Fundamentals of Higher Mathematics

A good part of the past century of mathematics has been devoted to finding deep connections between seemingly disparate subjects: algebra and topology, geometry and algebra, number theory and analysis, analysis and combinatorics, and many more combinations of 2 or more subjects. A few topics in mathematics make appearances almost everywhere, and are often considered the most important for a mathematician to learn: linear algebra and topology. Basic techniques from abstract algebra make appearances almost everywhere as well. Analytic techniques make surprise appearances as well, but are especially important for understanding how real-life processes behave, such as light and heat. This year long sequence will be an introduction to some of the most basic and important techniques from linear algebra, abstract algebra, point-set topology, and real analysis.

This will be a unique class teaching some of the most important topics from a variety of subjects, and will focus more on breadth over depth. The goal will be to learn prerequisite material needed for our advanced classes such as Analytic Number Theory, Complex Analysis, Differential Geometry, Ergodic Theory, p-adic Analysis, and Ring Theory and Algebraic Geometry. This will make time for more beautiful mathematics in those specific subjects, with less seemingly unrelated prerequisite material covered in those courses than currently.
The classes are as follows:

Fall: Linear algebra and basic abstract algebra
Winter: Point-set topology
Spring: Real analysis

We strongly recommend that students take all three classes as a sequence.

These classes will be slightly less challenging than our most advanced classes, but more challenging than our intermediate classes, and much more challenging than school classes. Introduction to proofs, at the level of our Transition to Proofs sequence, is a prerequisite. Although students in this course may be comfortable with many proof techniques, as we move on to point-set topology and analysis, the proofs might have a different flavor than students are used to, and therefore might be more challenging. Students who take this sequence and master the material in it will continue to develop the mathematical maturity needed for our most advanced classes.

These classes are tentatively scheduled to meet on Tuesday evenings from 5:30–6:30 PM Pacific time and Friday afternoons from 4–6 PM Pacific time, from September 21–December 3 with a break the week of November 23, but this is subject to change. It remains to be seen whether classes will be held online or in-person.

Applications for Fall 2021 are due July 25th. After that, we will continue to accept applications on a rolling basis while spots remain. Click here to apply!