Special Event: Dr. Brendan Harding on MathJobs and more.

Abstract: I'll give an overview of mathjobs and how to use it to find and apply for jobs. I'll also give some tips on putting together an application, including things to focus on in your cover letter and some suggestions for building an academic CV. I'll show some examples of applications I have submitted in the past.

Speaker: Daniel Wrench

Title: Intro to Bayesian Statistics

Abstract: The Bayesian approach is a very popular way of doing statistical inference and rests upon a simple theorem of conditional probability. Daniel will be discussing the basics of this mathematical method of updating your beliefs in light of new information.

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: Joseph Wilson

Title: Geometric algebra and an application to special relativity

Abstract: Real Clifford algebras are affectionately called “geometric algebras” by physicists for their utility and generality. They provide an elegant framework for describing rotations in arbitrary dimensions, including 3+1 dimensional spacetime. This leads to a simple and novel formula for the composition of Lorentz transformations in terms of their infinitesimal generators.

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.