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

Week 2: Wednesday 15th January

Alex Bowring Mathematics in the industry

In this talk I’ll give an overview of the work I carry out at as a Senior Mathematical Consultant at the Smith Institute, where we work alongside industry to tackle large-scale mathematical problems that affect everyday life. I’ll also discuss my experience in transitioning from a role in academia to a role in industry. You’ll hear about the variety of problems I have worked on at Smith Institute, before I zoom into our work on the Dynamic Reserve Project, where we collaborated with National Grid to build a machine-learning that predicts national energy reserves. Finally, I’ll wrap up with information about job and internship opportunities currently open at Smith Institute.

Week 3: Wednesday 22th January

Marco Milanesi Persistent homology

Persistent homology is a powerful notion in topological data analysis that allows us to understand the essential topological features of an object. Persistent homology is gaining increasing attention in a variety of applications, including biology and chemistry, astrophysics, automated image classification, sensor analysis, and social networks. In this talk, we will explore a very recent paper on this topic, and I will give you all the mathematical ideas behind it (it turns out that there are some).

Week 4: Wednesday 29th January

Teo Petrov Almost full transversals in equi-n squares

In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in latin squares, conjecturing that every equi-n-square (an n × n array filled with n symbols where each symbol appears exactly n times) has a transversal of size n − 1. That is, it should have a collectio of n − 1 entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-n-squares always have a transversal with size at least 2n/3. In 2019, Pokrovskiy and Sudakov disproved Stein’s conjecture by constructing equi-n-squares without a transversal of size n − (log n)/42 , but asked whether Stein’s conjecture is approximately true. I.e., does an equi-n-square always have a transversal with size (1 − o(1))n? We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-n-squares with no transversal of size n − Ω(√n) and that every equi-n-square contains n − n^(1−Ω(1)) disjoint transversals of size n − n^(1−Ω(1)).

Week 5: Wednesday 5th February

Alejandro Vargas De Leon Introduction to hyperplane arrangements

A hyperplane arrangement is a finite collection of hyperplanes in the affine space K^n, where K for us will be a field that is either finite, the reals, or the complex numbers. This talk is a quick tour through hyperplane arrangements, from its origins to some current developments. We mostly take a combinatorial point of view to explore some geometric questions regarding the union of all the hyperplanes, and also of its complement. The main example of this talk is the braid arrangement, and subarrangements of it. We relate this to graphs and graph invariants. Another combinatorial structure that naturally appears is matroids, as intersection lattices. We also describe the characteristic polynomial, which holds topological information about the complement of the arrangement, and satisfies some recurrence relations known as "deletion and contraction".

Week 6: Wednesday 12th February

Joseph Harrison Sums of irrational powers, o-minimality and functional transcendence

The set of all sums of perfect powers of a fixed irrational number has size asymptotically best possible. This is established using ideas at the intersection of model theory and Diophantine geometry, namely the theory of o-minimality. We will present the proof modulo a technical lemma. Time permitting, we will prove this lemma, which requires a powerful theorem in functional transcendence.

Week 7: Wednesday 19th February

Spyridon Garouniatis Characterizing Compact Subsets of D([0,1]) in the Skorokhod Topology: A Functional Approach

Many stochastic processes, such as Lévy processes and Poisson processes, exhibit discontinuous sample paths, making classical function spaces like the space of continuous functions on [0,1], C([0,1]) inadequate for their analysis. We introduce a metric on the space of right continuous functions, with left limits on [0,1], D([0,1]) the so-called "Skorokhod Topology", which provides a powerful framework for studying the convergence of such processes by allowing controlled deformations of time. This topology is fundamental in probability theory, underpinning key results like the Functional Central Limit Theorem for jump processes and the weak convergence of stochastic processes.

A central question in the study of D([0,1]) is the characterization of compact subsets under the Skorokhod topology. Unlike in C([0,1]), where compactness is governed by uniform equicontinuity (via the Arzelà-Ascoli theorem), compactness in D([0,1]) involves both uniform control on function values and constraints on jump discontinuities. In this talk, we explore necessary and sufficient conditions for compactness in D([0,1]).

Week 8: Wednesday 26th February

Miriam West Introduction to Supernovae

Undoubtably one of the most beautiful, complex and awe-inspiring natural phenomena in the known Universe, supernovae have captured the imagination of scientists and lay folk alike as they provide not only stunning visual displays of the beauty and wonder of the Universe, but challenge the borders of comprehended physics with explosion remnants as remarkable as black holes, neutron stars and magnetars. They act as cosmic probes to test the expansion of the universe and are thermonuclear power houses that chemically enrich their entire environments, providing a birth place for newer generations of stars.

What are supernovae? How do they form? How do their environments affect progenitors and remnants, and how is this field expanding into the future?This talk will address some, none and maybe all of these questions and will be pitched at mathematicians and designed to make you cry as you are exposed to the gross assumptions and abuse of mathematical laws that enable astronomers to conduct science on a cosmic scale.

Week 9: Wednesday 5th March

Ali Sadreddin (Imperial College) Hermitian Decomposition Lattices and Module-LIP

We are going to see particular rank 2 lattices over complex multiplication fields with some symmetries that are used in decomposition of algebraic integers in Hermitian squares. Then we will go through a reduction from Module Lattice Isomorphism Problem in rank 2 over a totally real number field or a Complex multiplication field to find a square basis for these lattices.

Week 10: Wednesday 12th March

Mark Chambers Theta lifting, and how it can "solve" the congruent number problem

Theta lifting is an example of how you can go between automorphic forms/representations for different groups. In a specific instance this is a relation between integral and half-integral weight modular forms, and is known to have many links to special L-Values (which occur in important conjectures such as that of Birch & Swinnerton-Dyer). We aim to explain this story via the example of the congruent number problem, which is an ancient question that asks which n can be written as the areas of Pythagorean triangles.

Term 3

Week 2: Wednesday 30th April

Andrey Chernyshev Normalization flow and Poincaré-Dulac theory

In this talk, we will present a new approach to the normal form theory for systems of ODEs near an equilibrium point. Traditional normalization procedure is step by step: non-resonant terms in the Taylor expansion of the vector field are eliminated first at degree 2, then at degree 3 with another change of variables, and so on. We propose a different method. We consider an infinite-dimensional space of all vector fields with a singular point (equilibrium) at the origin. In this space, we construct a flow generated by a certain differential equation with the following properties. Shifts along the trajectories of this flow correspond to changes of variables, and the flow moves in the direction of the normal form subspace. As a result, the normalization process becomes continuous. The formal aspect of the theory, as in the traditional approach, presents no difficulties. The analytic aspect and the problems of series convergence, as usual, remain nontrivial.

Week 3: Wednesday 7th May

Elena Maini

Week 4: Wednesday 14th May

Cameron Heather

Week 5: Wednesday 21th May

Week 6: Wednesday 28th May

Week 7: Wednesday 4th June

Week 8: Wednesday 11th June

Week 9: Wednesday 18th June

Week 10: Wednesday 25th June