# Alex

## PhD Doctor of Philosophy candidate

### Project ideas

#### Knot theory

A knot is simply a closed loop of string. You'll learn about: How to represent knots on a page. How knots can be combined, and how to find knots that can't be created by combining other knots. Techniques for determining whether or not two knots are distinct, in the sense that neither can be deformed to match the other. Related objects such as links and braids. Applications of knots in physical sciences.

#### Spectral graph theory

A graph is just a finite set of vertices, with some pairs of vertices connected by edges. Graphs can be described using matrices. What can the eigenvalues and eigenvectors of these matrices tell you about the graph? A lot, as it turns out! Students will need to know some linear algebra for this project.

#### Number theory

How do we know that there are no nonzero integers x, y, and z such that x^4 + y^4 = z^4? How can you tell whether a number is prime? Given numbers n and m, how can you tell whether n differs from some square number by a multiple of m? Can complex numbers tell us anything about ordinary integers? How can you send messages that an eavesdropper won't be able to decipher? These are some of the questions that you'll be able to answer when you learn number theory.