Complexity Cluster
Some of the speakers, participants and organisers at the Complexity Cluster workshop
On Friday, 8 March, Keble hosted a Complexity Cluster workshop, bringing together mathematicians from a wide range of research areas to share their work. The event continues a successful series of workshops at Keble, and was organised by Keble Professorial Fellows Gui-Qiang G Chen and Helen Byrne, and Keble Research Associate Dr Yaron Ben-Ami. Four mathematicians gave talks in which they presented their research in an accessible and engaging way. The audience consisted of a mix of professors, fellows, postdocs, DPhil students, and undergraduates from multiple branches of mathematics.
Max Anderson Loake
The first talk, entitled “ODDRIN: Oxford Disaster Displacement Real-Time Information Network”, was given by Statistics DPhil student Max Anderson Loake. Max’s work is motivated by the need for precise and thorough delivery of humanitarian aid after a natural disaster. Utilising variables such as hazard, exposure and vulnerability, Max has developed a statistical model to predict damage and human displacement after an earthquake. This information can then be used to inform humanitarian relief strategies. By using a Bayesian framework to estimate the parameters of his model, Max has simulated the damage, impact and human displacement caused by the earthquake that occurred in Morocco on 8 September 2023.
Dr Simon Martina-Perez
Mathematical modelling is a valuable tool that can be used to study biological phenomena. Dr Simon Martina-Perez gave a talk entitled “Inferring cell-cell interactions during collective invasion” where he explained how he uses mathematical techniques to identify the mechanisms that underpin cell invasion during wound closure. Through a combination of mathematical modelling and approximate Bayesian computation, Simon was able to fit experimental data from a series of experiments to predict how specific genes affect cell motility and proliferation. In another study, by using deep attention networks to analyse dynamic, single cell tracking data, Simon was able to explain how cells interact with each other during invasion. This work helped to identify cell invasion traits that could either enhance wound healing or inhibit metastasis.
Dr Sungkyung Kang
Dr Sungkyung Kang spoke about his work in the field of knot theory. This branch of mathematics seeks to understand and classify ‘knots’, which are twisted embeddings of a circle in three-dimensional space. One may wish to systematically catalogue all possible knots and to identify those which are ‘equivalent’. This problem usually involves the study of knot ‘invariants’, quantities which remain unchanged if a knot is continuously deformed. This subject has a rich history, dating back to the 1860s, when Lord Kelvin postulated that atoms were knots in the ether. While this hypothesis has since been proved false, it opened the door to a fascinating and mystifying mathematical world. Over time, many famous mathematicians have contributed to knot theory, introducing new tools that combine principles from geometry, topology, and algebra. The subject of Dr Kang’s talk concerned a variant of the classical knot theory setup: he studies knots in four-dimensional space and considers the surfaces which bind those knots. One can ask many questions, for instance: What is the minimal genus (number of holes) for such a surface? Can we construct ‘exotic’ surfaces which are equivalent in one sense (topologically isotopic) but not in another (smoothly isotopic)? Dr Kang’s work has provided some answers to these and other questions. However, open questions abound in knot theory! Moreover, although it has long been viewed as a ‘pure’ mathematical field, it has recently been fruitfully applied in biology to understand the tangling of DNA. Dr Kang’s talk gave a sense, in an approachable way, of the beauty and the potential of this challenging field.
Professor Tao Wang
A vortex sheet is an interface between two inviscid fluids across which there is a jump in velocity. They arise in a range of systems, including the plasma physics governing gaseous stars and the passage of air over the wings of fast-moving aircraft. Many such systems are described by systems of partial differential equations which are highly nonlinear and hyperbolic and, therefore, tend to develop discontinuities. Physical solutions to these equations must also satisfy “entropy conditions”. For compressible flows, vortex sheets are the building blocks of general entropy solutions. Therefore, it is of great mathematical and physical interest to understand their existence and stability (i.e. whether they will persist in time if the flow is slightly perturbed). Professor Tao Wang from Columbia University spoke about his work on the stability question. This question has its origins in work by Helmholtz (1868) and Kelvin (1894) who showed that every vortex sheet for incompressible fluids without surface tension is unstable. Subsequently, Landau (1944) and Miles (1958) showed that two-dimensional vortex sheets for compressible flow can be linearly stable, if the relative Mach number is high enough. However, the rigorous mathematical theory was developed much more recently, beginning with Coulombel and Secchi in 2008. Professor Wang is continuing to develop this research with the aid of sophisticated nonlinear analysis techniques.
Note: This material is based on notes taken by two DPhil students: Luke Heirene and Isaac Newell.
