Abstract

What does it mean to do a PhD in mathematics? Beyond advanced courses and seminars, a doctorate is about stepping into the unknown, asking questions, and developing the tools to answer them. It is training in the scientific method: observing, conjecturing, testing, and proving.

In this seminar, I will illustrate this process through my research in celestial mechanics. The n-body problem, which models the motion of several masses under gravity, looks simple but hides extraordinary complexity. Of special interest are periodic orbits — elegant solutions where the bodies follow repeating patterns. These orbits are mathematically rich and practically relevant for space science.

The process can be summarised in three steps: 

• See it: computational and variational methods reveal candidate orbits.

• Say it: we classify and characterise them, investigating their stability using tools like the Morse index.

• Sorted: we rigorously prove their existence with computer-assisted tools.

This cycle mirrors the essence of research: curiosity, structure, and proof.