Assembles the available tools and methods in a new, coherent theory, building from foundational material
Invites the reader to explore a wide open research area
Provides thought-provoking mixture of mathematical ideas
Originating from graduate topics courses given by the first author, this book functions as a unique text-monograph
hybrid that bridges a traditional graduate course to research level representation theory. The exposition includes
an introduction to the subject, some highlights of the theory and recent results in the field, and is therefore
appropriate for advanced graduate students entering the field as well as research mathematicians wishing to
expand their knowledge. The mathematical background required varies from chapter to chapter, but a standard
course on Lie algebras and their representations, along with some knowledge of homological algebra, is necessary.
Basic algebraic geometry and sheaf cohomology are needed for Chapter 10. Exercises of various levels of difficulty
are interlaced throughout the text to add depth to topical comprehension.
The unifying theme of this book is the structure and representation theory of infinite-dimensional locally reductive
Lie algebras and superalgebras. Chapters 1-6 are foundational; each of the last 4 chapters presents a self-contained
study of a specialized topic within the larger field. Lie superalgebras and flag supermanifolds are discussed in
Chapters 3, 7, and 10, and may be skipped by the reader.
Provides an introduction to geometric group theory based on the unifying theme of Gromov theorem
Shows the connections between a wide range of topics in geometric group theory
Collects together, for the first time, results previously scattered throughout the literature
With a Foreword by Efim I. Zelmanov
This book provides a detailed exposition of a wide range of topics in geometric
group theory, inspired by Gromov pivotal
work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of
the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters
and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups.
The results are unified under the common theme of Gromovfs theorem, namely that finitely generated groups of
polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which
is still active today.
The purpose of the book is to collect these naturally related results together in one place, most of which are scattered
throughout the literature, some of them appearing here in book form for the first time. In this way, the connections
between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas
surrounding Gromov's theorem.
The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with
basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects
of infinite groups.
Covers both the complete mathematical theory and examples of how to use the theory in specific situations
Combines the synthetic and the analytic method
Places some topics in the context of higher mathematics, opening a window toward further studies
This textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based
perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on
some classical examples and theorems. It is based on the combined teaching experience of the authors
(coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive
theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples
and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged
to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in
competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics
competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations.
The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods
and ideas to the attention of the public.
Talented high school and middle school students seeking to improve their problem-solving skills can benefit from
this book, as well as high school and college instructors who want to add nonstandard questions to their courses.
People who enjoy solving elementary math problems as a hobby will also enjoy this work.
https://doi.org/10.1142/11927-vol2 | March 2022
Pages: 446
ISBN: 978-981-124-197-0 (hardcover)
ISBN: 978-981-124-198-7 (softcover)
Yes, this is another Calculus book. However, I think it fits in a niche between the two predominant types of such texts. I
t could be used as a textbook, albeit a streamlined one ?Eit contains exposition on each topic, with an introduction,
rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution
guide ?Ebecause it contains full written solutions to each of the hundreds of exercises posed inside.
But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a
refresher ?Ewith its conversational tone, its "get right to it" content structure, and its inclusion of complete solutions
to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.
Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are
accompanied by "You Try It" problems, which are similar to the solved examples; the students use these to see
if they're ready to move forward. Then at the end of the section, there are "Practice Problems": more problems
similar to the You Try It problems, but given all at once. Finally, each section has Challenge Problems ?Ethese lean
to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.
My goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at
humor along the way. Because we have large engineering and meteorology programs at my institution, and they make
up the largest portion of our Calculus students; naturally, then, these sorts of STEM students are the target audience.
The Integration Dojo
The Mathematics Chainsaw Massacre
Round and Round We Go: Solids of Revolution
Approximation
The Fear of All Sums
A Change in Graph Paper
Solutions to All Practice Problems
Solutions to All Challenge Problems
Index
Undergraduate students currently taking or refreshing themselves on Calculus.
https://doi.org/10.1142/12586 | January 2022
Pages: 372
ISBN: 978-981-124-835-1 (hardcover)
Description
Algebraic Topology is a system and strategy of partial translations, aiming to reduce difficult topological problems
to algebraic facts that can be more easily solved. The main subject of this book is singular homology, the simplest
of these translations. Studying this theory and its applications, we also investigate its underlying structural layout
- the topics of Homological Algebra, Homotopy Theory and Category Theory which occur in its foundation.
This book is an introduction to a complex domain, with references to its advanced parts and ramifications.
It is written with a moderate amount of prerequisites ?Ebasic general topology and little else ?Eand a moderate
progression starting from a very elementary beginning. A consistent part of the exposition is organised in the form of
exercises, with suitable hints and solutions.
It can be used as a textbook for a semester course or self-study, and a guidebook for further study.
Introduction
Introducing Algebraic Topology
Singular Homology
Relative Singular Homology and Homology Theories
Singular Homology with Coefficients
Derived Functors, Universal Coefficients and Products
An Introduction to Homotopy Groups
Complements on Categories and Topology
Solutions to the Exercises
Graduate students, PhD students and Researchers in Mathematics and Physics interested in Algebraic Topology.
https://doi.org/10.1142/12584 | January 2022
Pages: 480
ISBN: 978-981-124-829-0 (hardcover)
The author presents three distinct but related branches of science in this book: digital geometry, mathematical morphology
, and discrete optimization. They are united by a common mindset as well as by the many applications where they are
useful. In addition to being useful, each of these relatively new branches of science is also intellectually challenging.
The book contains a systematic study of inverses of mappings between ordered sets, and so offers a uniquely helpful
organization in the approach to several phenomena related to duality.
To prepare the ground for discrete convexity, there are chapters on convexity in real vector spaces in anticipation
of the many challenging problems coming up in digital geometry. To prepare for the study of new topologies introduced
to serve in discrete spaces, there is also a chapter on classical topology.
The book is intended for general readers with a modest background in mathematics and for advanced undergraduate
students as well as beginning graduate students.
Preface
Introduction
Sets, Mappings, and Order Relations
Morphological Operations: Set-Theoretical Duality
Complete Lattices
Inverses and Quotients of Mappings
Structure Theorems for Mappings
Digitization
Digital Straightness and Digital Convexity
Convexity in Vector Spaces
Discrete Convexity
Discrete Convexity in Two Dimensions
Three Problems in Discrete Optimization
Duality of Convolution Operators
Topology
The Khalimsky Topology
Distance Transformations
Skeletonizing
Solutions
Bibliography
Author Index
Subject Index
Advanced undergraduate and graduate students, researchers.