Course subject: Mathematics (MATH)

This course will explore the basic properties of probability focusing on both discrete and continuous random variables. Topics include: Laws of probability, discrete and continuous random variables, probability distributions, mean, variance, generating functions, Markov chains, problem solving, history of probability.

This course explores the foundations of linear algebra and some of its applications. An emphasis will be placed on the development of mathematical thinking and the importance of proof in mathematical teaching. Topics include: matrices, linear mappings, vector spaces, determinants, diagonalization, inner products, the Fundamental Theorem of Linear Algebra, and the method of least squares.

This course explores the intersection between mathematics and computer science by examining general methods for solving problems efficiently. We will study a variety of algorithm design techniques for problems in diverse application areas and use mathematical models to compare algorithms. Students will be introduced to "big ideas" from computer science, such as the generalization and reuse of previous solutions, and the notion of limits to the power of computation.

This course will use mathematics to study the foundations of problem solving and computation. A decision problem (answering "yes" or "no") can be characterized as a set of strings of characters encoding inputs that yield "yes" answers, and a computer can be characterized as a simple mathematical machine model that processes strings of characters. By studying properties of classes of problems of increasing complexity, we will be able to establish relationships among classes, membership in classes, and hard problems for classes. The course will culminate in the examination of classes and models that correspond to the power of computers, and the demonstration that there exist problems that cannot be solved.

This course explores the foundations of integral calculus and the use of series in approximating the basic functions of mathematics. Topics include: Understanding the Riemann Integral and its flaws, the idea of Lebesque, the geometric meaning of the Riemann-Stieltjes integral, the Fundamental Theorem of Calculus, numerical integration, numerical series, uniform convergence of functions and the extraordinary nature of power series, Fourier Series.

This course is designed to allow the student to discover fundamental facts about geometry through the interactive use of mathematical software. Possible topics include: An introduction to affine, projective and non-Euclidean geometry, conic sections in projective geometry, inversion in circles, the Theorems of Desargues, Pappas and Pascal.

This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. The course will consist of either a one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.

This course aims to develop the student's mathematical problem solving ability. Common heuristics such as problem modification, patterning, contradiction arguments and exploiting symmetry will be examined. A wide range of challenging problems from various branches of mathematics will provide the medium through which these important principles and broad strategies are experienced.

Objectives: To develop the vocabulary, techniques and analytical skills associated with reading and writing proofs, and to gain practice in formulating conjectures and discovering proofs. Emphasis will be placed on understanding logical structures, recognition and command over common proof techniques, and precision in language. Topics Include: rules of formal logic, truth tables, role of definitions, implications, sets, existential and universal quantifiers, negation and counter-example, proofs by contradiction, proofs using the contrapositive, proofs of uniqueness and induction.

The capstone course is designed to give students an opportunity to showcase the knowledge they have gained and to provide a forum for bringing that knowledge into their own classroom. As part of this course students will design a mini-course on an approved subject in mathematics.