The following online courses are offered during the Fall, Spring, and Summer semesters. Visiting students not affiliated with JHU are welcome to apply.

## Precalculus (110.105)

This course is a precalculus course and provides students with the background necessary for a study of calculus. Includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools is introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. This course provides a mathematically sound foundation for students who intend to study calculus.

Prerequisite: None.

Credits: 4 credits

Required Text: *PreCalculus,* by Faires and DeFranza., 5th Ed., ISBN-13: 978-0-84006862-0 ISBN-10: 0-8400-6862-X

## Calculus II – Biological and Social Sciences (110.107) (Summer Only)

Syllabus

This is a second course in the calculus of functions of one independent variable. However, instead of continuing with standard calculus topics, this semester includes an introduction to differential equations, the basic structure of functions of several variables, an introduction to linear systems and linear algebra, and applications for systems of linear differential equations and probability distributions. Applications to the biological and social sciences will be discussed, and the course is designed to meet the needs of students in these disciplines.

This course is part of a two course sequence and succeeds AS.110.106 Calculus I (Biology and the Social Sciences). Students planning to take this course must demonstrate a proficiency in some form of first semester university calculus, either through the AP system resulting with an AB score of 5 or a BC score of 3 or better, or a course like AS.110.106 Calculus I. It is possible to gain access to this course via an adequate score on the Placement Exam II offered by the Mathematics Department, but that also requires permission form the department. This sequence of courses are considered terminal and are typically not to be considered adequate preparation for higher mathematics. This sequence satisfies a core requirement of two semesters of single variable calculus for both the major and minor in mathematics.

Credits: 4 credits

Required Text: *Calculus for Biology and Medicine*, *4 ^{th} Edition,* C. Neuhauser and M. Roper, New Jersey: Prentice Hall, January 2018, ISBN-10: 0134070046, ISBN-13: 978-0134070049

## Calculus I – Physical Sciences and Engineering (110.108)

Syllabus

This is the first of a two course sequence in the differential and integral calculus of single variable functions. Topics include the basic analytic geometry of graphs of functions, and their limits, integrals and derivatives, including the Fundamental Theorem of Calculus. Also, some applications of the integral, like arc length and volumes of solids with rotational symmetry, are discussed. Applications to the physical sciences and engineering will be a focus of this course, as this sequence of courses is designed to meet the needs of students in these disciplines. 4 credits

The calculus course sequence is considered foundational to all higher-level courses in mathematics. This course satisfies the core requirement for the first of two semesters of single variable calculus for both the major and minor in mathematics.

Prerequisite: Students planning to take this course must demonstrate a proficiency in pre-calculus, either through the successful completion of a prior course in pre-calculus (such as AS.110.105 or similar) or by achieving an adequate score in the Placement Exam I offered by the Mathematics Department.

Credits: 4 credits

Required Text: *Single Variable Calculus: Early Transcendentals*, by James Stewart, 8th Ed., ISBN: 978-1-305-27033-6.

## Calculus II – Physical Sciences and Engineering (110.109)

Syllabus

This is the second of a two course sequence in the differential and integral calculus of single variable functions. Topics include techniques of integration, applications of integrals, polar coordinates, parametric equations, Taylor’s theorem and applications, infinite sequences and series. Some applications to the physical sciences and engineering will be discussed, as this sequence of courses are designed to meet the needs of students in these disciplines. 4 credits

Prerequisite: Successful completion of AP Calculus AB, Calculus II, or equivalent.

Credits: 4 credits

Required Text: *Single Variable Calculus: Early Transcendentals*, by James Stewart, 8th Ed., ISBN: 978-1-305-27033-6.

## Calculus III (110.202)

Syllabus

This is a course in the differential and integral calculus of several variables. Topics include vectors in two and three dimensions, analytic geometry of three dimensions, parametric curves, partial derivatives, the gradient, optimization in several variables, multiple integration with change of variables across different coordinate systems, line integrals, surface integrals, and Green’s Theorem, Stokes’ Theorem, and Gauss’ Divergence Theorem.

Prerequisite: Successful completion of AP Calculus BC, Calculus II, or equivalent.

Credits: 4 credits

Required text:* Vector Calculus* by Marsden & Tromba, Freeman, 6th Ed., ISBN 9781429215084

## Linear Algebra (110.201)

Syllabus

This course is an introduction to the techniques of linear algebra in Euclidean space. Topics covered include matrices, determinants, systems of linear equations, vector spaces, linear transformations, complex numbers, and eigenvalues and eigenvectors. Diagonalization of matrices and quadratic forms, as well as applications of these topics to the biological, physical and social sciences to are also included.

Prerequisite: Successful completion of Calculus I. Recommended: Calculus II.

Credits: 4 credits

Required text: *Linear Algebra with Applications* by Bretscher, Prentice, 5th Edition, ISBN9780321796974

## Probability (110.275)

This course follows the actuarial Exam P syllabus and learning objectives to prepare students to pass the SOA/CAS Probability Exam. Topics include axioms of probability, discrete and continuous random variables, conditional probability, Bayes’ theorem, Chebyshev’s Theorem, Central Limit Theorem, univariate and joint distributions and expectations, loss frequency, loss severity and other risk management concepts. Exam P learning objectives and learning outcomes are emphasized.

## Introduction to Proofs (110.301)

This course will provide a practical introduction to mathematical proofs with the aim of developing fluency in the language of mathematics, which itself is often described as “the language of the universe.” Along with a library of proof techniques, we shall tour propositional logic, set theory, cardinal arithmetic, and metric topology and explore “proof relevant” mathematics by interacting with a computer proof assistant. This course on the construction of mathematical proof will conclude with a deconstruction of mathematical proof, interrogating the extent to which proof serves as a means to discover universal truths and assessing the mechanisms by which the mathematical community achieves consensus regarding whether a claimed result has been proven.

Prerequisite: None.

Credits: 4 credits

## Differential Equations with Applications (110.302)

Syllabus

This is an applied course in ordinary differential equations, which is primarily for students in the biological, physical and social sciences, and engineering. Techniques for solving ordinary differential equations are studied. Topics covered include first order differential equations, second order linear differential equations, applications to electric circuits, oscillation of solutions, power series solutions, systems of linear differential equations, autonomous systems, Laplace transforms and linear differential equations, mathematical models (e.g., in the sciences or economics).

Prerequisite: Calculus II.

Credits: 4 credits

Required text: *Elementary Diff Equations & Boundary Value Problems *by Boyce & DiPrima, Wiley, 10th edition, ISBN 9780470458310

## The Mathematics of Politics, Democracy, and Social Choice (110.303)

This course is designed for students of all backgrounds to provide a mathematical introduction to social choice theory, weighted voting systems, apportionment methods, and gerrymandering. In the search for ideal ways to make certain kinds of political decisions, a lot of wasted effort could be averted if mathematics could determine that finding such an ideal were actually possible in the first place.

The course will analyze data from recent US elections as well as provide historical context to modern discussions in politics, culminating in a mathematical analysis of the US Electoral College. Case studies, future implications, and comparisons to other governing bodies outside the US will be used to apply the theory of the course. Students will use Microsoft Excel to analyze data sets. There are no mathematical prerequisites for this course.

## Elementary Number Theory (110.304)

Syllabus

This course provides some historical background and examples of topics of current research interest in number theory and includes concrete examples of some of the abstract concepts studied in abstract algebra. Topics include primes and prime factorization, congruences, Euler’s function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley’s theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, and Dirichlet’s theorem on primes.

Prerequisite: Linear Algebra.

Credits: 4 credits

## Methods of Complex Analysis (110.311)

This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Topics include functions of a complex variable and their derivatives; power series and Laurent expansions; the Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.

Prerequisite: Linear Algebra and Calculus III.

Credits: 4 credits

Required text: *Fundamentals of Complex Analysis (with Applications to Engineering and Science), 3 ^{rd} Edition*, E. B. Saff & A. D. Snider. Prentice Hall, January 2003, ISBN-10: 0139078746, ISBN-13: 978-0139078743

## An Introduction to Mathematical Cryptography (110.375)

An Introduction to Mathematical Cryptography is an introduction to modern cryptography with an emphasis on the mathematics behind the theory of public key cryptosystems and digital signature schemes. The course develops the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Other topics central to mathematical cryptography covered are: classical cryptographic constructions, such as Diffie-Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures. Fundamental mathematical tools for cryptography studied include: primality testing, factorization algorithms, probability theory, information theory, and collision algorithms.

A survey of important recent cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography are included as well. This course is an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography.

Prerequisite: Linear Algebra and Calculus III.

Credits: 4 credits

## Introduction to Abstract Algebra (110.401)

A first introduction to abstract algebra through group theory, with an emphasis on concrete examples, and especially on geometric symmetry groups. The course will introduce basic notions (groups, subgroups, homomorphisms, quotients) and prove foundational results (Lagrange’s theorem, Cauchy’s theorem, orbit-counting techniques, the classification of finite abelian groups). Examples to be discussed include permutation groups, dihedral groups, matrix groups, and finite rotation groups, culminating in the classification of the wallpaper groups.

Prerequisite: Linear Algebra.

Credits: 4 credits

Required text: *Groups and Symmetry,* Armstrong, M.A., Springer-Verlag New York, 1988. ISBN: 978-1-4757-4034-9

## Real Analysis I (110.405)

Syllabus

This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.

Prerequisite: Linear Algebra and Calculus III.

Credits: 4 credits

Required text: *Way of Analysis*, Jones & Bart, ISBN: 9780763714970

## Introduction to Topology (110.413)

Topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits.