Postgraduate Seminars Autumn 2018

These seminars (unless otherwise stated) will take place on Thursdays at 12pm in Math Room 706 on an almost weekly basis.

See the map for further details. Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

27 September 2018

Speaker: Luis Alberto Abrego Rangel
Supervisor: Alexey Zaikin

Title: Integrated information measures in coupled genetic repressilators

Abstract:
Intercellular communication and its coordination allow cells to exhibit multistable dynamics as a mechanism of adaptation. This conveys information processing from intracellular signalling networks enabling self-organization between cells. How information is integrated in a functional manner and its relationship with the different cell fates is still unclear. In parallel, drawn originally from studies on neuroscience, integrated information, proposes an approach to quantify the balance between integration and differentiation in the causal dynamics among the elements in any interacting system. In this work, such an approach is considered to study the dynamical complexity in a genetic network of repressilators coupled by quorum sensing. Several attractors under different conditions are identified and related with integrated information to have an insight into the collective interaction and functional differentiation in cells. This research particularly accounts to the open question about the coding and information transmission in living systems.

04 October 2018

Speaker: Jessie Renton
Supervisor: Karen Page

Title: Evolution of cooperation on an epithelium

Abstract:
The prevalence of cooperation in nature poses a problem for Darwinian evolution. While mutual cooperation enhances overall group fitness, there is an individual fitness advantage for defectors, and natural selection functions on the individual level. Why then is cooperation selected for? Evolutionary game theory provides a framework for modelling competition between cooperators and defectors and it has been established that cooperation can thrive under certain conditions if there is population structure. While there are clear applications of this cooperator vs. defector paradigm in understanding the social behaviour of animals, it is also relevant to evolution on the cellular level. Many of the hallmarks of cancer (mutations required for cells to become malignant) for example, can be considered cooperative. We thus consider how the population structure arising from cellular arrangement in tissues affects the evolution of cooperation using two models. The first using a traditional evolutionary graph theory framework representing population structure as a static graph and the second using a Voronoi tessellation model to consider a more realistic dynamic population structure.

11 October 2018

Speaker: Francesco Di Giovanni
Supervisor: Jason Lotay

Title: A crash course on Ricci Flow

Abstract:
The Ricci flow is a geometric heat-type evolution for Riemannian metrics which was first introduced by Richard Hamilton for analysing three-manifolds with positive Ricci curvature. Recently the Ricci flow has played a pivotal role in proving several long-standing open problems, such as the Poincare’ and geometrization conjectures and a series of sphere theorems.

In this talk I will give a general overview of the main features of Ricci flow, with focus on the underlying geometric ideas rather than on the technical difficulties involved; accordingly, formal statements will be often replaced by pictures and examples.

If I have time at the end, I will briefly describe the problem I am working on and its general motivating questions.

The talk should be accessible to anyone, but I aim to make it mostly beneficial to people that have some familiarity with basic notions of at least one of the following areas: differential geometry/Riemannian geometry/topology/analysis of PDEs.

Spoilers: Spheres, black holes, Kahler structures and His Almighty Perelman will be mentioned.

18 October 2018

Speaker: Fabian Lehmann
Supervisor: Jason Lotay

Title: Curvature and Symmetry

Abstract:
In my talk I will explain how the concept of a geometric space is formulated in Riemannian geometry and what the most basic and important questions are. We will see that curvature and symmetry are two of the main themes in differential geometry. To find ‘nice’ geometric spaces means to look for spaces with particular curvature conditions. Mathematically this will lead to a system of partial differential equations. If the space has symmetries then this system takes an easier shape.

As the talk will be a leisurely introduction to differential geometry and no prior knowledge is required.

25 October 2018

Speaker: Udhav Fowdar
Supervisor: Jason Lotay

Title: An Introduction to Holonomy groups

Abstract:
This talk is meant to give a short and quick exposition to the idea of holonomy groups which has been touched upon in the last two talks.

Holonomy groups (essentially just N x N matrices) provide a way of classifying “interesting geometries” and have connections with several branches of physics; General relativity, Twistor theory, String theories, M-theory and F-theory. I will introduce the concept of holonomy, describe the classification and the links with physics. I will only assume the audience is familiar with basic linear algebra.

01 November 2018

Speaker: Benjamin Millard
Supervisor: Lars Louder

Title: Commutator Length in Free Groups

Abstract:
The Commutator length of an element, g, in a group is the minimum number of commutators needed such that their product is g. In general computing commutator lengths is a difficult problem. There is however, an equivalent geometric definition of commutator length, and for free groups this can be used to provide an algorithm for calculating the commutator length of any element. I will explain the ideas behind this through pictures and describe a question asking how commutator length differs in a subgroup of a free group. This talk should be accessible to all.

08 November 2018

Speaker: Kian Cheng
Supervisor: Rod Halburd

Title: Degree growth and special solutions of discrete equations

Abstract:
Using the method introduced by Halburd, the exact degree dn of each iterate yn as a rational function of y0 and y1 of a second-order rational mapping can be calculated easily through singularity confinement analysis. It allows us to compute the degree growth of the iterates, and further the algebraic entropy which is associated with integrability. The degree growth varies depending on the choice of initial conditions. In some situations, an otherwise non-integrable equation may have special integrable sub-classes. Some special initial conditions produce solutions that can be expressed in terms of solutions of discrete linear equations. In this talk, initial conditions will be classified for several discrete equations based on the rate of degree growth of the solution and this will be used to identify integrable sub-systems.

15 November 2018

SEMINAR CANCELLED

22 November 2018

Speaker: Giada Grossi
Supervisor: Sarah Zerbes

Title: Local to global principles in number theory

Abstract:
Many interesting problems in number theory arise from simple questions like determining whether some equations have rational solutions. This problem can be solved in the case of conics, but it turns out that in order to do that one needs to look at rational numbers under “the lights of prime numbers’’. A conic indeed has rational solutions if and only if it has a solution over the real numbers and over p-adic numbers for every prime p. In this talk I will give an introduction on p-adic numbers, discuss about the result on conics and show that in general (e.g. in the case of cubic curves) one of the two implications does not hold.

29 November 2018

Speaker: Sally Said
Supervisor: Frank Smith

Title: Modelling the combustion of explosives

Abstract:
High Explosives (HE) store energy which can have disastrous effects if released accidentally. Thus safe handling and storage of HEs is a matter of utmost concern. When a HE is subjected to significant heating it reacts, i.e. it burns to form gaseous products. The interaction of the burning solid and gaseous products formed is not well understood. More specifically, the process of the gaseous product heating explosives is yet to be explored in detail. Mathematical modelling aims to track the heat flow in the explosive and the gas response. This work begins by modelling single step kinetics using the simple Arrhenius model. An alternative asymptotic approach is also employed. There is agreement between the full reaction-diffusion problem and the asymptotic problem, which are both solved using finite difference methods. The model is then extended to include three step kinetics. Further work includes the motion of gas being incorporated in the existing model with temperature and pressure distributions considered.