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

## Apply Now

To apply for an upcoming course, please fill out the **Application form.**

## College Algebra (110.102)

This introductory course will create a foundational understanding of topics in Algebra. An emphasis will be on applications to prepare students for future courses like Precalculus or Statistics. After a review of elementary algebra concepts, topics covered include: equations and inequalities, linear equations, exponents and polynomials, factoring, rational expressions and equations, relations and functions, radicals, linear and quadratic equations, higher-degree polynomials, exponential, logarithmic, and rational functions.

Prerequisite: None.

Credits: 3 credits

Required Text: *College Algebra*, by Larson, 11th Edition. Online Homework Platform—WebAssign ISBN-13: 9780357454404

**Note: **This textbook is offered through WebAssign to pair with the online assignments. It includes seamless access to the eBook as well as study tools to use throughout the course. If you would like to purchase a physical copy of the textbook you are more than welcome to do so, however WebAssign access is still required.

## Precalculus (110.105)

This course provides students with the background necessary for the study of calculus. It begins with a review of the coordinate plane, linear equations, and inequalities, and moves purposefully into the study of functions. Students will explore the nature of graphs and deepen their understanding of polynomial, rational, trigonometric, exponential, and logarithmic functions, and will be introduced to complex numbers, parametric equations, and the difference quotient.

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)

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)

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)

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 I, or equivalent.

Credits: 4 credits

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

## Foundational Mathematics of Artificial Intelligence (110.110) (Summer Only)

As artificial intelligence models like ChatGPT become increasingly capable and part of our everyday life, the need to understand their inner workings intensifies. This course introduces the mathematical and statistical principles behind machine learning and AI technologies. Students will assimilate basic concepts including math models and performance measurement. They will apply software to build machine learning applications that serve as AI building blocks including linear regression, classification trees, neural networks, and reinforcement learning. Participants will be challenged to assess the quality of their analyses to better understand the opportunities for, and the limitations of AI.

Prerequisites: None

Credits: 4 credits

Required Text: *Fundamentals of Artificial Intelligence: Volume 1 (Introduction to Artificial Intelligence)*, by Talagala, Nisha and Ghanta, Sindhu, ISBN: 979-8795777597.

## Introduction to Data Analysis (110.125) (Summer Only)

This online course introduces students to important concepts in data analytics across a wide range of case studies. Students will learn how to gather, analyze, and interpret data to drive strategic and operational success. They will explore how to clean and organize data for analysis, and how to perform calculations using Microsoft Excel. Topics include the data science lifecycle, probability, statistics, hypothesis testing, set theory, graphing, regression, and data ethics.

Prerequisite: None

Credits: 4 credits

Required Text: None

## Mathematics for Sustainability (110.126) (Summer Only)

Mathematics for Sustainability covers topics in measurement, probability, statistics, dynamics, and data analysis. In this course, students will analyze, visually represent, and interpret large, real data sets from a variety of government, corporate, and non-profit sources. Through local and global case studies, students will engage in the mathematics behind environmental sustainability issues and the debates centered on them. Topics include climate change, natural resource use, waste production, air and water pollution, water scarcity, and decreasing biodiversity. The software package R is used throughout the semester.

Prerequisite: Comfort with algebraic expressions and functions. No prior experience in coding is required.

Credits: 4 credits

Required Text: Mathematics for Sustainability, Roe, DeForest, JamShidi

## Linear Algebra (110.201)

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

## Calculus III (110.202)

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

## Mathematics of Data Science (110.205)

This course is designed for students of all backgrounds to provide a solid foundation in the underlying mathematical, programming, and statistical theory of data analysis. In today’s data driven world, data literacy is an increasingly important skill to master. To this end, the course will motivate the fundamental concepts used in this growing field. While discussing the general theory behind common methods of data science there will be numerous applications to real world data sets. In particular, the course will use Python libraries to create, import, and analyze data sets.

Prerequisite: None

Credits: 4 credits

Required (free) text: *Introduction to Statistical Learning* by James, Witten, Hastie, Tibshirani

## Introduction to 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, transformations of random variables and moment generating functions. Exam P learning objectives and learning outcomes are emphasized.

Prerequisite: Calculus II.

Credits: 4 credits

Required text:* * P*robability and Statistical Inference *by Hogg, Tanis, and Zimmerman, 10th edition, ISBN 9780135189399

## Introduction to Financial Mathematics (110.276) (Summer only)

This course is designed to develop students’ understanding of fundamental concepts of financial mathematics. The course will cover mathematical theory and applications including the time value of money, annuities and cash flows, bond pricing, loans, amortization, stock and portfolio pricing, immunization of portfolios, swaps and determinants of interest rates, asset matching and convexity. A basic knowledge of calculus and an introductory knowledge of probability is assumed.

Prerequisite: Calculus I or equivalent.

Credits: 4 credits

Required text:* * I*nterest Theory: Financial Mathematics and Deterministic Asset Valuation* by Francis and Ruckman. ISBN-13: 978-0998160405

## Introduction to Proofs (110.301)

This course will provide a practical introduction to mathematical proof, both as they have been done for centuries, and using a modern technological theorem prover. The course begins with the basic building blocks of mathematics: propositional logic, set theory, functions, and relations. These foundational tools lead to answers to questions that are surprisingly difficult, like “what are numbers?” Students will be exposed to mathematical notation and how to create it in digital documents, as well as an “artificially intelligent” proof assistant. The course will conclude with a consideration of the role of A.I. in pure mathematics, particularly as it applies to proofs.

Prerequisite: None.

Credits: 4 credits

## Differential Equations with Applications (110.302)

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, 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.

Credits: 4 credits

## Elementary Number Theory (110.304)

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

Required Text: *Number Theory*, by George E. Andrews; ISBN: 978-0-486-68252-5

## 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, 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 preferred.

Credits: 4 credits

Textbook: *An Introduction to Mathematical Cryptography*, by Hoffstein, Pipher, and Silverman. ISBN-13: 978-1441926746

## Introduction to Abstract Algebra (110.401)

A first introduction to abstract algebra through ring theory and group theory, with an emphasis on concrete examples. The course will introduce basic notions (rings, subrings, groups, subgroups, homomorphisms, quotients) and prove some foundational results (Division algorithm, Lagrange’s theorem, Isomorphism theorems). Examples to be discussed include integers modulo n, matrix rings, polynomial rings, permutation groups, and dihedral groups.

Prerequisite: Linear Algebra.

Credits: 4 credits

Required text: *Abstract Algebra, An Introduction (3rd ed.) *by Hungerford; ISBN: 978-1111569624

## Real Analysis I (110.405)

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

## Real Analysis II (110.406)

This course continues AS.110.405 with an emphasis on the fundamental notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem, functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral.

Prerequisite: Real Analysis I

Credits: 4 credits

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

## Honors Algebra II (110.412)

This is a continuation of Algebra I. Topics include: principal ideal domains, structure of finitely generated modules; introduction to field theory, linear algebra over a field, and Field extensions; splitting field of a polynomial, algebraic closure of a field; Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals.

Credits: 4 credits

Required Text: TBD

## Introduction to Topology (110.413)

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

Credits: 4 credits

Required Text: *Topology*, 2nd Ed., Munkres, J., New Jersey: Prentice Hall, January, 2000, ISBN-10: 0131816292, ISBN-13: 978-0131816299