# COURSES

### 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

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.

# Group Projects

• The Ham Sandwich Theorem
• Applications of Topology – Persistent Homology
• The Characterization of Natural Numbers in terms of the Sum of Three Squares

# SPEAKERS

Shira Faigenbaum
Research Assistant Professor – Computational Palaeography

Shira Faigenbaum-Golovin is a Phillip Griffiths Assistant Research Professor at Duke University’s math department as well as at the Rhodes Interdisciplinary Initiative. Her research ranges between the analysis of high-dimensional data from a geometrical perspective and addressing research questions raised by Digital Humanities (archeology, art, evolutionary anthropology, geophysics, and texts). During her master’s and Ph.D., Shira studied the enigma of ostraca (texts written in ink on ceramic potsherds). She was fascinated by these ancient, difficult-to-read texts that are more than 2600 years old. At that time digital humanities and especially computerized paleography were in their infancy. Our research group developed mathematical and statistical methods that provided a toolbox to study these inscriptions. Using these tools, we unveiled hitherto invisible texts, shed light on the level of literacy in 600 BCE, and helped to address long-debated questions.