Dr. Ben Adcock (adcockb@sfu.ca)

Project: Optimal sampling strategies for surrogate model construction in high dimensions: From Christoffel to Kadison-Singer

Mathematical models of complex physical processes involve many parameters. Parametric modelling, in which one tries to determine how the parameters of a model affect its output, is a crucial task when trying to interpret, optimize or otherwise analyze physical processes. This is typically done by constructing a surrogate model, using sample data generated from simulations of the model. Surrogate modelling is challenging, due to the high dimensionality of the model and the high computational cost in generating the data.

The goal of this project is to develop novel, and ideally near-optimal, sampling strategies for approximating high-dimensional functions arising from parametric PDE models. We will focus on scenarios where the parameters range in infinite intervals, rather than bounded intervals. This project will commence by considering polynomial surrogate models using the so-called Christoffel function, before investigating more recent adaptive sampling strategies for deep neural network models trained via deep learning. Finally, we will also investigate the practical use theoretical advances related to the solution of the Kadison-Singer index problem. In theory, these results can be used to obtain formally optimal sampling procedures. Yet little is known about their practical performance, especially for problems arising from parametric PDEs.

Requirements: Linear algebra, basic analysis, numerical analysis (MACM 316 or equivalent) and proficiency with MATLAB are essential.  

Dr. Ben Ashby (bashby@sfu.ca)

Project: Modelling evolution in the host microbiome

Microbiomes play a key role in host health, yet we know little about the ecological and evolutionary processes that govern these communities. In particular, we know very little about how microbes evolve to be more harmful (parasitic) or helpful (mutualistic) to their hosts. Studying microbial evolution along the parasitism-mutualism continuum has important implications not only for our understanding of fundamental ecological and evolutionary processes, but also for future therapeutics to treat infectious diseases through manipulation of host microbiota. 

In this project, you will use mathematical models to understand the ecological and evolutionary dynamics of microbes in the host microbiome. You will be expected to adapt and analyse systems of ODEs to answer key questions about how microbial traits evolve. This work forms part of a longstanding research program about microbial evolution along the parasitism-mutualism continuum.

Requirements: Previous experience with dynamical systems (ODEs) and coding in MATLAB or Python.

Dr. Cedric Chauve (cchauve@sfu.ca) and Dr. Mahsa Faizrahnemoon (mfaizrah@sfu.ca)

Project: optimization-based method to detect plasmids

Plasmids are small circular pieces of DNA found in bacterial cells that are important due to their role in the spread of antimicrobial resistance (AMR), a major public health concern. Therefore, the problem of discovering plasmids in sequencing data from bacterial pathogens is a very active research area in bioinformatics. Bioinformatics is a research field aimed at designing computational methods for the analysis of biological data.

The problem of discovering plasmids from sequencing data can be addressed through combinatorial optimization approaches that analyze a graph known as the assembly graph, that is generated when sequencing data is assembled. Recently we have designed such a method based on an iterative optimization framework which is implemented in a Mixed Integer-Linear Program (MILP). Our method outperforms existing methods, although its iterative nature leads to well characterized errors in the predicted plasmids.

The project we propose is to develop a post-processing method, based on local search in this graph in order to refine and improve the solution obtained by the iterative MILP. The post-processing algorithm will be evaluated on a large-scale real data set for which the ground truth (actual plasmids) is known.

Requirements: The student should have a keen interest in the application of mathematical and computational methods in biology as well as working with large data. Strong programming skills (in python) and a good background in graph theory and optimization is required. No specific biology background is required although it is expected the student will develop some knowledge about genomics and plasmids during the course of the project. This project offers an excellent opportunity to discover the application of mathematical methods in the context of data science.

Dr. Nadish de Silva (nadish_de_silva@sfu.ca)

Project: Classical simulation of quantum computation via stabiliser methods

Small quantum computers are currently under construction.  How will we verify that they are working correctly?  We might try to simulate their operation on an existing conventional supercomputer.  This would be slow and difficult, however, as quantum computers, by design, perform tasks beyond the abilities of conventional computers.  Thus, clever schemes have been devised to classically simulate quantum computers as efficiently as possible within limited regimes.  Beyond their anticipated practical use, the study of classical simulations of quantum computation gives us insight into the poorly understood mechanisms that drive quantum computational power.

This project will centre on studying the most widely utilised schemes: those based on the stabiliser formalism.  The goal will be to better understand the rich lattice-theoretic geometry underlying the stabiliser formalism and to utilise it towards advancing state-of-the-art techniques for verifying near-term quantum computers.

Requirements: Strong background in mathematics and computer science.  Coding skills in Python could be an asset but are not strictly necessary.

Dr. Razvan Fetecau (van@sfu.ca)

Project: Emergent collective behaviours on nonlinear spaces

This project concerns mathematical models of swarming and self-collective behaviour. The topic has received a great amount of interest in recent years due to applications of such models in a variety of areas, including population biology (chemotaxis of cells, swarming or flocking of animals), physics and chemistry (self-assembly of nano-particles), robotics and space missions, and opinion formation. The models have demonstrated emergence of very complex behaviours as a consequence of individuals following simple interaction rules, without any leader or external coordination.

More specifically, the project will consider applications to swarming on nonlinear spaces, such as surfaces in 3 dimensions. The student will implement numerical methods to simulate the models on various surfaces (eg., sphere, hyperboloid). The major interest is to investigate numerically (and possibly analytically) the emergent behaviours, in particular the qualitative properties of the swarm equilibria.

Requirements: Strong background in differential equations (MATH 260/310, MATH 314), and numerical methods for differential equations (MACM 316). MACM 416 and/or MATH 418 are not required, but they would be great assets. Some experience with Matlab would be very helpful.

Dr. Razvan Fetecau (van@sfu.ca) and Dr. Steve Ruuth (sruuth@sfu.ca)

Project: Numerical methods for partial differential equations on surfaces

The project aims to investigate numerically and analytically certain partial differential equations (PDEs) set up on surfaces. The class of PDEs that we will consider include the p-Laplacian operator, with the infinity Laplacian as a special case (as p tends to infinity). Such operators and PDEs have applications in image processing and game theory.

The student will review relevant existing literature, will implement various numerical methods to simulate these PDE's on simple surfaces such as a sphere or a cylinder, and will investigate numerically (and possibly analytically) these methods.

Requirements: Students are expected to have taken and performed well (with grades in the A range) in MACM 316 and MATH 314. Having taken MACM 416 and MATH 418 are great assets. Students need to be proficient with scientific computing using Matlab.

Dr. Ailene MacPherson (ailene_macpherson@sfu.ca) and Dr. Lloyd Elliott (lloyde@sfu.ca)

Project: Estimation of Disease Induced Extinction

In addition to their threat on human health, infectious diseases are an important threat to wildlife and a major driver of extinction. Smith et al. (2006) found that infectious diseases contributed to ~4% of recent extinctions and is a conservation threat for 8% of species known to be critically endangered. While disease is undoubtedly important, there is one important limitation to our current understanding of disease induced extinction risk. What if the 8% of species at risk from disease are all frogs? How would that change our understanding rather than assuming it is a random collection of all animals? This would have huge implications for our understanding of the role disease plays in conservation.

In this USRA project you would determine a robust estimate of extinction risk due to disease using "phylogenetic comparative methods" which are statistical methods developed for use on the tree of life. You would begin by curating data from the International Union on the Conservation of Nature (IUCN) Red List on extinction risks across the animal kingdom. You would then use this data to calculate an overall "evolution aware" estimates for the conservation risk of disease and associate epidemiological and biological correlates. This USRA is designed to be carried out on a full-time basis over the summer of 2023 with the expectation of presenting your work at the Eco-Evo retreat in Squamish BC in the following fall term. The project is co-supervised by Dr. Ailene MacPherson (Math) and Dr. Lloyd Elliott (Stats).

Ref: Smith et al. (2006).

Requirements: Stats 240 (Introduction to Data Science) or analogous background

Dr. David Muraki (muraki@sfu.ca)

Project: Computation of Fluid Models for Atmospheric Science

Students are invited to join a research effort that uses computational models to understand the fluid mechanics of the weather.  There are active projects that investigate a variety of atmospheric phenomena.

One current area of interest involves projects connected with the question, "What is the shape of a cloud"?  We have developed a new mathematical model for the motion of cloud edges --- one that has
already confirmed the behaviour of "lenticular" clouds caused by airflow over mountains, and somewhat rare phenomenon known as a "holepunch" cloud (search for images!).

Requirements: Students should be independently motivated undergraduates in the third or fourth year of their degree. Some background in a differential equations is essential, as is proficiency in a computational environment such as Matlab. Participants should have a natural curiosity for the workings of the Earth's atmosphere.

Dr. Nilima Nigam (nna29@sfu.ca) and Dr. James Wakeling (wakeling@sfu.ca)

Project: Muscles: from MRI data to biomechanics

Muscles - in concert with tendons and other connective tissues - are critical to locomotion, and exhibit fascinating mechanical properties. There's great interest in the fields of muscle physiology as well as neuromuscular science around how the structure and composition of actual muscle tissue impacts muscle function. Mathematical models can provide a great deal of insight, and that's where our work lies. Medical imaging data (especially MRI data) of muscle-tendon units need to be translated into hexahedral meshes which are used as part of a discretization strategy for the equations governing the dynamics of muscle.  Several questions arise: what are the mathematical conditions on allowable meshes? How do we ensure the meshes are topologically consistent? How does one robustly identify the directions of muscle fibres from MRI data? How does one (if at all) find thin sheets of connective tissue from the data?

You'll learn about both the physiological/medical questions which are driving these investigations, and the state-of-the-art discretization strategies we're using. Your work will be at the interface of mathematics, computer graphics and biomechanics, and you'll be part long-term collaboration with the Neuromuscular Lab in BPK. You'll be given access to code libraries to start building your ideas on.

Up to 2-3 students are invited to join this project.

Requirements: Prior experience in computing is required (Matlab OK, C++/Python preferred). Candidates should be very comfortable with the contents of  strong undergraduate course in  numerical analysis (316 minimally, 416 ideally) and ideally have some background in basic mechanics (either a physics, engineering or BPK course); a PDE course would be an asset. Since this is a highly interdisciplinary project, it's pretty unlikely anyone will have all the necessary background; learning it is part of the project outcome. The ideal candidates will be curiosity-driven, motivated, and willing to read/learn independently.

Dr. Nilima Nigam (nna29@sfu.ca)

Project: Spectral geometry for the Steklov-Maxwell system

If you've taken a first PDE course, you've likely encountered the so-called Dirichlet eigenvalue problem for the Laplacian (the eigenfunctions are used in the the technique of separation-of-variables). In these, the eigenvalues depend quite strongly on the spatial domain. In Steklov eigenvalue problems, we seek eigenmodes of a given PDE operator (could be the Laplacian!) in which the eigenvalue relates the Dirichlet and Neumann data. In this project we'll first look at a mathematical statement of the Steklov eigenvalue problem for Maxwell's equations. Next, we'll frame and answer questions about how the geometric properties of the domain influence the spectrum.

Requirements: A strong background in vector calculus, analysis and PDE (Math 320 minimum, Math 314 minimum, Math 418 an asset). The ideal candidate will have had some exposure to electromagnetics, though this is not a requirement; a willingness to learn and read independently is a must.

Dr. Alexander Rutherford (arruther@sfu.ca)

Project: Simulation Modelling for Pandemic Preparedness

The recent pandemic demonstrated the importance of using mathematical models to plan and evaluate strategies for responding the major infectious disease events. This project will involve developing mathematical models of epidemics and integrate them with health system models to evaluate strategies for pandemic response. It will build on ongoing research with the BC Centre for Disease Control and the BC Ministry of Health to model the impact of the COVID-19 pandemic on the critical system. This project will focus on developing a hybrid simulation model to evaluate the impact inequities within the healthcare system on pandemic response. The student in this project will works with researchers at The Pacific Institute on Pathogens, Pandemics and Society (https://pipps.ca).

Requirements: This project is well-suited to a student interested in applications of mathematical modelling to epidemics and public health. A course in probability models, simulation modelling, or operations research would be beneficial (MATH 208W, MATH 348, or Math 402W). Programming experience in R, MATLAB, or python is required.

Dr. Manfred Trummer (trummer@sfu.ca)

Project: High-order solvers for differential equations

Many natural, scientific, and economic phenomena can be described in terms of differential equations.

The project investigates spectral methods and radial basis function methods for solving such equations.  Implementing these methods often involves the solution of linear systems of equations. The matrices arising in these systems have very interesting structure and properties that can be exploited for efficiency. These techniques are also useful in other areas of science and technology. Topics include designing effective preconditioners for iterative solvers, and investigating new methods for differential-difference equations (equations with a delay). 

Requirements: Accessible to students who have completed a first course in numerical analysis.

Dr. Paul Tupper (pft3@sfu.ca)

Project: Extending diversities to infinite sets of points

Given a set X, a diversity d on X is a function from the set of finite subsets of X to the nonnegative reals. d must satisfy three conditions to be a diversity: it must give zero to the empty set and singletons and positive values to all other sets; it must be monotone (adding more points to a set does not increase d); and it must be subadditive on intersecting sets (if A and B intersect, that d(A union B) is less than or equal to d(A) + d(B).

Diversities are an extension of the idea of a metric space, in that when we restrict a diversity d to pairs of points, we get a metric space.

There are many set functions that arise naturally in geometry that are diversities, including the diameter, the circumradius (radius of smallest enclosing ball) and the length of the minimal traveling salesman circuit of a set of points. 

A natural question is whether we can extend diversities to arbitrary infinite subsets of the base set X. What are the natural axioms for such an object? It turns out that there are multiple non-equivalent ways to extend the definition, partly depending on which equivalent definition of diversities for finite sets you start with. The student will investigate these alternate possible definitions, and try to determine which has the best theoretical properties, and includes the most interesting examples.

Requirements: Students must have taken Math 242 and performed well in a third-year pure math course such as Math 320 or 340.

Dr. JF Williams (jfwillia@sfu.ca)

Project: Efficient Fast Fourier Transforms in the presence of symmetry

In this project we will work on minimizing functionals which describe problems from material science. Solving these minimization problems requires taking many Fast Fourier Transforms (FFTs). But, the structures we are interested in resolving posses crystallographic symmetries such as reflection symmetry, two-fold or three-fold rotational symmetry, shears, and so on. This allows us to accurately represent solutions with a small number of Fourier modes. However, traditional FFT algorithms do not take advantage of this simplification. The goal of this project is to use the group properties to reduce either the time or memory requirements of taking FFTs to solve more difficult and challenging problems.

Requirements: Prior experience in computing in MATLAB is required. A PDE course such as 314 or 418 would be an asset.

Dr. Jessica Stockdale (jessica_stockdale@sfu.ca) and Dr. Caroline Colijn (ccolijn@sfu.ca)

Project: Modelling and simulation of outbreaks and their virus genomes: can we use genomes to understand the force of infection?

The “force of infection” in an outbreak describes the portion of infections coming from different groups. For example, if younger or older individuals, or individuals in one location or another, are more at risk of infecting someone else, the “force of infection” coming from that group will be higher. It is further affected by the contact patterns and thereby transmission dynamics between infectors and infectees: not just who infection is coming from, but who it is going to. This force of infection is very hard to measure directly, because we do not usually observe transmission events, but it is fundamental to infectious disease modelling. Knowing more about the force of infection can help us write better models containing better descriptions of transmission. 

In this project, we will explore how we can use virus genomes (collected from infected individuals) to understand this force of infection. You will build simulations of outbreaks that incorporate different ages or locations, so that we know the true (simulated) force of infection, with accompanying virus genomes.  We will investigate under what circumstances virus genomes can give us information about the force of infection, vs when they cannot. This sets the stage for improving how we use sequence data to understand infections and their spread, and how we can begin to use sequence data for forecasting.

Requirements: Some experience in modelling, computation and/or simulation, along with knowledge of a programming language such as R, Python, or MATLAB. 

Dr. Caroline Colijn (ccolijn@sfu.ca)

Project: Useful and less useful metric spaces for trees 

Background:
Phylogenetic trees are the primary tool for visualizing and analysing patterns of ancestry. The tree of life is one example: genetic data from many species are analysed together to understand which species are most closely related, when their common ancestors were, and which of these ancestors evolved from each other. In infectious disease, phylogenetic trees describe information about how infections spread from person to person, or at a wider scale, across geographical regions. The trees also include information about evolution, for example showing the ancestry and spread of antibiotic-resistant forms of a bacteria, or of new viral variants. Phylogenetic trees are key for our understanding of how infections evolve and spread. 

Phylogenetic trees are usually defined as binary trees: graphs, with no cycles, with leaves (nodes of degree 1) and internal nodes (of degree 3). Leaves correspond to organisms (for which we have data), and internal nodes indicate shared ancestry (which we seek to discover). Rooted trees have a special node with degree two, and edges are directed away from that root, and towards the tips -- forward in time.  Reconstructing phylogenetic trees from data is challenging. Usually, DNA or RNA sequences from a set of organisms (like viruses, bacteria, plants, animals or other taxa) are the input, and two organisms are placed close to each other on a tree when they are closely related. But sequence data doesn't usually uniquely define a tree, and for decades, researchers have been building and improving methods for tree reconstruction. Bayesian methods proceed by sampling many trees, capturing not just one tree from data, but a collection of trees sampled from the posterior. But it is very hard to tell when this approach is done: have we sampled enough trees?

The project:
In this project, we will use ideas from discrete mathematics to explore metric spaces of trees, with a focus on which metric spaces are "good", and how they could contribute to this challenge. A number of metrics -- in the sense of true distance functions -- exist for phylogenetic trees. But which metric spaces are useful, and for what? Are there more metric spaces we could create that would be more useful? One way to think about whether a metric space for phylogenetic trees is a good one is to ask: are trees that are closer together in the metric also similar in how well they describe the data? Why or why not? Having good metrics would help us to know whether we have sampled enough trees to capture the collection of trees supported by data. In other words, good metrics can form the basis of heuristics to determine whether a Bayesian method has converged. This would have a wide range of applications.

With this motivation, we will explore the usefulness of current and new metrics on phylogenetic trees, in collaboration with colleagues who are analysing data from COVID-19, tuberculosis and other organisms. The project has quite a bit of flexibility for creative thinking (new metrics and heuristics) and independence, and a very solid jumping off point with a clear question, data and existing software for making trees and for computing distances under different metrics. 

Up to 2 students are invited to join this project.

Prerequisites:
This project requires some background in discrete mathematics, probability and statistics. It requires a curiosity about biology and evolution, but you do not need formal background in them. The project requires programming in R and the ability to work independently.