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Courses

Mathematics

Cryptography
How can two people communicate secrets back and forth, even when a third party can eavesdrop on everything they say? Modern cryptography is an increasingly important field which helps to answer this question. It draws from a variety of mathematical and math-related fields, such as number theory, computational complexity, and the theory of algorithms. Topics may include some basic encryption-breaking techniques, the discrete logarithm problem, integer factorization and primality testing, probability and combinatorics, elliptic curves, complexity theory, and P versus NP. The course will introduce several modern techniques, such as Diffie-Hellman key exchange, ElGamal encryption, RSA, and probabilistic prime tests. While this is not a computer programming course, those interested in the subject may find this course an illuminating introduction to some theoretical aspects of the field.

Session 1

June 25 - July 14

Prerequisite(s)

Completion of an algebra course.

Age and grade requirements:

1. 9th - 10th grade at the time of application.
2. age 14 - 16 on the first day of the session.
Introduction to Logic
This course is an introduction to Symbolic Logic. It shows how to encode information in the form of sentences in Symbolic Logic; it shows how to reason with information in this form; and it provides an overview of logic technology and its applications -- in mathematics, science, engineering, business, law, and other fields. Topics include the syntax and semantics of Propositional Logic, Relational Logic, and Herbrand Logic, validity, contingency, unsatisfiability, logical equivalence, entailment, consistency, natural deduction (Fitch), mathematical induction, resolution, compactness, soundness, completeness.

Session 1

June 25 - July 14

Prerequisite(s)

Completion of an algebra course, as well as comfort with concepts of sets and set operations, such as union and intersection.

Age and grade requirements:

1. 8th - 9th grade at the time of application.
2. age 13 - 15 on the first day of the session.
Number Theory
Number theory, the study of properties of integers, has attracted the interest of mathematicians for over 4000 years. This branch of mathematics continues to be an area of intrigue and active research. For some, the attraction is the possibility of solving a problem that has remained unsolved for hundreds of years; for others it is the pure beauty of a branch of mathematics where the basic concepts are easy to understand, yet the techniques are deep and intricate. Number Theory is also important for its applications in cryptography, which are routinely applied to insure the secure transmission of information over the Internet. In this course, students learn about unique factorization, the Euclidean Algorithm, congruence arithmetic, the Fermat/Euler Theorem, Diophantine Equations, Fibonacci Numbers, continued fractions, and quadratic reciprocity. Students will be given the opportunity to explore a subtopic of their choosing in greater depth as part of a culminating project of the course.

Session 1 & 2

June 25 - July 14
July 17 - August 05

Prerequisite(s)

Completion of an algebra course.

Age and grade requirements:

1. 10th - 11th grade at the time of application.
2. age 15 - 17 on the first day of the session.
Mathematical Puzzles and Games
For centuries, mathematicians have invented, analyzed, and been stumped by a huge variety of challenging puzzles and games. This course offers students the opportunity to study some of the greatest hits, all while learning new math, expanding problem-solving abilities, and having fun. The course's two main subjects of focus are permutation puzzles and impartial two-player strategy games, though students will also study logic puzzles, dissection puzzles, advanced combinatorics, and a variety of other notable mathematical diversions. To fully understand these topics, students will develop analytical skills and explore rich areas outside the standard math curriculum, including numeric bases, mathematical invariance, the golden ratio, and group theory. For the part of the course devoted to permutation puzzles, the main object of study is the Rubik's cube, and it should be noted that while all students will learn to solve the puzzle, they will not do so using typical methods. Instead, students will learn how to create their own unique solutions, and in so doing they will learn generalizable methods for solving any similar puzzle. All experience levels with the cube are welcome: while most students enter the course with little or no prior exposure, those who have already learned to solve the cube master the same generalized methods as they discover solutions to an assortment of more complex puzzles.

Session 1

June 25 - July 14

Prerequisite(s)

Completion of an algebra course.

Age and grade requirements:

1. 9th - 10th grade at the time of application.
2. age 14 - 16 on the first day of the session.
Discrete Mathematics
Discrete mathematics encompasses a broad range of mathematical fields centered on discrete (non-continuous) mathematical structures with an eye toward applications in applied and theoretical computer science. Topics include number theory, set theory, logic, graph theory, and combinatorics. Problems encountered in this field range from easy to very difficult, so this course provides an opportunity to hone mathematical problem-solving skills. Additionally, the course will help students develop proof-writing skills, and it will enable them to build a strong mathematical background for future study in computer science. The course will include applications in the analysis of computer algorithms.

Session 2

July 17 - August 05

Prerequisite(s)

Participants must have completed math courses through pre-calculus.

Age and grade requirements:

1. 9th - 10th grade at the time of application.
2. age 14 - 16 on the first day of the session.
Logic and Problem Solving
This course is for those who delight in solving challenging math problems and who would like to further develop both their problem-solving and their logical-reasoning skills. Problem solving is the activity of the mathematician, and logical reasoning is the framework for this activity. Here we give an introductory course in logic, drawing from examples outside of mathematics but focusing on the use of logic within mathematics. Students are introduced to the basics of propositional and first-order logic, and this gives them access to formal notions of familiar logical methods. Additionally, students discover how their formal understanding can be used directly to help solve certain mathematical problems. But logical reasoning is not all there is to problem solving. Good problem-solving skills include ingenuity, creativity, and the ability to apply a variety of strategies and techniques. In this course, students are taught fundamental tools and standard techniques for problem solving, and they are given the opportunity to develop their mathematical ingenuity through practice on problems in a wide range of difficulty. The mathematical subject areas that the problems are drawn from include set theory, number theory, and combinatorics - none of which require more background than algebra.

Session 2

July 17 - August 05

Prerequisite(s)

Completion of an algebra course.

Age and grade requirements:

1. 8th - 9th grade at the time of application.
2. age 13 - 15 on the first day of the session.
The Mathematics of Symmetry
Symmetry plays an important role throughout mathematics and the sciences, and this course will lead students on an exploration of symmetry as it takes on different meanings in a range of contexts. The familiar concept of geometric symmetry can be used to understand properties of geometric objects, and it plays an important role in art and architecture as well as science and engineering. Symmetry can also be explored mathematically in the context of number systems, algebraic structures, and mathematical puzzles. In addition to an investigation of geometric symmetry, this course will explore a class of popular mechanical puzzles, as well as algebraic properties of numbers. While studying symmetry, students will learn how artists like MC Escher created his intriguing designs and planar tessellations. Students will see how symmetry operations have properties shared by numeric operations and also by puzzles such as Rubik's Cube. The course will introduce students to the abstract mathematical objects known as a group, which is used to capture the essence of these various kinds of symmetry. Group Theory is a branch of mathematics that grew out of geometry, algebra, and number theory and has a wide range of applications other branches of mathematics, as well as many scientific and engineering fields. This course will develop the basic terminology and tools of group theory and apply them to symmetry, puzzles, number systems, and other mathematical problems. The course will help students develop problem solving skills through challenging and engaging problems on a range of topics.

Session 2

July 17 - August 05

Age and grade requirements:

1. 9th - 10th grade at the time of application.
2. age 14 - 16 on the first day of the session.