge in calculus, linear algebra, and differential
equations. Elements of point set topology and measure theory are
introduced as needed.
This book is a result of the MASS course in analysis at Penn State
University in the fall semester of 2008.
This book is published in cooperation with Mathematics Advanced Study Semesters
Undergraduate and graduate students interested in dynamical systems and fractal geometry.
[F]or a student with a reasonable background in topology and measure theory this is a very useful book covering many of the main ideas in fractal geometry and dynamical systems in an accessible way, with a particular emphasis on dynamically-defined fractals.
-- Ian Melbourne, Mathematical Reviews
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