Supervisor: Professor Christopher Bergevin, Department of Physics and Astronomy, York University
Co-Supervisor: Professor Natasha Mhatre, Department of Biology, Western University

Project Location: Fields Institute (Toronto, ON)

Project Overview

Our ears possess remarkable characteristics that allow us to sense the world around us, such as selectively amplifying incident sound to boost detection. However, much remains unknown about precisely how these processes arise, and how to best fix them when they break (i.e., hearing loss). This project will be interdisciplinary and broadly explore the rich nonlinear features at play in the biomechanics of the ear. Combining applied mathematics, biophysics, sensory physiology, and acoustics, students will explore their interests using theoretical, computational, and/or empirical tools. The broad goal is to develop a small group of students who will each work on their own unique project that synergistically ties together with the others. Expected themes include stochastic-driven coupled nonlinear oscillators, fluid mechanics, nonlinear time series analysis (including spectral methods), and energy transport (e.g., diffusive versus wave propagation). Students will gain/hone a range of skills as they work on novel hypothesis-driven research questions.

Supervisor: Professor Annie Lee, Ontario Tech University

Project Location: Fields Institute (Toronto, ON)

Project Overview

URIEL is a world language database that provides vector representations and distance measures for 8,171 languages of the world. Due to the significance of our recent contributions, URIEL+ was accepted in Coling 2025  https://arxiv.org/abs/2409.18472. With URIEL++, we are addressing the remaining gaps: feature importance and exploration, graph algorithms, and confidence metrics.

Feature importance enables us to identify the most and least important features both within individual datasets and across the entire URIEL+ knowledge base.  We are investigating which categories of features (specifically linguistic typological features) - such as syntactic, phonological, inventory, and morphological - are most important for the knowledge base. Our goal is to prioritize the addition of features from the most critical categories, enriching the knowledge base with more data.

In addition, to better understand the relationships between features, we have developed graph algorithms that demonstrate how the value of one feature can be inferred from others. In the future, we aim to expand this approach to include all features within the knowledge base, to fill in more features and further enhance the feature coverage of URIEL++.

Lastly we are incorporating different statistical and informatic theoretic measures to quantify the significance of these features and their distances for users.

Supervisor: 

Project Location: Fields Institute (Toronto, ON)

Project Overview

Supervisor: 

Project Location: Fields Institute (Toronto, ON)

Project Overview

Supervisor: 

Project Location: University of Ottawa (Ottawa, ON)

Project Overview

Supervisor: Professor Hadi Salmasian, Department of Mathematics and Statistics, University of Ottawa

Project Location: University of Ottawa (Ottawa, ON)

Project Overview

The Weyl algebra is a fundamental object of noncommutative ring theory, with a wide range of applications in geometry and analysis. The simplest example of a Weyl algebra is the algebra of linear operators on functions in one variable $t$, with two generators: multiplication by $t$, and differentiation in $t$. Despite its simple description, this algebra plays a key role in quantum mechanics: it yields a mathematically rigorous description of the phase space of the quantum harmonic oscillator. From this point of view, the Weyl algebra is closely related to problems in the field of representation theory that connect number theory, harmonic analysis, and algebra. The goal of this project is explore analogous features of certain deformations of the Weyl algebras that were recently introduced by the project supervisor and his co-authors.

The first step of the project involves reading parts of a research paper by the supervisor about the construction of a quantum deformation of the classical Weyl algebra. Along the way, the student(s) will also learn about the required algebraic tools such as quantized enveloping algebras, bialgebras, and R-matrices.

The second step of the project is to perform computations to investigate a conjectural connection between representations of certain non-commutative algebras known as i-quantum groups and the quantum Weyl algebra. The supervisor will explain nough background to the student(s), and the computation will be reduced to a concrete problem that can be tackled by methods of linear algebra and basic ring theory.

The third step is to write the findings of the project as an expository report (10-15 pages). Significant results will lead to a short research paper that can be submitted to a peer-reviewed publication venue.

Supervisor: Professor Stacey Smith?, Department of Mathematics and Statistics, University of Ottawa

Project Location: University of Ottawa (Ottawa, ON)

Project Overview

The Buruli ulcer is a neglected tropical disease with a paucity of mathematical models. The mode of transmission is unknown, so this project will "reverse engineer" the usual idea of fitting models to data and instead attempt to use modelling to determine (or rule out) potential transmission routes.

The Buruli ulcer is an emerging mycobacterial disease caused by Mycobacterium ulcerans which has seen a steady increase in reported cases over the last 25 years. The disease typically affects the skin in the form of a painless ulcer resulting from the damage caused to the subcutaneous fat layer by the bacterium's necrotic toxins; untreated, it can also affect the bones and joints, causing permanent disfigurement and disability. However, the mode of transmission is currently unknown. Treatment is very costly and requires the removal of infected tissue by excision or amputation; there is no pharmacological treatment at this time. Furthermore, there are very few mathematical models of this disease. The project would involve creating a mathematical model to account for varying methods of transmission: direct from soil or water, between humans via mosquitos, or an animal reservoir. The general model will account for all these factors simultaneously. We will then compare with available data to attempt to match the model with data, in order to see if modelling can provide clues as to the transmission routes. This is the reverse of most modelling, which builds models based on the underlying mechanism of disease transmission. Instead, we would use the model to attempt to elucidate information about the mechanism. 

Students will examine existing data in the literature in order to determine as much as possible about the mechanism of transmission. They will then develop a generalised mathematical model for the disease, analyse the model and perform a variety of numerical simulations for possible scenarios. Once the basic model is functioning, the students will then develop economic models to investigate the cost-effectiveness of treatment and prevention. The economic model will also be analysed and numerical simulations performed to determine which parameters have the greatest effect on the outcome, what range of options may be viable, and so on. This project will contribute to a broad investigation of neglected tropical diseases that I have been undertaking in the past several years. Mathematical models are often at the forefront of new investigations and there have been a paucity of models for most tropical diseases.

Supervisor: 

Project Location: Western University (London, ON)

Project Overview

Supervisor: 

Project Location: Western University (London, ON)

Project Overview

Supervisor: 

Project Location: Western University (London, ON)

Project Overview