Speaker:Timotheus Keanu

Title: Elliptic Curves II: The Road to Mordell-Weil

Abstract: The Mordell-Weil theorem tell us that any elliptic curve over a number field is finitely generated. Not only this is an impressive result by itself, but the proof of this theorem highlights the rich structure that any elliptic curve has. In this talk, I will give a sketch of the proof and show how ideas from algebra, geometry, and number theory can work together.

Speaker:Samuel Bastida

Title: Friendship and Colouring In

Abstract: In 1941 R.L. Brooks investigated one of the most important problems that had been vexing preschoolers for centuries: "How many colours are required to colour something in?". As it turns out, answering this question in general is (to use a technical term here) "hard", however, Brooks was able to characterise when a graph with maximum degree k is k-colourable. In 2016 Stiebitz and Toft were able to expand Brooks' Theorem to characterise when a graph with maximum friendship k (sometimes called maximum local edge-connectivity k) is k-colourable. It is known that Brooks' Theorem extends to list-colouring (a more general form of colouring that is even "harder" than the original) and so we determine whether or not Stiebitz and Toft's Theorem also extends to list-colouring and obtain some rather surprising results.

Speaker: Johanna Franklin

Title: Algorithmic randomness and its connections to analysis

Abstract: Most people have an intuitive idea of what it means for an infinite sequence of coin flips to be random, but as mathematicians, we'd like to have a formal definition for this concept. Computability theory, a subfield of mathematical logic, allows us to do this. A point in a computable probability space is said to be random if it does not have certain easily-defined measure-zero properties, and different types of properties give rise to different definitions of randomness. In recent years, the resulting notions of randomness have been used to characterize the sets of points that satisfy various theorems in analysis. I will provide an introduction to computability theory and randomness and survey some of these recent results.

Speaker: Rezwana Razzaque Angana

Title: Regularization of Shallow Water Equations: The Role of Weak Nonlinearity and Dispersion in Controlling Singular Shock Waves.

Abstract: The regularization of shallow water equations is a critical area of study aimed at enhancing the physical realism and mathematical stability of models used to describe fluid dynamics in shallow regions, such as coastal areas and rivers. This paper focuses on the impact of weakly nonlinear and dispersive effects on singular shock waves within these systems. By introducing regularization techniques, such as incorporating higher-order dispersive terms and weakly nonlinear corrections, the study seeks to smooth out sharp gradients and manage singularities that arise in traditional shock wave models. The analysis reveals how these modifications lead to the formation of weakly singular shock waves, characterized by continuous profiles with discontinuous derivatives, and explores their stability and propagation behaviour. The findings have significant implications for improving the accuracy of numerical simulations and enhancing our understanding of complex wave phenomena in shallow water environments, particularly in scenarios where traditional models fail to capture the nuanced interplay between nonlinearity, dispersion, and shock wave dynamics.

Speaker: Timotheus Keanu

Title: Stone Duality

Abstract: The duality between two objects is one of the most important principles in mathematics. In this talk, we will discuss a specific example of duality between particular classes of algebraic structures and classes of topological spaces, based on Marshall Stone's works. I will explain the related algebraic and topological structures, how to construct a duality between them, and give some basic examples.

Speaker: Roy Jansen

Title: The Two-Headed Snake Groupoid: Steinberg Algebras and Ideals

Abstract: The n-headed snake groupoids are often used as examples of
non-Hausdorff groupoids. This (very casual!) talk will give
entry-level background to topological groupoids and Steinberg algebras
over a field K, with the example of the two-headed snake. We will look
at ideals of singular functions and their independence from K. If we
have time I might introduce the three-headed snake, where the
characteristic of K does weird things to the ideals ...

Speaker: Ellen Hammatt

Title: Structures Computable Without Delay

Abstract: In this talk we investigate what happens when we take concepts from computable structure theory and forbid the use of unbounded search. In other words, we discuss the primitive recursive content of structure theory. This central definition is that of punctual structures, introduced by Kalimullin, Melnikov and Ng in 2017. We investigate various concepts from computable structure theory in the primitive recursive case. A common theme is that new techniques are required in the primitive recursive case. We also discuss a degree structure within punctual presentations which is induced by primitive recursive isomorphisms. This degree structure is a new concept that does not arise in computable structure theory.

Speaker: Jago Edyvean

Title: Spiders?

Abstract: Spiders have long been the subject of many myths and internet rumours but are any of them actually true. These myths include "are daddy long-legs truly the most poisonous spiders?" ; "do we really swallow an average of eight spiders a year while asleep?" ; and "was Shelob right all along?" These questions and more will be asked in this talk.

Speaker: Daniel Wrench

Title: Who gives a toss? The statistics of coin-flipping

Abstract: "Many people have flipped coins, but few have stopped to ponder the statistical and physical intricacies of the process." So begins the abstract to a remarkable study on coin-flipping published last year. In this seminar I will be exploring those statistical and physical intricacies, as well as the history of this iconic game. Take a chance and join me and some very meticulous experimenters on a journey of counter-intuitive probability.

Speaker: Ellen Hammatt

Title: The punctual degrees of the rational numbers

Abstract: In this talk, we investigate computable structure theory without unbounded search. We call such structures, punctual or informally, structures computable without delay. We use the concrete notion of primitive recursion to formalise computable without delay, this notion is nice to work with since it has a reduced version of the Church-Turing thesis. The inverse of a primitive recursive function is not necessarily primitive recursive. This induces a natural degree structure within the punctual presentations of a particular structure. We focus on investigating whether we can embed the atomless Boolean algebra into the punctual degrees of various linear orders, in particular the rational numbers.

Note that this is a talk for me to practice for an overseas conference, so it is for an expert audience. However it is a 25 min talk, so we can use the rest of the time to clarify and discuss some aspects. I will also start with an extra introduction to clarify a few definitions to hope to make the talk more understandable!

Speaker: Timotheus Keanu

Title: Mordell-Weil Groups over Elliptic Curves

Abstract: Elliptic curves is a type of algebraic curves that has many applications in number theory due to its arithmetic nature. They have lots of interesting properties. For example, the Mordell-Weil theorem tell us that an elliptic curve over the rational numbers is a finitely generated abelian group. In this talk, we will explore this result in more detail and discuss the challenges faced in the current research.

Speaker: Grace Jacobs Corban

Title: Hexaflexagons are the bestagons

Description: Lets flexigate our way to happiness with the most fun you can have on six sides! In this seminar we will introduce the most influential mathematical object of the 20th century, that some call Feynman's greatest contribution to humanity, and go over how to make a 3-sided, 6-sided, and maybe 12-sided hexaflexagon. Hexaflexagons are paper folding toys, where 'flexing' reveals a new side and a new colour. How does this work? Where do the colours come from?!?! Come along to find out 🙂

Speaker: Sam Bastida

Title: The Mathematics of Juggling

Abstract: A mathematical model of juggling, known nowadays as site swaps, was developed around 1985 independently by groups in both the United States and United Kingdom. In this model numbers are assigned to the common throws jugglers use and a sequence of such numbers describes a juggling pattern or site swap. Not all site swaps are valid and the sequences that are valid correspond to affine permutations which are relevant for the construction of affine symmetric groups. In this talk I will discuss some of the mathematical ideas behind site swaps as well as exploring the various ways they have been extended to apply to different juggling tricks, I will also give some live demonstrations.

Speaker: Mashfiqul Huq Chowdhury

Title: EM Algorithm and Gaussian Mixture Model

Abstract: We usually aim to fit a suitable probability distribution to the observed data using maximum likelihood estimation (MLE). In practice, the complex structure of the data motivates us to include latent variables in our model. However, finding the MLE of parameters, in this case, is challenging because it involves non-convex optimization. In such a setting, the EM algorithm provides an efficient approach for finding the MLE of the parameters by constructing the lower bound (ELBO) of the likelihood of the data. The EM algorithm iteratively optimizes ELBO objective function until convergence. The Gaussian mixture models (GMM) are widely used for unsupervised clustering tasks in Machine learning. The application of the EM algorithm on the GMM will also be discussed. This talk will finish by showing an illustration of GMM for clustering purposes.

Speaker: Mark Bishop

Title: Effects of observed projections on turbulence statistics in the intracluster medium

Abstract:The total mass of a cluster is one of its most fundamental properties. Measurements of the galaxy cluster mass often relies upon assuming hydrostatic equilibrium. However, this is often invalidated as the intracluster medium (ICM) is continuously disturbed by mergers, feedback processes, and motions of galaxies. These processes generate gas motions that contribute nonthermal pressure; typically turbulence, that leads to an underestimation of the mass by as much as 30\%. We can measure turbulence through indirect probes that come in the form of fluctuations in the X-ray surface brightness and Sunyaev-Zeldovich effect maps. These are projected characteristics, encoding the 3D structure of the turbulence in the ICM. This project will be an analysis of the 3D to 2D projections of the intracluster medium and its effect on the retrieved statistical measures commonly used in turbulence analysis like the power spectrum by using numerical simulations appropriate to galaxy clusters.

Speaker: Jago Edyvean

Title: Hermite Transforms and Plasma Turbulence

Abstract: Transforms are a useful and powerful method for investigating and understanding systems. In collisionless plasma turbulence, the governing equation. the Vlsov Equation, is very hard to solve numerically and so transforms reduce the complexity of this. Transforming the Vlasov equation also yields interesting physical interpretations of the system. The Hermite transform is one such method.

Speaker: Sapir Ben-Shahar

Title: Introduction to Computable Metric Spaces

Abstract: This talk will begin with an introduction to metric spaces and some notions of computability, including presentations of spaces, and computably enumerable (c.e) sets.
From there we will build up to the idea of computable metric spaces, with the goal of presenting a space that is not computable, but in an unusual, almost computable way: it has both a left-c.e presentation and a right-c.e presentation but no computable presentation. This is surprising because normally we would expect something that is both left-c.e. and right-c.e. to be computable, with the left-c.e. and right-c.e. presentations each giving us, in a sense, half of the information that we need for a computable presentation. Knowledge of computability and metric spaces is NOT assumed.

Speaker: Ellen Hammatt

Title: Punctually 1-Decidable Boolean Algebras

Abstract: In my talk I will discuss my recent work on online (punctual) presentations of boolean algebras. The main result is: There exists a 1-decidable boolean algebra such that is not computably isomorphic to any punctually 1-decidable boolean algebra. All terms and definitions will be clarified in the talk.

Speaker: Sam Bastida

Title: List Colouring and Maximal Local Edge Connectivity

Abstract: A (proper) k-colouring of a graph assigns one of k colours to each vertex such that no two adjacent vertices have the same colour. A fundamental theorem in graph theory, Brooks' Theorem, states that a graph with maximum degree k (a graph where each vertex has at most k neighbours) has a proper k-colouring unless it is a complete graph or a cycle with an odd number of vertices. More generally one can consider graphs in which each pair of vertices have at most k edge-disjoint paths between them, such graphs are said to have maximal local edge connectivity k. The question of which graphs with maximal local edge connectivity k can be k-coloured was answered by Stiebitz and Toft in 2018. One can also consider a generalisation of k-colouring where each colour must be chosen from a list of colours of size k for each vertex. A graph is k-list colourable if such a colouring exists for each assignment of lists of size k to vertices. In this talk, we describe our progress towards an attempt to extend the theorem of Stiebitz and Toft to k-list colouring. Certainly, a graph that is k-list colourable is k-colourable, but the converse may not be true - this makes the problem of characterising which graphs are k-list colourable more difficult than the problem for k-colouring.

Speaker: Flynn Owen

Title: Finite Mixture Models - A Divide and Conquer Approach

Abstract: Within the field of data clustering, methods are commonly referred to as either ‘distance-based’ or ‘model-based’, with each having certain limitations. Distance-based methods are fast, but lack an underlying parametric model. Model-based methods are interpretable and generalise easily to distributions other than Gaussian, but are slow to estimate, especially when the number of clusters is large. In this thesis we take inspiration from divisive hierarchical clustering and develop four ‘divide and conquer’ model-based algorithms to rapidly estimate mixture models with large numbers of components. Our algorithms are: 1) The Hierarchical No Turning Back (HNTB) algorithm, 2) The Hierarchical Single Reallocation (HSR) algorithm, 3) The Hierarchical Reallocation Loop (HRL) algorithm, 4) The Hierarchical algorithm that Reallocates During execution (HRD). Using Akaike Information Criterion (AIC) as our measure of model quality, our algorithms work by recursively fitting a data matrix with R = 1 and R = 2 components, and then partitioning this data-matrix dependent on cluster membership, halting when no sub-components can be partitioned further. We then adapt the ‘E-step’ from the Expectation-Maximisation (EM) algorithm to reallocate data points across each estimated sub-component of the data matrix. We perform a series of Monte Carlo simulation studies on a series of datasets generated from the Bernoulli distribution, and compare our results against an adapted version of well known ‘Wolfe’s method’ (Wolfe, 1971). We establish that ‘divide and conquer’ mixture modelling methods are able to identify mixtures with large numbers of components at a much faster rate than our adapted version of Wolfe’s method, and approximate both the true simulated data, as well as the results from our adapted method to a good degree. Algorithms introduced in this thesis provide suitable methods to address scalability with the ever increasing magnitude of data being stored and processed, while ensuring interpretability with the increasing demand for data-driven insights.

Speaker: Matthew Askes

Title: Strongly Online Graph Colouring

Abstract: An online algorithm is an algorithm that processes its input in pieces and must process the current piece of data before it gets the next. Online algorithms have wide-ranging applications from emergency vehicle dispatching, scheduling, and processing large data sets. In effect, any algorithm that needs to work on partial information can be represented as an online algorithm. However, online algorithms do not fully capture the nature of certain problems. Often you can see slightly ahead, or you may be able to delay an action temporarily until more data is available. In these situations the problem is not fully online. To this end, we introduce the notions of locally strongly online and strongly online presentations. In particular, we will discuss strongly online colourings of graphs and touch on the idea of strongly online path decompositions.

Speaker: Linus Richter

Title: On Homological Algebra, Group Extensions, and Descriptive Set Theory

Abstract: One way of thinking about group extensions is via short exact sequences in the category of groups. I will outline, using simplicial homology, how algebraic topology associates algebraic objects as topological invariants. Using cohomology and descriptive set theory, I will then briefly explain how the class of group extensions whose bonding morphisms are Borel is trivial in certain circumstances. This extends results by Kanovei and Reeken (2000).

Speaker: Linus Richter

Title: When the search for proofs turns futile

Abstract: It came as a shock when in 1931 Kurt Gödel proved that mathematics is not complete. We can find mathematical statements that are "independent"; they can neither be proven nor refuted. Surprisingly, while the original independent "Gödel sentence" is a somewhat contrived construction, meaningful mathematical questions can be independent, too.

For example, a problem long confounded set theorists: is there a subset of the real numbers that is larger than the set of integers yet smaller than the real numbers themselves?

The assertion that such intermediate sets do not exist is called the "Continuum Hypothesis" (CH), and many set out to prove CH. In 1963, Paul Cohen showed that those attempts had been futile -- he proved that CH is independent. Cohen did so by inventing the versatile method of "forcing", which has since been used to yield many more independence proofs.

I will introduce some mathematical logic, the notion of independence, and why we care about all this. Using set theory, I will then outline how forcing yields a proof of the independence of CH.

Speaker: Josh Baines

Title: The Painlevé-Gullstrand Form of Various Spacetimes

Abstract: Every spacetime is defined by its metric, the mathematical object which further defines the spacetime curvature. From the relativity principle, we have the freedom to choose which coordinate system to write our metric in. Some coordinate systems, however, are better than others. In this talk, we begin with a brief introduction into general relativity, Einstein's masterpiece theory of gravity. We then discuss some physically interesting spacetimes and the coordinate systems that the metrics of these spacetimes can be expressed in. More specifically, we discuss the existence of the rather useful Painlevé-Gullstrand coordinate system in these spacetimes. Using this useful coordinate system then allows us to conduct further analysis of these spacetimes, which we discuss.

Speaker: Michal Salter-Duke

Title: Tangles in networks and their application to describing communities

Abstract: Communities in networks are structurally or functionally distinct subgraphs. They can be disjoint or overlapping, depending upon definition; there are several definitions of communities, and many algorithms to identify them. We present a new definition based on the graph-theoretic concept of tangles, and investigate how well it identifies subgraphs that represent cohesive parts of the network, using two protein-protein interaction networks. This requires the development of an algorithm for finding tangles in graphs. We compare our results with standard methods from the literature based on metrics that use metadata to establish communities. Our results show that tangles provide a different view of communities that complements other methods, although they are computationally expensive to identify.

Slides: MichalSalter-DukeGradTalk.pdf

Speaker: Malcolm Jones

Title: An algebraic model in public health: the groupoid of bubbles

Abstract: Groupoids are a generalisation of groups that arose in the early 20th century in work on quadratic forms by H. Brandt. Since their inception groupoids have provided a fruitful framework in a variety of fields, including fundamental work by J. Renault in operator algebras in the 80s. Groupoids have even appeared in epidemiology as recently as 2014 as models of cognitive processes in biology. In this talk, we will review the elementary theory of groupoids, and we will discuss a widely applicable groupoid model of interactions in populations.