**CTY’s mathematics, science, and computer science courses are dedicated to Dr. Richard P. Longaker, Provost of Johns Hopkins University from 1979 to 1987, in recognition of his advocacy and guidance through CTY’s initial years.**

Mathematics can be described as a language, a tool, a science, and an art. Computer Science is an area of study that continues to gain importance for its rich theory and widespread applications in the physical and social sciences. In CTY’s math and computer science courses, students move beyond basic skills to gain greater understanding of both the underlying principles and the intriguing ways these concepts can be applied and extended to a range of contexts.

Economics is an essential tool for understanding the underpinnings of modern society. In CTY’s economics courses, students gain an understanding of the central concepts behind trade and finance on both the local and international scale.

In all of these courses, students have the opportunity to enrich or accelerate their studies. They investigate advanced concepts through a process of discovery and engagement that promotes a lifelong interest in these disciplines. Through hands-on, thought-provoking exercises, students learn to make connections between abstract ideas and their use in a range of fields, including science, engineering, and advanced mathematics.

Students who wish to receive credit or placement for CTY courses should consult with their home schools before registering to determine their schools’ policies and requirements because these vary widely.

**Math, Computer Science, and Economic Courses**

Review eligibility for minimum test score requirements for math, computer science, and economics courses.

**Mathematics**

- Cryptology
- Advanced Cryptology
- Probability and Game Theory
- Mathematical Logic
- Number Theory
- Topology
- Individually Paced Math Sequence

**Computer Science**

**Economics**

Sample syllabi for all courses are also available with each course description. Please note that the sample syllabi are meant to provide an idea of the level of the course and whether or not the content area interests you. CTY instructors are given guidelines within which they each develop their own syllabi. Choice and sequencing of topics within the content area, as well as specific activities, labs, and assignments, will vary.

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

Cryptology is the study of the codes and ciphers used to create secret writing. In this math course, students begin their journey with an exploration of many early techniques for creating secret writing, such as cipher wheels, the Caesar shift, polyalphabetic substitution, and the Vigenère cipher. They move on to learn about modern techniques including RSA public key cryptography, as students explore how data transmitted by computers can be secured with digital encryption. Discussions about the vulnerabilities of each encryption system enable students to attack and decrypt messages using techniques such as frequency analysis and cribbing. Students apply the concepts learn to encrypt and decrypt their own secret messages.

Though the course’s central focus is on the mathematics of cryptology, the historical context of cryptography and cryptographic devices is provided to further develop an understanding of this branch of mathematics. For example, students examine the design and fallibility of the Enigma Machine, one of the most important cryptographic devices in history.

**Sample text:** *The Code Book*, Singh.

**Session 1:** Carlisle, Hong Kong, Lancaster, Los Angeles, Saratoga Springs, Seattle**Session 2:** Carlisle, Lancaster, Los Angeles, Saratoga Springs, Seattle

**Prerequisite**: CTY Summer Programs’ Cryptology course.

- Sample syllabus 1 (PDF)

In today’s environment, data transmission and data security play an increasingly critical role in the global marketplace and in national security. Picking up where CTY’s Cryptology course leaves off, this class delves further into the rich mathematics behind the science of secret keeping.

Students build upon key concepts covered in the first-level course and progress into the study of more complicated cipher techniques and additional topics from number theory. Students also examine how other areas of mathematics and computer programming are applied to cryptology. For example, they employ computational techniques to learn about statistical attacks on cryptosystems.

In addition, students deepen their knowledge of historical cryptography systems. They develop a more nuanced understanding of the techniques used in breaking the Enigma devices used during World War II and investigate the inner workings of the M-209, a device of the same era that was used primarily by the United States military. Students leave this course with an advanced understanding of the mathematical basis and history of cryptology.

**Note:** A TI-84 graphing calculator is required for this course.

**Sample text:** Materials compiled by the instructor.

**Session 1:** Lancaster**Session 2:** Not offered

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I.

The study of probability and game theory allows students to apply math to real-world situations. Game theory is a branch of mathematics focusing on the application of mathematical reasoning to competitive behavior. In this course, students develop familiarity with some of the major tools of game theory, including dominance, mixed strategies, utility theory, Nash equilibria, and *n*-person games. In doing so, students also learn the necessary mathematical tools from probability and linear algebra to analyze and develop successful strategies in these games.

The games “played” by a budding game theorist in this course are abstract representations of real-life situations. For example, Nash equilibria have been used to solve questions of political competition and analyze penalty kicks in soccer, and the Prisoner’s Dilemma has been used to analyze the social networks of different populations. Class exercises involve individual and group work, and may also include class tournaments.

**Sample texts:** *Game Theory and Strategy*, Straffin; *Thinking Strategically*, Dixit and Nalebuff.

**Session 1:** Carlisle, Lancaster, Los Angeles, Saratoga Springs**Session 2:** Carlisle, Lancaster, Los Angeles, Saratoga Springs

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I.

In this course, students learn about and practice what most mathematicians spend their time doing: proving or disproving conjectures. Students are introduced to the techniques of formal proof and counterexample. While the focus is on mathematical proofs, the logical reasoning skills developed serve as the building blocks of intellectual inquiry.

Students learn about the key concepts of logic, including validity, soundness, consistency, and satisfiability. They develop systems of logic in formal symbolic languages, including the propositional calculus and first-order quantified logic. These systems allow students to test the validity of arguments and write formal proofs with precision. Students explore the syntax (rules of grammar) and semantics (meanings of the symbols) of these languages. They then engage in metalogic: reasoning logically about a system of logic. They examine soundness, asking whether all proofs in the system really prove their conclusion, and completeness, considering whether every valid conclusion can actually be proven using the rules of that system.

By the end of the course, students become proficient at a skill essential to mathematicians—the ability to write proofs accurately and rigorously. Most importantly, students develop strong problem-solving skills and learn to think analytically, traits vital for rigorous inquiry in any field.

**Sample texts:** *Logic: Techniques of Formal Reasoning*, Kalish, Montague, and Mar.; *Formal Logic: Its Scope and Limits*, Jeffrey.

**Session 1:** Baltimore, Hong Kong, Lancaster**Session 2:** Baltimore, Lancaster, Los Angeles

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisites:** Geometry and Algebra II.

Called “the queen of mathematics” by the great mathematician Carl Friedrich Gauss, number theory is the study of the natural numbers, the number system from which all others are derived. Despite the simplicity of the natural numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture, formulated in 1742, which posits that every even integer larger than 2 is the sum of two prime numbers, has defied all proof attempts.

In this proof-based course, students are introduced to the major ideas of elementary number theory and the historical framework in which these concepts were developed. While strengthening their ability to analyze and construct formal proofs, students explore topics such as the Euclidean Algorithm and continued fractions, Diophantine equations, Fibonacci numbers and the golden ratio, modular arithmetic, Fermat’s Little Theorem, RSA public key cryptography, and Fermat’s Two Square Theorem. Students leave the course with an appreciation of the elegance of theoretical mathematics and the ability to craft rigorous arguments.

**Sample text:** *Number Theory: A Lively Introduction with Proofs, Applications, and Stories*, Pommersheim, Marks, and Flapan.

**Session 1:** Lancaster**Session 2:** Lancaster

- No sample syllabus available; new course

**Prerequisites:** Geometry and Algebra II.

Topology is the mathematical study of shapes and space that considers questions such as, “what objects that visually seem quite different share the same properties?” One of the major fields of mathematics, topology possesses wide-ranging applications and beautiful theorems with far-reaching consequences. This course introduces students to point-set topology as they delve into bizarre notions of “space” and develop skill with rigorous, proof-based mathematics.

Students begin by tackling the core concepts of sets, topologies, and continuous mappings, before moving on to topological invariants such as compactness, connectedness, and the separation axioms. With these tools in hand, students understand why topology is the study of shape as they explore how to deform shapes and spaces without altering their fundamental properties. This knowledge allows students to see why it took 100 years for mathematicians to prove Poincaré’s 1904 conjecture about the nature of a sphere. The course concludes with a survey of applications of topology, such as how the study of knots influenced our understanding of proteins, or how the study of manifolds led to a deeper understanding of the topological shape of the universe.

**Sample text:** New course.

**Session 1:** Lancaster**Session 2:** Los Angeles

- No sample syllabus available

Individually Paced Math Sequence is designed for students with the interest and ability to accelerate their study of mathematics and who wish to receive credit or placement from their home school. It includes courses in Algebra I, Geometry, Algebra II, and pre-calculus topics including Functions, Trigonometry, Discrete Math, and Analytic Geometry.

Inspired by research conducted by Johns Hopkins University’s Study of Mathematically Precocious Youth (SMPY), this course is designed for students with strong independent learning skills, high self-motivation, and high mathematical ability. Math Sequence allows students to work independently at a pace commensurate with their individual abilities. **For most of the day, students work on their own, reading and solving problems from their textbooks.**

At the beginning of the session, instructors assess the knowledge base of each student with diagnostic testing. The results of this testing, together with information provided by students and their schools, determine the appropriate starting point for each student in our curriculum and allow students to focus their attention on unfamiliar material.

The instructor and teaching assistant monitor each student’s progress carefully and offer support and individualized instruction as needed. Students must demonstrate a mastery of the concepts and skills within each subject area before moving on to the next topic.

**Students who have not consulted with their home school regarding credit and placement should not register for this course.**

For more information, visit the Math Sequence Web Page.

**Sample texts:** *Algebra*, Smith et al.; *Geometry*, Jurgensen; *Algebra II with Trigonometry*, Smith et al.; *Advanced Mathematics*, Brown; *Precalculus with Trigonometry*, Foerster.

**Session 1:** Baltimore**Session 2:** Baltimore

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I.

More than just a programming course, Fundamentals of Computer Science introduces students to three major areas of the discipline: theory and algorithms, hardware systems, and software systems. In the theoretical component of the course, students learn about algorithms, Boolean algebra, binary mathematics, and theory of computation. While studying hardware systems, they gain familiarity with the physical components of computers, digital logic, computer architecture, and computer networks. As students investigate software systems, they are introduced to elements of programming languages, compilers, computer graphics, and operating systems.

**Note:** Students may apply and illustrate some concepts they learn through work on programming projects. Learning a particular programming language is not a goal of the course.

**Sample text:** Materials compiled by the instructor.

**Lab Fee:** $135

**Session 1:** Carlisle, Los Angeles, Saratoga Springs, Seattle**Session 2:** Carlisle, Los Angeles, Saratoga Springs, Seattle

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** CTY’s Fundamentals of Computer Science or at least a “B+” in a high school or college-level computer programming course from an accredited provider.

In order for a computer to find a solution to a particular problem, it is necessary to formalize the problem in terms of a mathematical model, find a suitable algorithm to solve the problem in that model, and then implement the algorithm in a particular programming language. In this class, students learn how to design, analyze, and implement these algorithms.

Students begin by studying data structures such as arrays, lists, stacks, queues, trees, and sets in order to learn different ways of organizing data. Students then analyze many sorting, searching, and graphing algorithms to determine their run-time efficiency. By examining these fundamental algorithms, students learn how design decisions can affect the efficiency and scalability of an algorithm. A series of programming assignments helps students learn how to put these abstract ideas into practice. By the end of this course, students acquire the conceptual tools necessary to model and analyze computational problems.

**Sample text:** *Data Structures and Algorithm Analysis*, Weiss.

**Session 1:** Lancaster, Los Angeles**Session 2:** Lancaster

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I and one of these CTY courses: Mathematical Logic, Number Theory, Fundamentals of Computer Science, or Data Structures and Algorithms.

Beyond programming lies a deeper understanding and appreciation of elegance in computing. In this advanced computer science class, students work with a series of mathematical models to discover the power and limits of computation as well as gain a robust understanding of complexity in computer science.

Starting with the simplest computer models, deterministic finite automata, students build up to increasingly complicated models, including Turing machines, which are equivalent in power to any algorithm for solving problems. They consider classes of computational problems such as regular languages, pushdown automata, and Turing machines. Once the basic framework from automata theory is established, students can move into more complex material from computability and complexity theory, including, for example, the undecidability of the halting problem, reducibility, the Church-Turing thesis, and the P vs. NP problem. A key objective of this course is to help students develop their ability to understand mathematical statements and formulate rigorous mathematical proofs.

**Note:** Students should take this course if they have either a solid foundation in programming and wish to explore more theoretical aspects of computer science or a solid foundation in mathematical thinking and wish to study computability or complexity theory. This is a theory-based course with some applications, not a programming course.

**Sample text:** Materials compiled by the instructor.

**Session 1:** Lancaster**Session 2:** Not offered

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I. Students who are eligible for CTY Intensive Studies humanities courses may also take this course, so long as they have satisfied the Algebra I prerequisite.

Economist Alfred Marshall described economics as “the study of mankind in the ordinary business of life.” How much are you willing to pay for ice cream on a hot summer day? Who is responsible for the cost of pollution? Is there such a thing as a perfectly competitive market? Why has Google been accused of monopolistic practices? Microeconomics provides insights into these questions as it examines how individual buyers and sellers make decisions about allocating limited resources.

In this course, students analyze microeconomic theory and consider it in the context of today’s economic climate. They begin by studying the fundamental concepts of supply and demand curves, price elasticities, market structure, public goods, and externalities. Students build on this foundation to explore topics from the broad range of microeconomics: competition, consumer choice, monopoly, oligopoly, and the role of government in promoting greater efficiency and equity.

By applying mathematical concepts and critical analysis to economic theory, students uncover how economists analyze and predict changes in the behavior of both consumers and producers. Students leave the course with a firm foundation in central topics in microeconomics and a better understanding of practical economic issues that affect us all.

**Sample text:** *Principles of Microeconomics*, Mankiw.

**Session 1:** Baltimore, Carlisle, Los Angeles, Saratoga Springs, Seattle**Session 2:** Baltimore, Carlisle, Seattle

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

**Prerequisite:** Algebra I. Students who are eligible for CTY Intensive Studies humanities courses may also take this course, so long as they have satisfied the Algebra I prerequisite.

What are the key indicators of an economy’s performance? How do governments craft monetary and fiscal policies to promote economic growth? What does it mean for a country to have a trade deficit? Analyzing economies at an aggregate level, macroeconomics—the study of economic systems—explores questions such as these, providing a bird’s-eye view of economic activity.

Students in this course explore fundamental concepts in macroeconomics including national income, economic growth, inflation, employment, money, banking, financial markets, and the role of public policy. Building upon this foundation, students consider the global economy and issues in international trade and finance. Students examine comparative advantage and balance of payments, along with exchange rates and foreign currencies. By applying mathematical concepts to economic theory, students explore how economists analyze and predict changes in the economy.

Through lectures, readings, discussions, simulations, and research, students gain a firm grounding in macroeconomics and an introduction to central concepts in international trade and finance. Throughout the course, they draw from this knowledge to better understand the state of the US and world economies today.

**Sample text:** *Macroeconomics*, Krugman and Wells.

**Session 1:** Baltimore, Carlisle, Hong Kong**Session 2:** Baltimore, Carlisle, Los Angeles, Saratoga Springs

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