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In the two-volume set 'A Selection of Highlights' we present basics
of mathematics in an exciting and pedagogically sound way. This
volume examines many fundamental results in Geometry and Discrete
Mathematics along with their proofs and their history. In the
second edition we include a new chapter on Topological Data
Analysis and enhanced the chapter on Graph Theory for solving
further classical problems such as the Traveling Salesman Problem.
This book gives an advanced overview of several topics in infinite
group theory. It can also be considered as a rigorous introduction
to combinatorial and geometric group theory. The philosophy of the
book is to describe the interaction between these two important
parts of infinite group theory. In this line of thought, several
theorems are proved multiple times with different methods either
purely combinatorial or purely geometric while others are shown by
a combination of arguments from both perspectives. The first part
of the book deals with Nielsen methods and introduces the reader to
results and examples that are helpful to understand the following
parts. The second part focuses on covering spaces and fundamental
groups, including covering space proofs of group theoretic results.
The third part deals with the theory of hyperbolic groups. The
subjects are illustrated and described by prominent examples and an
outlook on solved and unsolved problems.
This second edition gives a thorough introduction to the vast field
of Abstract Algebra with a focus on its rich applications. It is
among the pioneers of a new approach to conveying abstract algebra
starting with rings and fields, rather than with groups. Our
teaching experience shows that examples of groups seem rather
abstract and require a certain formal framework and mathematical
maturity that would distract a course from its main objectives. Our
philosophy is that the integers provide the most natural example of
an algebraic structure that students know from school. A student
who goes through ring theory first, will attain a solid background
in Abstract Algebra and be able to move on to more advanced topics.
The centerpiece of our book is the development of Galois Theory and
its important applications, such as the solvability by radicals and
the insolvability of the quintic, the fundamental theorem of
algebra, the construction of regular n-gons and the famous
impossibilities: squaring the circling, doubling the cube and
trisecting an angle. However, our book is not limited to the
foundations of abstract algebra but concludes with chapters on
applications in Algebraic Geometry and Algebraic Cryptography.
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