Intradisciplinary lecture series

In this general audience talk, we will combine tools from analysis (and in particular spectral analysis) and probability theory to investigate time decay properties of various physical systems (such as the spreading of heat inside some volume). It is well-known that a spectral gap is equivalent to a so called Poincaré inequality and to an exponential rate of decay. After explaining these notions, we will see how some of these results can be extended to cases where there is no spectral gap. This is joint work with Amit Einav (Graz).

This talk will be in two parts. In the first part I will give a brief overview of some of my research into how graph theoretical models can be used to solve (or at least approximately solve) some real-world operational research (OR) problems.

The second half will be a more detailed case study on the maximum happy vertices problem, stemming from some recent work carried out with colleagues in Australia [1]. This is a new type of problem that involves determining a vertex colouring of a graph such that the number of vertices assigned to the same colour as all of their neighbours is maximised. This problem is trivial if no vertices are precoloured, though in general it is NP-hard.

[1] Lewis, R., D. Thiruvady and K. Morgan (2019) 'Finding Happiness: An Analysis of the Maximum Happy Vertices Problem'. Computers and Operations Research, vol. 103, pp. 265-276.

An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. This talk will provide an elementary introduction to the combinatorics of ASMs, and will include an outline of a recent proof of a conjecture for the number of odd-order diagonally and antidiagonally symmetric ASMs.

I will give an overview and introduction to conformal symmetry. Structures exhibiting conformal symmetry are invariant with respect to changes of coordinates which preserve angles but not necessarily lengths. In two dimensions the algebra of local conformal transformations is infinite-dimensional and thus leads to rich mathematical structures with fascinating implications for physics, such as, exactly solvable quantum field theories.

The single-arm trial is a very simple design: everyone gets a treatment (mostly within oncology) and wait to see if it works, in terms of a binary response. It assumes that those not treated will not get better or have a small chance (p0) of getting better with the hope that the treatment improves response to the chance p1. One improvement to this design is an adaptive version where an interim analysis is carried out after observing the outcomes of the first, n1 participants. The intention of the interim analysis is to stop the trial early as the treatment is not working.

This adaptive version of the trial design is often labelled Simon’s two-stage design and it is perhaps the most used adaptive design in practice. However, from results of a review it shows how a design can be abused with poor reporting and adherence to a design.  I will describe different ways of improving this design such as doing many interims, being Bayesian, adding in biomarkers and comparing this approach to randomised designs. Perhaps in the end I might grow to like this design.

The talk touches upon several topics from different subjects. First, I will talk about uniformly distributed sequences and different characteristics of uniformity. Second, I will argue that, contrarily to a wide-spread belief, the sequences with good space-filling properties do not resemble uniformly distributed sequences, especially if the dimension of the space is large. Most of the time I will spend on the problem of covering of a high-dimensional cube by n balls and will demonstrate several unexpected phenomena of good covering schemes. I will also mention the problem of quantisation also known as the problem of optimal facility location.

Missing data are a commonly encountered problem when analysing data. Simple methods exist to deal with this problem but are not always ideal with the potential to introduce bias into the analysis. In this presentation I will review the area of the missing data and present some topics I have worked to address in this area. In particular, I will talk about the approach of multiple imputation as an appealing approach to deal with this problem. I will also discuss the approach of dealing with the problem of missing data at the design stage of an experiment.

Time permitting I will also talk about the problem of data confidentiality. In this area, there is a tension between data-holding organisations to release useful data to researchers and the public while also protecting the privacy of those individuals that provided the data. I will focus here on the area of synthetic data as a viable approach to address this problem.

I intend for this talk to be accessible to a wide audience so I will try not to dwell too deeply on the technical detail and instead attempt to present the research problems and methods in a non-technical way wherever possible.