Numerically Approximating Conformal Maps with the Zipper Algorithm

As with the previous post, I want to mention some other work I did related to my DPhil course. This project was about implementing an algorithm to numerically compute conformal maps between simply connected domains and the unit disk. By the Riemann Mapping Theorem we know that for every such domain, a map must exist, but finding it analytically is rarely easy if possible at all. A good discussion of these ideas can be found, for instance, in Terence Tao’s lecture notes on the subject. Given a domain, there are several ways to numerically construct a map which approximates the conformal map guaranteed by this Theorem, such as by approximating the domain by polygons and using Schwarz-Christoffel maps to map it to the unit disk.

I implemented the Geodesic or Zipper algorithm due to Don Marshall. The full code (in Matlab) can be found here, along with a few graphical tools to explore it. It has some precision problems for domains with “sharp” boundaries for the inverse map (that is, mapping the unit disk to a given domain) but overall I’m pretty happy with the results. I did not include all of the approximations one could to make the algorithm both more accurate and more efficient, but I may come back to this later and do that, as well as demonstrate some of the applications of these maps. If you’re interested, I would also check out this Thesis on the topic which I found to be useful. It also contains Python code for this algorithm as well as another approach using sphere packing. Below I will include a few examples of what this code can generate, but I encourage you to download it and play with it yourself. As always, comments and questions are appreciated!

Some Visualizations in Mathematical Physiology

“…the equations we deal with are probably more complicated than even most physical scientists are accustomed to. This is because the phenomena we are attempting to describe are generally more complex than most physical systems, although it may reflect our own ineptness in perceiving their underlying simplicity.” -James Murray, Mathematical Biology

Due to the complexity of modelling biological phenomena, analytical and numerical approaches are often used together to give a more thorough understanding of a model than either could provide alone. This is particularly true in the area of mathematical physiology, where there has been a dual development of increasingly sophisticated models, as well as analytical and numerical tools to analyze these models. In this post I want to describe some visualizations of simple models in this area, as well as reference some useful tools to explore these simulations further. These visualizations complement the lecture notes for a corresponding course that I tutored last year, which can be found here for the moment, and a much older version of the notes can be found here.

Non-Existence of Patterns in Reaction-Diffusion Systems

In writing up several recent papers I have spent some time reading about Reaction-Diffusion systems and the kinds of behaviour they can have. I will likely blog about these kinds of equations more often, as well as applications of them to chemical, biological, and other systems. Today I want to just record some interesting observations from the literature. This will be about the existence of large time asymptotic solutions to these equations. Roughly speaking, these must be either uniform or non-uniform steady states, or more exotic time-periodic solutions such as oscillations, spatiotemporal waves, and uniform or spatial chaos. Other interesting transient behaviours can occur, such as wave propagation, but here I’ll focus entirely on the long time dynamics, and specifically on the existence of stable equilibria. For a broad overview of these models in some detail see [1].

Dissipative Functions and Reaction-Diffusion Equations

I want to review some aspects of dynamical systems theory for a class of dissipative systems which are particularly simple. These are systems that posses a Lyapunov function. Dissipative systems are mathematical objects that evolve in time, but remain in some bounded region of state-space for all times. Below I will define the concept of a Lyapunov function, how it relates to the long-time dynamics of differential equations, and finally develop some intuition for why the (scalar) Reaction-Diffusion equation is relatively simple in terms of dynamics, while Reaction-Diffusion systems can have much richer structure.

Nonlinear Dynamics on Networks

Today I want to talk about dynamical systems on networks. A good review article can be found here. In this post I want to focus on the interplay between the topology of the underlying network, and the asymptotic dynamics of the system. I will focus on two aspects of this area that I have found particularly interesting.

Thoughts on Uncertainty and an Unpredictable Future

I have recently been reading about when certain methodologies or paradigms fail. My current research has taken the direction of when classical models of certain kinds of physics are not appropriate. Assuming the results of my proposed alternatives look good, I will share some of them here. I have also been reading things very much outside of my area, and it has made me think more about the nature of academic, government/industrial, and layperson thought. The rest of this post will be a very brief response to this article by Nassim Nicholas Taleb, as well as a few other pieces such as a book on Street-Fighting Mathematics, which is an excellent read for anyone interested in approaching mathematics from a very different perspective.

A Note on Multiple Scales: Homogenization Theory

As I have mentioned before, many problems in science can be profitably attacked from a large variety of different angles. Today I want to discuss one particular aspect of theoretical work in a technical way, in order to provide a bit more motivation to why a mathematical method of analysing a scientific problem provides insight that complements computational or experimental work. This is the technique of multiple-scales analysis, and specifically Homogenization. Most of the following was written as part of a broadening course I took in a Perturbation Methods lecture that is taught at Oxford for fourth-year undergraduates.