Advanced Topics in Mathematics

  • ETHS Advanced Topics in Mathematics 2021-22

    The Advanced Topics class in the Mathematics Department at Evanston Township High School is a college-level class that provides students who have completed Multivariable Calculus and Linear Algebra an opportunity to continue their studies at the high school.  The class is also open to selected students taking Multivariable Calculus and Linear Algebra concurrently, with the recommendation of their BC Calculus teacher.  The goal of the class is to increase students’ abilities to produce mathematically rigorous arguments, write proofs, and solve interesting problems.  The class meets three times each week at ETHS.  A Ph.D. instructor from Northwestern University leads the class two days each week.  An ETHS Mathematics Department teacher leads the class on the third day.  

    The topics covered vary from year to year, depending on the interest and availability of the Northwestern faculty.  The number of students in the class also varies, and is typically between 15 and 20.  The students participate in high-level discussions in class about mathematics, and work hard independently and collaboratively on challenging weekly problem sets.  One day each week is usually devoted to student-led individual and group presentations of solutions to selected problems.

    The topics covered in 2021-22 are Foundations of Higher Mathematics (Semester 1) and Number Theory (Semester 2).  The texts are Chapter Zero:  Fundamental Notions of Abstract Mathematics by Carol Schumacher (Semester 1), and Elementary Number Theory by David M. Burton (Semester 2).  Past topics include combinatorics, graph theory, topology, real analysis, complex analysis, elementary differential equations, abstract algebra, and non-Euclidean geometry, including introductions to spherical geometry, hyperbolic geometry and projective geometry.  All of the material is rich with challenging and interesting mathematics and well-suited to the goals of the course.