Representation of a Number as a Sum of Powers

Instructor: Rena Chu

In 2025, we will reach the first “square” year of this millennium and the only “square” year of this century. Indeed 2025=45^2, preceded by 1936 and followed by 2116. More common are years which are sums of two squares. For example, the last was 2020=42^2+16^2, and the next (not counting 2025) is 2026=45^2+1^2. What do these numbers have in common, if anything? Can 2023 be written as a sum of two squares? If not, what about a sum of three squares, or four? How can we be sure—must we check every possibility? Together we will explore these mysteries of sums of squares and their role in encoding the secrets of the integers.

Squeezing Shapes

Aygul Galimova photo

Instructor: Aygul Galimova

We’ll explore the field of topology, the study of shapes up to stretching and squeezing, comes into play here. Topology studies the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, and bending, but no cutting. Topology comes up in the study of knots and objects, billiards and pool, and even breast cancer detection. This course will cover how to play tic-tac-toe on a donut, glue spaces to get new ones, orientation, knots, the rent-sharing problem, and a few applications. No prerequisites except curiosity.