Term 1

Week 1: Wednesday 2nd October

Arjun Sobnack Delayed parabolic regularity for the Curve Shortening Flow

The Curve Shortening Flow (CSF) is a parabolic partial differential equation (PDE) applied to an evolving curve, which at each time insists that a curve moves to locally decrease its length in an optimally efficient way. Being a PDE, the CSF can be analysed via parabolic regularity, which roughly states that the smoothness of a solution to a parabolic PDE improves in time; classically, this improvement manifests instantaneously.

In this talk, I will introduce the italicised notions above and describe a novel phenomenon, whereby in contrast to the classical case, one must wait for a fixed, but geometrically understandable, length of time before parabolic regularity begins to kick in.

The content of this talk is joint work with P. M. Topping.

Week 2: Wednesday 9th October

Nicola Rosetti 

Week 3: Wednesday 16th October

Maryam Nowroozi Perfect powers in elliptic divisibility sequences

Let be an elliptic curve over the rationals given by an integral Weierstrass model and let be a rational point of infinite order. The multiple has the form where are integers with and coprime and positive. The sequence is called the elliptic divisibility sequence generated by . In this talk we answer the question posed in 2007 by Everest, Reynolds and Stevens: does the sequence contain only finitely many perfect powers?

Week 4: Wednesday 23th October

Eva Zaat Thinking Mathematically

Most people will have come across a maths problem that they could not immediately solve at some point. You have likely felt stuck on a maths exercise yourself. So, how do we get un-stuck?

In this talk, we’ll break down the problem-solving process we use whenever we tackle a maths problem by creating a simple rubric. We’ll also discuss general strategies you may already be using to explore your options when you get stuck. You might think, “I’m a postgrad in a maths-related field; I know how to do maths.” That’s true! But have you never wallowed in despair because you cannot solve the problem? If not, do you think that will not happen during the next few years? Even if you are that brilliant, can you teach someone else to do it? Clearly communicating your thought process is a valuable skill, especially when sharing (in progress) research. Or, if you’re a TA or supervising this year, helping your students think mathematically will greatly benefit them (and you if you are also marking their work). The aim of this talk is to help improve our mathematical thinking and communication skills.

Week 5: Wednesday 30th October

Alessandro Cigna (King's College) Surfaces and the Thurston norm in 3-manifolds

A classical strategy for studying the topology of a manifold is to analyze its submanifolds. The world of 3-manifolds is rich and diverse, and we aim to explore the complexity of surfaces contained within a given 3-manifold. After reviewing the fundamental definitions, we will introduce the Thurston norm, a seminorm on the second real homology of a compact orientable 3-manifold. Expect engaging visuals and detailed examples!

Week 6: Wednesday 6th November

Tommaso Faustini Moduli spaces of triangles

In this talk, we will explore the concept of moduli spaces, which serve as powerful tools for studying the behavior of mathematical objects and their deformations. We will begin by defining what a moduli space is and discuss its significance in the broader context of mathematics. Along our journey, we will address various challenges and misconceptions that arise, refining our definition of moduli spaces in the process.

To illustrate these concepts, we will use the toy example of moduli spaces of triangles, providing a concrete framework to visualize how different triangles can be categorized and deformed within this space.

Week 7: Wednesday 13th November

Darragh Glynn The Combinatorics of Rational Maps

Rational maps — the familiar quotients of complex polynomials that we all meet in school — are fundamental throughout mathematics. In this talk, we will explore the hidden combinatorics that govern these maps and explain how some long-standing problems (such as the famous Hurwitz Realisability Problem) can be approached entirely combinatorially. If time permits, we will also relate this to dynamics.

Week 8: Wednesday 20th November

Nicola Ottolini (Roma Tor Vergata) Unlikely intersections in Diophantine geometry

Starting from Mordell, many conjectures have been put forward (and proved) about how geometry influences the behaviour of diophantine problems. It turns out that many of them can be put in a common framework about varieties that for dimensional reasons we do not expect to intersect. Whenever they do we say that this intersection is "unlikely".

In this talk we will introduce the tools to state a quite broad conjecture of this type, due to Zilber and Pink, and look at some special cases and consequences.

Week 9: Wednesday 27th November

Dan Roebuck Time Dependent Knowledge

Imagine there is some mathematical object you're interested in. You do not know exactly what this object is, but you have a black box that every second tells you slightly more about the object. If in the limit (as time tends to ) the object is completely determined, then there is a sense in which you know what the object is. We will explore a way in which topology can be used to quantify the extent to which you know what the object is, and we will discuss how you can work with these partial descriptions instead of the object itself.

Term 2