## ## What is Arithmetic Topology?

Arithmetic Topology is motivated by an heuristic known as the "Knots and Primes" analogy. It was first noticed back in the 60s that certain objects in Number Theory seem to have similar properties to certain objects in topology. In particular, a number ring (e.g. the integers) is "like" a 3-manifold, and a prime of this number ring is "like" the embedding of a knot into this 3-manifold.

Of course naively this sounds ridiculous, how are the integers supposed to be like a manifold in any way? Essentially Mazur noticed that if you equip these arithmetic objects with something known as the étale topology, then the cohomologies of the arithmetic and topological objects look very similar and they obey very similar looking theorems.

An important note is that this analogy is purely a heuristic, so there is no mathematical proof that these objects are related. Arithmetic Topology uses this analogy to motivate potential theorems, results, or constructions. e.g. “There is some theorem Y in topology, what should the arithmetic analogue of this theorem look like?”. Ultimately in order to prove the arithmetic analogue one would still have to use number theory and not topology.

# Research

My research interests broadly include Algebraic Number Theory and Arithmetic Geometry, and my current research focuses on Arithmetic Topology. I am interested in arithmetic analogues of linking phenomena in topology. For example, I am interested in arithmetic invariants arising from Arithmetic Chern-Simons Theory as defined by Minhyong Kim and his collaborators, which can be used to define a notion of arithmetic linking numbers. I am also interested in triple symbols which can be viewed as the arithmetic analogue of milnor invariants, which are used in topology to detect links such as the Borromean Rings.

# Writeups

This section contains writeups and expository articles on various mathematical topics.

- Persistent Homology (2022) - A group project as part of the GlaMS PhD Training, written jointly with Malthe Sporring and Adrián Doña Mateo.
- Deforming Galois Representations (2022) - This is my Cambridge Part III Essay (Master's Thesis), written under the supervision of Professor Jack Thorne.
- On the q-analogue of Kostant's Partition Function (2021) - A summary of my undergraduate summer project supervised by Dr. Rong Zhou.
- More to come!