The ePoster Schedule

Polyspherical coordinates (one of which is the familiar spherical coordinates) provide a class of coordinate systems on $S^n$ for which the Laplace-Beltrami operator $\Delta_{S^n}$ is separable.

A way to define polyspherical coordinates is using a method outlined by Vilenkin that utilises binary rooted trees. The method identifies each polyspherical coordinate system on $S^n$ with a binary rooted tree with $n+1$ leaves. Given a tree, we can algorithmically construct the coordinate system on $S^n$ and the product of special functions that will define a basis of the eigenfunction space of $\Delta_{S^n}$. 

We will discuss how this method can be used to construct coordinate systems on $S^n$ as well as eigenfunctions, commuting operators and the corresponding joint spectrum. We will look at the combinatorics of equivalence classes of systems defined by unlabelled rooted binary trees and associated moment polytopes.

The inflated dynamic Laplacian introduced by Gary Froyland and Peter Koltai is a Laplace-Beltrami operator with respect to a fibred Riemannian metric generated by the flow of a dynamical system which acts on the time-expanded phase space. Level sets of its leading eigenfunctions provide a description of the emergence and decay of finite-time coherent sets in the  system. Since the dominant spectrum of this operator accumulates at zero and data-driven discretisations of the operator lead to  sparse systems, solving the eigenproblem is numerically challenging. Exploiting the fibred structure of the inflated dynamic Laplacian, we consider an approximation of its operator semigroup  via a Trotter-product  and investigate convergence properties of eigenvalues and eigenfunctions to solutions of the original problem.

Poincare famous recurrence theorem says that if (X,\mu,T) is a probability preserving system, i.e. (X,\mu) is a probability space and T: X \to X preserves the measure in the sense that \mu(A) = \mu(T^{-1}A) for all measurable sets A, then there exists a positive integer n such that \mu(A \cap T^{-1}A) > 0. One can make this theorem 'quantitative' in the sense that you can ensure n is bounded in terms of the measure of your set A. The Furstenberg-Sarkozy theorem says the same thing as Poincare’s recurrence theorem, except this time one can replace n with n^2. There has been much work generalising and strengthening these types of results, but it appears that nobody knows whether or not one can ensure that n^2 is bounded in terms of the measure of A, i.e. nobody knows if there is a quantitative Furstenberg-Sarkozy theorem.

We will discuss the attempt to apply the heat kernel signature (HKS) on time-evolving manifolds. The HKS is proposed by Sun, Ovsjanikov and Guibas where it is used to extract features from static objects. We are interested in applying the HKS on manifolds that are changing with respect to some given dynamical system. To that end, we hope to construct a dynamic HKS with respect to the dynamic Laplacian introduced by Froyland. 

In this short presentation, I will show to the audience my ideas and thoughts about Riemann's hypothesis proof. I will observe nodal lines of the Riemann Zeta function and assume, in one (or maybe two) special case(s), that some nodal lines intersect outside the critical line generating zeroes of the function. Then I will derive contradiction(s).

Cancelled!

Delay differential equations are an interesting class of non-local equations that involve a function and its derivatives evaluated at different points in time. By introducing a new class of functions, we have been able to provide solutions for linear time-delay partial differential equations. In this talk, we study the solution to the time-delay heat equation and demonstrate its finite time blow-up. 

Cancer is a potentially fatal disease. In 1994, Kuznetsov et. al. presented a cell-mediated model responding to a tumour cell population, where they looked at the mechanisms of "sneaking through", tumour dormancy and immunostimulation.

In this talk, a mathematical analysis of the cell-mediated model, through the lens of geometric singular perturbation theory (GSPT) will be presented. The bifurcation structure arising from the variation of the basal effector supply rate will be discussed and partitioned into three regimes. In each of these regimes, the model exhibits different qualitative behaviour. 

I will present some possible reference scales one can choose in the different regimes, where resulting ODEs will be related by a rescaling of variables.  We then apply tools from GSPT to reduce these ODEs and look at how these reduced models in the different regimes reflect the full model.

Transmissions trees are an important way to understand and visualise how contagious a disease is. Knowing with more accuracy its structure and dynamics, we can better know how dangerous a virus is and which measures we have to apply to combat the spread of the disease. Recently, thanks in part to technological improvements, we can understand better these transmission trees. Even so, the most used methods used to infer these transmissions networks only have been used to explore the most plausible scenarios, ignoring other scenarios that even if they are not very plausible, they still can happen. For that reason, we think that a deeper inspection of all plausible scenarios has to be done. Here we are going to present some preliminary results applied to a virus that behaves similarly as COVID-19, and also, we want to discuss how we want to continue this study using genetic data.

Lagrangian coherent structures are a concept in fluid dynamics used to identify the most consistent flow patterns and behaviour observable within a dynamic velocity system over a finite period of time. Many different methods have been conceptualised for the numerical detection of these structures, with each one carrying a different definition of what constitutes a Lagrangian coherent structure. One inhibiting factor stopping the more mainstream use of Lagrangian coherent structures in real world settings is the lack of understanding over how these methods are able to produce reliable results in flow systems represented by velocity data sets with considerable uncertainty. Hence, this talk aims to address this issue by focusing on quantitative analysis of the impact of noise and uncertainty on a handful of these Lagrangian coherent structure detection methods; undertaken as part of my PhD research.

Cognitive control, through the frontal cortex and subthalamic nucleus (STN) circuit, is responsible for suppressing automated actions to facilitate behavioural adaptation. Experiments have shown that STN exhibit increased spiking and theta band power during such response conflict conditions. The electrophysiological mechanisms underlying these neural signatures remain poorly understood. To address this lacuna, we constructed a novel large-scale, biophysically principled model of the subthalamopallidal network and examined the mechanisms that give rise to theta power and spiking in response to cortical input. Simulations confirmed that theta power does not emerge from intrinsic network dynamics but is robustly elicited in response to cortical input as burst events representing action selection dynamics. NMDA, but not AMPA, currents were necessary and sufficient for theta power expression which was related to a triphasic STN response characterized by spiking, silence and bursting periods. The slower NMDA kinetics afforded a window for integrating such cortical information.