Mathematical Systems

This program is built around intensive study of several fundamental areas of pure mathematics, including Abstract Algebra, Real Analysis, and other areas such as Geometry and Topology. The program will also include seminars on the culture, history, and philosophy of mathematics.

The work in this advanced-level mathematics program is quite likely to differ from students' previous work in mathematics, including Calculus, in a number of ways. We will emphasize the careful understanding of the definitions of mathematical terms and the statements and proofs of the theorems that capture the main conceptual landmarks in the areas we study. Hence, the largest portion of our work will involve the reading and writing of rigorous proofs, often in the context of axiomatic systems. These skills are valuable not only for continued study of mathematics but also in many areas of thought in which arguments are set forth according to strict criteria for logical deduction. Students will gain experience in articulating their evidence for claims and in expressing their ideas with precise and transparent reasoning.

In addition to work in core areas of advanced mathematics, we will devote seminar time to looking at our studies in a broader historical, philosophical, and cultural context, working toward answers to such critical questions as: Are mathematical systems discovered or created? In what sense do mathematical objects actually exist? What is current mathematical practice? How did the current mode of mathematical thinking come to be developed? What are the connections between mathematics and culture?

This program is designed for students who intend to pursue graduate studies or teach in mathematics and the sciences, as well as for those who want to develop a detailed understanding of mathematical thinking at a high level. Program activities will include lectures, workshops, seminar discussions and essays, weekly homework assignments, and regular quizzes and exams in each area of study. The material will be challenging and will require a very substantial time commitment outside of class in addition to consistent in-class participation. Students will be expected to complete several substantial homework assignments each week and to bring that learning to the classroom through active engagement with classmates. A strong emphasis will be placed on developing an inclusive learning community which fosters mutual support and success.