Urban mobility simulations use agent based modelling to estimate the daily activities of individuals within a community. The output from these simulations can be used to generate a detailed synthetic social network that carries information about the duration and venue of all contact events within a city on a given day, as well as the key demographic information (age, occupation, etc) of the participants in each event. We have been working with The Black Arcs, a Fredericton area technology company, to incorporate synthetic social networks generated by their software into network based models of COVID-19 and related diseases. Our goal is to make quantitative predictions about the impact of various public health interventions on disease spread. For example, we are interested in questions like: Is it more effective to shut down schools or retail businesses to control disease spread? What is the effect of different testing strategies and delays on hospitalizations? And, which policies do the best job of protecting vulnerable demographic groups?
For a recent research project on discrete spacetimes, I have been using finite element methods to solve the Helmholtz equation on various interesting 2-dimensional triangulated manifolds. A little while ago, I realized that a neat by-product of these calculations is the ability to easily solve the free particle Schrodinger equations in these geometries. In the physics literature, this problem is sometimes called “quantum billiards” because it is the quantum mechanical analogue of studying the motion of billiard balls (on oddly shaped tables).
Here is an example:
This movie shows the position probability density of a free particle confined to an elliptical cavity (i.e. the modulus squared of the position space wavefunction). Here is a version of the same movie with the trajectory of a classical particle with the same initial position and velocity as the quantum wavepacket superimposed:
At the initial time, the particle is localized near the centre of the ellipse and has a velocity directed up and to the right. The particle’s wavepacket scatters off the walls of the ellipse several times. Each collision caused the wavepacket to spread out in space, and, by the end of the movie, the particle is de-localized over most of the ellipse. Interference patterns are formed as portions of the reflected wavefunction from different collisions interact with one another.
In the above movie, you should be able to see that the ellipse is actually made up of a bunch of small coloured triangles. This is because I am not actually solving the Schrodinger equation within a continuous elliptical region, I am rather solving for a discrete version of the wave function defined on a triangulation of the ellipse. By making the triangles smaller one gets a better and better approximation to the continuous case. But the catch is that as the triangles get smaller, the computational time to generate the movies gets longer. The movies on this page are the result of simulations that take a few hours on my laptop.
The Bunimovich Stadium is essentially a rectangle with semi-circular caps. It is interesting because a classical particle contained within it exhibits ergodicity. That is, if you consider a classical billiard ball in this region with a random initial position and velocity, is trajectory will (almost always) eventually fill up the entire stadium uniformly. The above simulation is possibly hinting at the quantum analogue of this classical ergodicity, with the final wavefunction configuration being even more dispersed than the elliptical case.
The next example I looked at was meant to be similar to the famous double slit experiment:
In this movie, the particle lives in a circular arena with a triangular obstacle in the middle. The obstacle cleaves the particle’s wavefunction in two, essentially meaning that there is an equal probability of measuring the particle taking a path above or below the triangle. After the splitting, we can see the development of intricate interference patterns, just like in the double slit experiment.
Here is another example of a wavepacket interacting with a 2-dimensional barrier:
In this case, the particle undergoes a glancing collision with a circular obstacle in a hexagonal arena.
The aperiodic monotile has a remarkable property: it can be used to tile a 2-dimensional plane in a completely non-repeating way. It is pretty unclear to me how the tiling properties of the monotile might relate to the trajectory of confined quantum particles, but it does make for a pretty movie…
In lieu of a final exam there was a final project where students could study any topic of interest using the methods learned throughout the term. Part of the final submission was a poster similar to those presented at a scientific conference. I think these turned out great, and I’m happy to share a selection (with the students’ permission) below. Enjoy!
The award of University Research Scholar honours members of the active faculty complement of the University of New Brunswick (UNB) who have demonstrated a consistently high level of scholarship, and whose research is, or has the potential to be, of international stature.
I’d like to sincerely thank all my colleagues who nominated me for this and wrote letters of support. I’d also like to congratulate Aurora Nedelcu from the Department of Biology at UNB (Fredericton) for also being named to this position.
Last week was very busy in terms of the public release of COVID modelling I have been working on for various governments in Canada. First off was a press conference from the Government of Prince Edward Island on Wednesday, January 26:
About 35 minutes in, you can find Dr Heather Morrison (Chief Health Officer) discussing recent forecasts for new cases and hospitalizations under current conditions, and various hypothetical scenarios considering the effects of the January 19 circuit breaker in PEI.
I have also been doing some modelling for the Northwest Territories (NWT). On the government’s website, you can find a webpage called The Use of Mathematical Modelling in the Northwest Territories (NWT). This page contains a concise summary of NWT modelling results as of Wednesday, January 26. We plan to update this webpage regularly as more data becomes available.
Finally, Dr Jennifer Russell (Chief Medical Officer of Health of New Brunswick) presented a modelling slide at a Thursday, January 27 press conference (around 5 minutes in):
This chart compared model projections for hospitalizations during New Brunswick’s level 3 restrictions and actual data.
For over a year now, my main research focus has temporarily shifted away from gravitational physics towards the mathematical modelling of COVID-19 (for obvious reasons). Although this might seem like a big change, the two areas actually share a lot in common. One of the main techniques for understanding cosmological dynamics as well as the spread of infectious disease is the theory of dynamical systems, which has played a key role in my research for many years. Similarly, Bayesian statistics are extremely useful for analyzing both astrophysical observations and COVID case counts.
I have also been interested in how various phenomena spread on graphs. Several years ago, I made this video on how memes go viral on the internet:
The idea is that there is a social network online describing the connections between individuals. If one person becomes interested in something (back then, the “ice bucket” challenge was in vogue), they might share it with their contacts, who in turn might share it with other people, and so on.
You might be thinking that this process sounds an awful lot like how an infectious disease spreads. The purpose of the above video is to push this analogy as far as we can by creating a disease model of how a “meme” spreads on the internet.
How about actual infectious diseases? Adapting the techniques in the video to model COVID-19 in New Brunswick was the topic of a problem at the 2021 AARMS Industrial Problem Solving Workshop. I presented this problem in collaboration with The Black Arcs, a local Fredericton company with expertise in detailed computer simulations of daily life in cities and towns. This project is ongoing, and is starting to yield some exciting results.
Several years ago (for reasons that I can’t really remember) I became interested in 4-dimensional polyhedra. I made some animations that I like looking at, and I wanted to try out my new blog, so here we are…
4-dimensional polyhedra are the higher dimensional generalizations of three dimensional objects such cubes and tetrahedrons. You can think of them as solid objects that are highly symmetric; that is, their edges all have the same length, their faces are all congruent, and the angles at all of their vertices are the same. A familiar 3-dimensional example is a cube whose consists of six identical 2-dimensional regions, each of which is a square. The 4-dimensional version of cube is called a tesseract (or 8-cell). It’s boundary consists of 8 identical 3-dimensional regions, each of which is a cube.
How can you visualize a 4-dimensional object? Well, strictly speaking this is not something that human ought to be able to do, since our brains are trained to operate in a 3-dimensional world. But we can play a simple trick to draw pictures of 4 dimensional polyhedra by looking at their shadows. In the real world, the shadow of 3-dimensional object on a wall is 2-dimensional, and can therefore be accurately represented on a piece of paper. So, it is therefore possible to represent a 4-dimensional object by imagining what its shadow would look like on a 3-dimensional “screen”.
That’s exactly what the animated gifs shown below demonstrate: a 3-dimensional rendering of the shadow of several examples of these objects. Each of the polyhedra is rotating in 4-dimensional space.