# Warwick Imperial Mathematics Conference (WIMP)

**Autumn Warwick and Imperial Mathematics Conference 2020**

**Organised by Ziad Fakoury and Jacob Stephenson of the University of Warwick, and Gautum Chadhuri and Calle Sönne of Imperial College London**

**28 ^{th} – 29^{th} November **

**Warwick and Imperial Mathematics Conference, Autumn 2020**

**What is WIMP?**

The Warwick Imperial Mathematics Conference (WIMP) is a biannual conference primarily aimed at late undergraduates and early graduate students, an opportunity for students to exchange work and see what is happening in related fields. We will host 9 talks from on a range of topics in both applied and pure mathematics. Each talk is 1 hour in length with breaks in between.

**Attending WIMP**

The event is open to all and is free to attend. The Autumn meeting for WIMP 2020 will take place virtually on the 28th and 29th of November, and the talks will be streamed on Microsoft Teams. This year we have the opportunity to host speakers and attendees from around the world and to specialise segments of the conference to suit attendees’ interests.

To sign up, please use this form, so we can add you to the MS Teams chat. WIMP is organised and led by students of the University of Warwick and Imperial College London. Speakers are either current students or recent graduates of these universities.

Guidance for how to attend can be found and the bottom of this page.

**Abstracts **

Quirin Vogel: Can we make flat Swiss cheese? 10 am, 28^{th} Nov

In our talk we will give a gentle introduction into the random walk and properties of its range. We will point out several features of the random walk, amongst the end, we will briefly introduce our work for the planar case and point out some open questions.

Sean Thrasher: Adiabatic Transitions in Quantum Mechanics 11 am, 28^{th} Nov

The study of adiabatic perturbation theory is an exciting and rich field in quantum mechanics, which, since its founders Born and Fock in the 1930s, has developed into a theory that has wide uses in molecular chemistry and solid state physics. The prototypical problem addressed by the theory involves computing the probability of transitions in a quantum system, when the energy of the system is slowly changed. The history of this problem is rich and interesting, where two main approaches are taken: older papers tend to investigate the problem of adiabatic quantum transitions by guessing the correct form of the solution and improving it, whereas the modern approach is to emphasise effective equations of motion throughout all stages of the construction. The latter approach has the advantage of being more applicable since it can provide one with much better physical insight. However, papers such as the one written by Betz and Teufel in 2005 use an intricate process known as iteration and get an approximate answer. This project, on the other hand, takes a novel approach, aiming to improve the methods used in the literature. This new approach could shed light on 3 a class of problems in physics and chemistry known as adiabatic problems. The Born Oppenheimer model, key to understanding molecular dynamics in chemistry and physics, is a prime example where the theory could be applied in real life. There are also adiabatic problems in solid state physics, which could have uses in technology and industry.

Tom Slattery: On Fibonacci Partitions 1:30 pm, 28^{th} Nov

The number of partitions of a number 𝑛 is defined to be the number of ways to write n as a decreasing sum of positive integers, for example the number of partitions of 4 is 5. Similarly we define the number of Distinct Fibonacci Partitions to be the number of ways to write n as the sum of decreasing distinct Fibonacci numbers. We prove an exact formula for OEIS A000119, which counts the number of Distinct Fibonacci Partitions. We also establish an exact formula for its mean value, and determine the asymptotic behaviour.

Lilybelle Cowland Kellock: The Herbrand-Ribet Theorem on the Class Number of Cyclotomic Fields 2:30 pm, 28^{th} Nov

We will look at a strengthening of Kummer’s theorem that “𝑝 divides the class number of ℚ(𝜁𝑝)if and only if 𝑝 divides the numerator of some Bernoulli number 𝐵𝑛 ”, for 0 < 𝑛 < 𝑝 − 1, known as the Herbrand-Ribet theorem.This result is initially surprising, as it relates something defined algebraically (ideal class groups), to something defined analytically (Bernoulli numbers). We will see how the Herbrand-Ribet theorem gives a construction as to why this theorem of Kummer’s is true. We will study the proof of Herbrand’s theorem, which invokes Kummer’s congruences, which has connections to 𝑝-adic 𝐿- functions. Then we will talk about how Ribet’s converse can be proved and how this fits into the context of the Main Conjecture of Iwasawa theory, which reveals a deep relationship between 𝑝-adic 𝐿-functions and ideal class groups of cyclotomic fields.

Kevin de Jong: Pro-𝑝 groups and their properties 4:00 pm, 28^{th} Nov

In this talk we shall examine pro-𝑝 groups, and prove some interesting facts about them (as cleverly hinted at in the title). We shall first start off with some basic definitions and proofs from group theory, as well as quickly going over some of the properties of the 𝑝-adics, so everyone is clear on notation etc. We shall then move on to finite powerful 𝑝-groups, before introducing the concepts of a profinite group and that of a pro-𝑝 group. We shall then prove some pretty topological and group theoretic properties of pro-𝑝 groups, before sketching a proof of their correspondence with Lie lattices.

Ken Lee Schemes: The Manifolds of Algebraic Geometry 10:00 am, 29^{th} Nov

Schemes were first introduced in their current generality by Grothendieck with the goal of uniting number theory with geometry. In this talk, we will first discuss what we are looking for in a “geometric object”, with subsets of Euclidean space as motivation, leading us to the definition of a locally ringed space. At this point, we will have enough to state the Spec −Γ adjunction, which roughly says that there is a functor turning rings into spaces, Spec, that is “inverse” to taking the ring of global functions on a space, Γ. The spaces yielded by Spec are what are called affine schemes, and spaces that are locally affine schemes, are the schemes. This is analogous to how manifolds are defined to be spaces that are locally Euclidean. Along the way, I will showcase some basic but important ideas such as infinitesimals, which under the language of schemes generalise to seemingly non-geometric situations. The approach taken is light on sheaf theory, so it won’t be like choking on Hartshorne.

Nicolas Manrique: Adventures in the Cobordism Category 11 am, 29^{th} Nov

Have you ever looked at a torus and thought “Hey, that sure looks like multiplication by 2”? No? Are you wondering what that even means? Then this could be the talk for you. A cobordism between two 𝑘-manifolds 𝑀 and 𝑁 is an (𝑘 + 1)-manifold whose boundary is the disjoint union of 𝑀 with 𝑁 – a recurring example will be the topological pair of pants. These things are of interest in their own right, but they also form the collection of morphisms in a certain “cobordism category”. In recent times these categories have become natural homes for invariants in low-dimensional topology, most particularly through the use of so-called topological quantum field theories, or TQFTs. This talk will serve as an accessible and picture-driven introduction to the language of these categories, the application of some simple TQFTs, and the invariant properties we can get out of them. By the end, you too will be able to draw pants.

Tudor Ciurca: Model Categories and Whitehead’s Theorem 1:30 pm, 29^{th} Nov

Model categories provide an abstract framework to do homotopy theory. Whitehead’s theorem, a hallmark of algebraic topology, tells us that the homotopy group functors are enough to identify when continuous maps between CW complexes are homotopy equivalences. I will go over the general details of model categories and how one can prove Whitehead’s theorem by constructing a certain model structure on the category of topological spaces. The constructions involved are some of the most elegant pieces of mathematics I have seen over the past year, and you should definitely give this talk a go if you are a fan of category theory, algebra or topology. A key reference is Mark Hovey’s book on model categories.

Sara Veneziale: Homology, Cohomology and the ICSS 2:30 pm, 29^{th} Nov

Given a map 𝑓 ∶ 𝑋 → 𝑌 of topological spaces, can we deduce the homology and cohomology groups of 𝑓 (𝑋 ) from those of 𝑋? What about the cup product structure? One answer is given by the Image Computing Spectral Sequence constructed on the multiple point spaces of 𝑓 (Mond and Goryunov). In this talk we will go through the building blocks of the ICSS, seeing its power with a practical example, and go through the reason why calculating the cup product with this method can give us some problems.