Dr. David Muraki (muraki@sfu.ca) and Dr. JF Williams (jf_williams@sfu.ca) 

Project: Mathematics of the Weather

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!).  Many of the beautiful observed cloudscapes display a distinctive pattern that are generally believed to be formed by atmospheric instability.

Other areas of research involve the study of larger-scale weather flows, and the evolution of midlatitude storms.  Most of the projects involve the use of a research-quality numerical forecast model.

1 student is invited to work on this project.

Requirements: Students should be independently motivated undergraduates in the third or fourth year of their degree.  A strong interest in differential equations is essential, as is proficiency in a computational environment such as Matlab/Python.  Applicants should note that Math 462 Fluid Dynamics is offered this Spring term. Participants should have a natural curiosity for the workings of the Earth's atmosphere.

Dr. Cedric Chauve (cchauve@sfu.ca)

Project: Ambiguity and conflict in ancestral gene orders reconstruction

Ancestral gene orders reconstruction aims to infer putative chromosomes (gene orders, not DNA sequence) of extinct species based on the gene orders of sequenced descendant species. This is a computational biology problem that relies on using mix of algorithms from phylogenetics (reconstructing the evolutionary history of genes) and genome mapping. As in all evolutionary biology problems, the true ancestral gene orders are not known and one can merely infer hypothesis about them.

While phylogenetics has well established methods to handle this uncertainty (especially through Bayesian sampling), genome mapping methods do not (as they were developed to handle existing data for which the ground truth can be estimated).

The goal of this project is to revisit two recent high profile methods for ancestral gene orders reconstruction (EdgeHOG, https://www.nature.com/articles/s41559-025-02818-0 and AGORA https://www.nature.com/articles/s41559-022-01956-z) with a focus on the uncertainty and ambiguity in the ancestral gene orders they do provide.

The work will involve:

• data analysis (re-analyzing the data used in the papers cited above),
• method development (developing and implementing a measure of distance between alternative ancestral gene orders for a given species, work in progress currently).

Required skills: solid background in graph theory (e.g. MATH345), solid programming skills (python), interest for biological questions; no specific preliminary background in biology is required.

Dr. Razvan Fetecau (van@sfu.ca)

Project: Emergent collective behaviours modelled by differential equations

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 machine learning. 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 of self-collective behaviour 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. A particular model of interest is an interacting particle system used in large language models.

Requirements: Strong background in differential equations (MATH 260, MATH 314), and numerical methods for differential equations (MACM 316). MACM 416, MATH 418 and/or Math 467 are not required, but they would be great assets. Some experience with Matlab would be very helpful.

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

Project: Simulating and correcting quantum computers via the stabiliser formalism

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.  Studying this question also gives us insight into the poorly understood mechanisms that drive quantum computational power.

In the longer term, how will we protect highly sensitive quantum data from errors while operating on them?  As quantum data cannot be copied or directly observed, more clever schemes have been devised to enable quantum computation in the face of environmental noise.

This project will centre on the mathematics used to answer both these questions: the stabiliser formalism.  The goal will be to better understand the stabiliser formalism and to utilise it towards facilitating quantum error correction and verifying near-term quantum computers.  

The precise question tackled in the project, and the balance between theory and numerical computations, will depend on the interests and skills of the student.

1-2 students are invited to work on this project.

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

Dr. Imin Chen (ichen@sfu.ca) 

Project: Transformers and mathematics

Transformers have become the leading architecture for artificial intelligence applications because of their scalability and ability to effectively implement large language models. This project will investigate the use of transformers to machine learn a variety of arithmetic invariants in number theory, for example, the rank of an elliptic curve over the rationals. 

1 student is invited to work on this project.

Requirements: Interested applicants should have sufficient course work in algebra and number theory and clearly indicate their familiarity with Python and the PyTorch package.

Dr. Weiran Sun (weirans@sfu.ca) and Dr. Wuyang Chen (wuyang@sfu.ca)

Project: Lean Formalization for PDEs

Lean formalization has seen rapid development in recent years, with substantial progress made in algebra, topology, number theory, and analysis.
Partial differential equations (PDEs), however, remain largely absent from this landscape, despite their central role in modern mathematics, physics, and engineering. This project seeks to bridge the current gap by formalizing the foundations of PDE theory in Lean. We are looking for up to two motivated students who are interested in PDEs and have taken Math 418. The project will be based on Rustum Choksi’s PDE textbook used in Math 418. 

Participants will contribute to:

• Formalizing core PDE concepts
• Building reusable Lean libraries for functional analysis and PDEs
• Translating textbook proofs into fully verified Lean code
• Developing infrastructure that future PDE formalization can build upon

The outcome of this project will be integrated into existing PDE Lean code (weiran-sun.github.io/pde). Students will work closely with Prof. Wuyang Chen (SFU CS), Prof. Weiran Sun (SFU Math) and their team members.

DR. CAROLINE COLIJN (CCOLIJN@SFU.CA)

Project: Mathematics and music

Mathematics and music are deeply intertwined: from the ratios that define musical intervals to the construction of tuning systems to the symmetry and patterns that shape harmony and rhythm. The connections arise fundamentally because music has a tremendous amount of mathematical structure, and also because both mathematics and music appeal to our sense of beauty.

In this project, we will research how to use mathematical representations in music to help in the task of music learning. We will first ask: what mathematical frameworks most effectively capture and illuminate musical structures, and how can they be organized into a useful theory for learners?

There are two directions this project could take. One option is to focus on researching how to represent and illuminate musical structures using mathematics, and organize this into a theory for learners interested in math and music. This theory would feature, for example, mathematical definitions of tuning systems, examples of different tuning systems and how they sound, and harmonic progressions and their placement within tuning systems. It would then move to mathematical representations of harmonic progressions and common, simple tunes. Mathematics arises heavily in acoustics, and there are accessible acoustic applications (such as sound wave amplification and cancellation) that are readily treatable with undergraduate mathematics. Finally, the mathematical structures and patterns in music have, for decades, made automatic generation of music feasible. Now, as AI technologies develop, there is increasing interest in automatically-generated music.

The second direction is to focus on music learners. Here, we will ask: how can we use mathematical representations of music to develop gamified systems for music learning? For example, sight reading is a challenging task for music learners, and it involves the rapid recognition of common harmonies and harmonic progressions. By representing these mathematically, could we map sight reading tasks to computer games, in the same manner that (for example) learning to type is mapped to driving games? Or could the necessary sight reading tasks be encoded into scrolling platform games? The same questions could be asked for musical improvisation, which, while it sounds very free, typically comes with some well-accepted constraints (for example, which scales go with which chords, and how chords tend to change along harmonic progressions). In this direction, the goal would be to use mathematical representations of music to develop a computer game that helps with the task of music learning.

Requirements: This project requires a student with a strong interest and background in music and an interest in establishing, articulating and using mathematical representations of music. For the gamified systems, a student who is comfortable writing computer games would be ideal. In this case, this project can host two students.

Up to 2 students are invited to work on this project.

Dr. Paul Tupper (pft3@sfu.ca) 

Project: Improving Clustering Algorithms with Diversities

Clustering is the task of assigning a large number of unlabeled data points to a small number of classes. Typically, the data are points in a vector space and a metric is defined on the vector space, leading to a distance matrix for the data. When clustering, the metric is used to break up the data into a number of groups. But using just the metric may throw out a lot of information about the placement of the points in the space. 

Diversities are an extension of the idea of a metric space, where not just pairs of points but all finite subsets of points are assigned a value. The diameter of a set of points in a metric space is a diversity, but there are many other natural examples. In this project the student will investigate using the perspective of diversities to devise new clustering algorithms, investigating what works well computationally on large data sets.

Requirements: Programming skills. Strong background in mathematics, including knowledge of metric space theory. 

Up to 2 students are invited to work on this project.

Dr. JF Williams (jf_williams@sfu.ca) 

Project: Machine learning for time-stepping in differential equations

Are you interested in exploring how machine learning can be used to solve differential equations? This research project investigates whether neural networks can learn to adaptively select timesteps in numerical methods by observing solution behavior and predicting errors. Traditional time-stepping methods rely on carefully designed heuristics and error estimators, but what if a neural network could learn these strategies directly from data? You’ll develop and train networks that analyze the local dynamics of DE solutions to make intelligent decisions about when to take larger or smaller steps, then benchmark their performance against established adaptive methods. 

This project sits at the exciting intersection of numerical analysis and machine learning, offering hands-on experience with both computational mathematics and modern ML frameworks. You’ll gain practical skills in implementing numerical solvers, designing neural network architectures, and conducting rigorous computational experiments.

Pre-requisites: MACM 316 and experience with either Matlab or Python.

DR. CAROLINE COLIJN (CCOLIJN@SFU.CA)

Project: Prevent the next pandemic with data science 

During the COVID-19 pandemic, sequencing took off as a major tool in infectious disease. By sequencing (reading the DNA or RNA of a virus or pathogen), we can learn how a virus is spreading and evolving, and be prepared for what might come next.

But the benefits of virus sequencing depend on data sharing, which requires databases where public health labs can upload sequences, download sequences from other places, understand how their data fits in and what’s new in it, and how it relates to previous versions of their virus. The main database for sequence sharing has been GISAID, but its data use agreements are restrictive, and its owners can remove data and method providers’ access at any time. This caused serious problems during the COVID-19 pandemic.

Pathoplexus is a global, open, and community-driven platform designed to support real-time sharing and analysis of sequences. It provides an infrastructure that enables researchers, public health agencies, and clinicians to deposit, curate, analyze, and visualize sequences alongside relevant metadata. It has open and transparent governance and access control and aims to improve equity. Pathoplexus is new, and is going to integrate tools for phylogenetics, surveillance, and outbreak investigation.

To prevent the next pandemic, we need real-time data sharing and interpretable outputs that anyone can see. That’s what Pathoplexus, based in Zurich, is aiming to build.

In this project we will support the Pathoplexus team, for example: building the capability to generate visualizations like a map of selected viral sequences, lineages over time; visualize selected sequences in an alignment viewer; using LLMs to curate geolocation data and then implementing the solution into Pathoplexus. We will collaborate with the Pathoplexus team.

This project requires a data scientist (comparable expertise) and an interest (but not necessarily background!) in viruses, genetic data, and preventing pandemics. 

DR. CAROLINE COLIJN (CCOLIJN@SFU.CA)

Project: The mathematics and statistics of antimicrobial resistance (AMR)

Many species of bacteria (and sometimes other microbes) have become resistant to multiple antibiotics or treatments, making infections difficult or even impossible to treat. Overuse or misuse of antibiotics in healthcare, agriculture, and the environment can cause rises in resistance, and lead to superbugs. These so-called “superbugs” pose a major global public health threat: they lead to longer illnesses, higher healthcare costs and even deaths.

In this project we will explore how antibiotic consumption influences multidrug resistant strains vs drug-susceptible strains across Europe and in other settings. We will use ECDC antimicrobial resistance surveillance data to quantify how E. coli and other key pathogens  acquire and spread resistance to different antibiotic classes. Biological differences in resistance mechanisms—such as mutation-driven fluoroquinolone resistance versus gene-acquisition-driven carbapenem resistance—are expected to produce distinct statistical signatures. We will develop and apply statistical models to disentangle de novo resistance emergence from resistance that is spreading because it is being transmitted. We will identify pathogen–antibiotic combinations at highest risk of causing steep rises in dangerous antibiotic resistance. 

This project requires a student interested in AMR, with some background in biology, and strong knowledge of statistics or modelling.