“Numerical methods for differential equations” online poster session (2024)

Another session of my course on Numerical methods for the solution of differential equations has concluded, and, as usual there were some exceptional student final project. Please enjoy a selection of the research posters summarizing the excellent work!


Fredericton Fall Classic 2023

I participated in my very first organized run last weekend: the 5K race in the Fredericton Fall Classic 2023. Pretty happy with my time, and I won a jar of blueberry jam!


Slides: A classical-quantum approximation for bipartite systems

I massively underestimated the length of my talk today at the 2023 Canadian Association of Physicists Congress, and I didn’t get to do the second half. So, I thought I’d post the slides here in case they are useful (animations are not really working, but you can view them on YouTube). If you are interested, much more detail is available from our recent preprint.

COVID-19 Talks

Pandemic planning using urban mobility simulations

I am a bit tardy posting this, but here is a recording of a talk I gave in the Mathematics for Public Health Colloquium series on November 15, 2023:

And here is the abstract:

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?


Quantum billiards

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.

Here is another movie in a related geometry, the Bunimovich Stadium:

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.

Finally, this movie shows a quantum particle confined to the newly discovered aperiodic monotile:

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…


“Numerical methods for differential equations” online poster session (2022)

I had the great pleasure of teaching the course Numerical methods for differential equations for the fourth time in the winter of 2022. This time, it was part of the AARMS Advanced Course Program, which allowed students from outside of the University of New Brunswick to participate.

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!


University Research Scholar

I got some good news the other day: I have been selected to be a University Research Scholar at the University of New Brunswick from July 1, 2022 – June 30, 2024 (pending final approval by the Board of Governors).

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.

UNB website

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.

Here is a list of past award winners.

EDIT: Guess this is official now.


Thanks for the shout-out Dr Russell!

It was fun for UNB to get a shout out from New Brunswick’s Chief Medical Officer of Health Dr Jennifer Russell during the February 8 COVID-19 briefing:

This was at around the 6:45 mark. It has truly been a pleasure working with the folks at NB public health on COVID modelling, and I’m very much looking forward to our future collaborations.


Recapping a busy COVID modelling week

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.

Forecast of hospitalizations in the NWT from Wednesday, January 26

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.

I am very pleased to acknowledge that all the above modelling was done in cooperation with government scientists from multiple jurisdictions. The underlying modelling framework is the result of a yearslong effort in collaboration with the New Brunswick Department of Health. The work has also been supported by the National Sciences and Engineering Research Council of Canada (NSERC), the Canadian Institutes of Health Research (CIHR), the New Brunswick Health Research Foundation (NBHRF), Mathematics for Public Health (MfPH), the Atlantic Association for Research in the Mathematical Sciences (AARMS), and the Government of New Brunswick.



A retrospective of the pumpkins since Halloween 2013. The stranger choices were definitely due to my daughters.