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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 13 - August 1)
Prerequisite(s): Participants must have completed math courses through pre-calculus.
Age and grade requirements: 9th, 10th or 11th grade in Spring 2009, and age 14 - 17 on July 13, 2009.
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Mathematical 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 1 (June 21 - July 10) This course is full for 2009.
Session 2 (July 13 - August 1)
This course is full for 2009. Prerequisite(s): Completion of an algebra course.
Age and grade requirements: 8th or 9th grade in Spring
2009, and between age 13 and age 15 at start
of session.
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Non-Euclidean Geometry
This course is for students who have completed a course in geometry and would like to learn more. The course challenges students to stretch their logical, mathematical, and creative abilities. Topics include a review of the axiomatic development of Euclidean geometry, advanced problem solving, and, centrally, an introduction to non-Euclidean geometries, specifically projective, elliptic geometry and hyperbolic geometry. We shall also investigate some of the curious history of the parallel postulate, the ancient, problematic axiom of Euclid that led to the rise of non-Euclidean geometry. We shall also look at how the discovery of non-Euclidean geometries in the late 1800s shattered the traditional conception of geometry as the true description of physical space. This discovery was fundamental to Einstein's development of the theory of relativity, and it has played an essential role in modern theories of the universe. This course provides an excellent opportunity to reinforce a solid foundation in geometry, build geometry problem-solving skills, and gain a taste of some of the most fascinating developments in the field.
Session 2E (July 13 - August 8)
Prerequisite(s): Completion of courses in algebra and geometry.
Age and grade requirements: 10th or 11th grade in Spring 2009, and age 15 - 17 on July 13, 2009.
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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, and other topics.
Session 1 (June 21 - July 10)
Prerequisite(s): Completion of an algebra course. Students who have taken second year algebra, pre-calculus, or other courses beyond algebra may be placed in a more advanced section.
Age and grade requirements: 10th or 11th grade in Spring 2009, and age 15 - 17 on June 21, 2009.
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Topology: Knots and Surfaces
Is a donut the same as a coffee cup? From a topological perspective, the answer is ''yes.'' The mathematical field of topology is concerned with properties of geometric structures that are preserved when the structure is deformed by stretching and bending without cutting or gluing. In this way, a coffee cup can be topologically transformed into a donut where the handle of the cup becomes the hole of a donut. This course introduces basic concepts from the field of topology, first through the study of knots. For most people, knots are about shoelaces, fishing line, ribbons and ropes, but not about mathematics. However, knot theory is a very important branch of mathematics that is a subfield of topology. Furthermore, knot theory is an area of current mathematical research, and it has applications in physics, chemistry and biology. Beginning with the study of knots, and leading to the study of surfaces, this course will cover topics such as Reidemeister moves, graphs and networks, Euler characteristic, homeomorphism, and classifications of knots and surfaces.
Session 1 (June 21 - July 10)
Prerequisite(s): Participants must have completed courses in algebra, geometry, and trigonometry.
Age and grade requirements: 10th or 11th grade in Spring 2009, and age 15 - 17 on June 21, 2009.
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