This collection features books and book contributions written by faculty in the Department of Mathematics at Fairfield University.

Quantum Computing for Everyone
Christopher R. Bernhardt
An accessible introduction to an exciting new area in computation, explaining such topics as qubits, entanglement, and quantum teleportation for the general reader.
Quantum computing is a beautiful fusion of quantum physics and computer science, incorporating some of the most stunning ideas from twentiethcentury physics into an entirely new way of thinking about computation. In this book, Chris Bernhardt offers an introduction to quantum computing that is accessible to anyone who is comfortable with high school mathematics. He explains qubits, entanglement, quantum teleportation, quantum algorithms, and other quantumrelated topics as clearly as possible for the general reader. Bernhardt, a mathematician himself, simplifies the mathematics as much as he can and provides elementary examples that illustrate both how the math works and what it means.
Bernhardt introduces the basic unit of quantum computing, the qubit, and explains how the qubit can be measured; discusses entanglement—which, he says, is easier to describe mathematically than verbally—and what it means when two qubits are entangled (citing Einstein's characterization of what happens when the measurement of one entangled qubit affects the second as “spooky action at a distance”); and introduces quantum cryptography. He recaps standard topics in classical computing—bits, gates, and logic—and describes Edward Fredkin's ingenious billiard ball computer. He defines quantum gates, considers the speed of quantum algorithms, and describes the building of quantum computers. By the end of the book, readers understand that quantum computing and classical computing are not two distinct disciplines, and that quantum computing is the fundamental form of computing. The basic unit of computation is the qubit, not the bit.

Geometry and Discrete Mathematics: A Selection of Highlights
Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger, and Dennis Spellman
This twovolume set collects and presents many fundamentals of mathematics in an enjoyable and elaborating fashion. The idea behind the two books is to provide substantials for assessing more modern developments in mathematics and to present impressions which indicate that mathematics is a fascinating subject with many ties between the diverse mathematical disciplines. The present volume examines many of the most important basic results in geometry and discrete mathematics, along with their proofs, and also their history.

A course in mathematical cryptography
Gilbert Baumslag, Benjamin Fine, Martin Kreuzer, and Gerhard Rosenberger
Book description: Cryptography has become essential as bank transactions, credit card information, contracts, and sensitive medical information are sent through insecure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area.

Turing's Vision: The Birth of Computer Science
Christopher R. Bernhardt
Book description: In 1936, when he was just twentyfour years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing’s Vision, Chris Bernhardt explains the theory, Turing’s most important contribution, for the general reader. Bernhardt argues that the strength of Turing’s theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, “The sheer simplicity of the theory’s foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory.” Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing’s theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing’s later work, and the birth of the modern computer. Publisher description

Number theory : an introduction via the density of primes, Second edition
Benjamin Fine and Gerhard Rosenberger
Book description: Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of padic numbers, Hensel's lemma, multiple zetavalues, and elliptic curve methods in primality testing. Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The userfriendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both selfstudy and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.

Contemporary Mathematics: Algebra and Computer Sciencevolume 677
Delaram Kahrobaei, Bren Cavallo, David Garber, Chi Sing Chung, Benjamin Fine, Anja IS Moldenhauer, Gerhard Rosenberger, and Xiaowen Zhang
Benjamin Fine is a contributing author, "On secret sharing protocols" with Chi Sing Chum, Anja IS Moldenhauer, Gerhard Rosenberger, and Xiaowen Zhang.
Book description: This volume contains the proceedings of three special sessions: Algebra and Computer Science, held during the Joint AMSEMSSPM meeting in Porto, Portugal, June 1013, 2015; Groups, Algorithms, and Cryptography, held during the Joint Mathematics Meeting in San Antonio, TX, January 1013, 2015; and Applications of Algebra to Cryptography, held during the Joint AMSIsrael Mathematical Union meeting in TelAviv, Israel, June 1619, 2014. Papers contained in this volume address a wide range of topics, from theoretical aspects of algebra, namely group theory, universal algebra and related areas, to applications in several different areas of computer science. From the computational side, the book aims to reflect the rapidly emerging area of algorithmic problems in algebra, their computational complexity and applications, including information security, constraint satisfaction problems, and decision theory. The book gives special attention to recent advances in quantum computing that highlight the need for a variety of new intractability assumptions and have resulted in a new area called groupbased cryptography.

The Elementary Theory of Groups: A Guide Through the Proofs of the Tarski Conjectures. Vol. 60
Benjamin Fine, Anthony Gaglione, Alexei Myasnikov, Gerhard Rosenberger, and Dennis Spellman
After being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. Both proofs involve long and complicated applications of algebraic geometry over free groups as well as an extension of methods to solve equations in free groups originally developed by Razborov. This book is an examination of the material on the general elementary theory of groups that is necessary to begin to understand the proofs. This material includes a complete exposition of the theory of fully residually free groups or limit groups as well a complete description of the algebraic geometry of free groups. Also included are introductory material on combinatorial and geometric group theory and firstorder logic. There is then a short outline of the proof of the Tarski conjectures in the manner of Kharlampovich and Myasnikov.Publisher's Description

An Introduction to Partial Differential Equations with MATLAB, 2nd edition
Matt Coleman
An Introduction to Partial Differential Equations with MATLAB^{®}, Second Edition illustrates the usefulness of PDEs through numerous applications and helps students appreciate the beauty of the underlying mathematics. Updated throughout, this second edition of a bestseller shows students how PDEs can model diverse problems, including the flow of heat, the propagation of sound waves, the spread of algae along the ocean’s surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom.
Suitable for a twosemester introduction to PDEs and Fourier series for mathematics, physics, and engineering students, the text teaches the equations based on method of solution. It provides both physical and mathematical motivation as much as possible. The author treats problems in one spatial dimension before dealing with those in higher dimensions. He covers PDEs on bounded domains and then on unbounded domains, introducing students to Fourier series early on in the text.
Each chapter’s prelude explains what and why material is to be covered and considers the material in a historical setting. The text also contains many exercises, including standard ones and graphical problems using MATLAB. While the book can be used without MATLAB, instructors and students are encouraged to take advantage of MATLAB’s excellent graphics capabilities. The MATLAB code used to generate the tables and figures is available in an appendix and on the author’s website.  Publisher description.

Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography
Celine Carstensen, Benjamin Fine, and Gerard Rosenberger
 Providing an accessible account of the theoretical foundations
 Covering topics not found in competing works: Free groups, module theory, extensions of rings
 Also including cryptography
 Comes with end of chapter problems
Aims and Scope:
A new approach to conveying abstract algebra, the area that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, that is essential to various scientific disciplines such as particle physics and cryptology. It provides a well written account of the theoretical foundations; also contains topics that cannot be found elsewhere, and also offers a chapter on cryptography. End of chapter problems help readers with accessing the subjects.
This work is copublished with the Heldermann Verlag, and within Heldermann's Sigma Series in Mathematics.