Mathematics of Euler

The Basel Problem

In Mathematics of Euler (Summer Session 2, 2023), we will study the work of Leonhard Euler, one of the greatest mathematicians in history and also the one after whom Euler Circle is named. Since Euler was arguably the most prolific mathematician in history, we will not have a chance to look at all of his work, but we will look at some of his most interesting papers and results. Euler was a pioneer in so many branches of mathematics, from topology to graph theory to analysis to Lie groups to modular forms to the calculus of variations, and as a consequence he was able to discover some of the beautiful early results in these fields before anyone else got to them: he saw mathematics when others did not realize it was there. In this class, we will be motivated by Euler’s work to introduce some of these topics

Euler’s papers, at least the ones that have been translated into English, are wonderfully refreshing to read. While modern authors typically hide their thought process and write only about the things that worked, Euler was eager to tell his readers what he was thinking about, and even what he wasn’t able to do. We can all learn a lot from reading the thoughts of a mathematician at that level; we would do well to follow Laplace’s advice: “Read Euler: he is the master of us all!” And so we shall.

Modern standards of rigor in mathematics were codified in the nineteenth century. Thus Euler, an eighteenth-century mathematician, was not bound by them and was not aware of how rigorous and precise mathematics would become the following century. Thus he was happy to play fast and loose with infinities and divergent series, in ways that modern mathematicians are admonished never to do. But we should never forget how fun it is to play with mathematics and determine what is true and how things work, without worrying too much about what the precise rules of the game are. For example, Euler was the first to make the now-famous and controversial observation that 1+2+3+4+\cdots=-\frac{1}{12}. This isn’t true in the sense of convergence of series, a nineteenth-century concept. But it hints at the very deep and important notion of analytic continuation. All of Euler’s tricks are hints of deep concepts that remain very important in modern mathematics, and we will look at (a small sample of) what they are and where Euler’s work has led in more recent times.

In this class, we assume that students are familiar with calculus, either from having taken a class on it or from studying it on their own.

This class will run for 5 weeks, from July 17 through August 18, meeting Mondays, Tuesdays, Thursdays, and Fridays from 5:00–7:00 PM Pacific time. There may be additional optional (but strongly encouraged) sessions at other times. Students will be expected to do a considerable amount of work outside of class; this is expected to be the students’ primary activity for the duration of the class. The class will primarily be held on Zoom, but we may schedule optional extra in-person sessions for local students if there is sufficient interest.

Applications for this class are due April 9. Click here to apply!