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: Professor Yang Xu, Department of Computer Science, Unviersity of Toronto
Co-Supervisor: Ms. Aida Ramezani, Department of Computer Science, Unviersity of Toronto

Project Location: Fields Institute (Toronto, ON)

Project Overview

Can machines tell right from wrong? Recent advances in artificial intelligence show that large language models possess knowledge of human moral values. However, this knowledge is identified to be biased and does not represent the diverse range of moral values across cultures. In this project, the students will develop mathematical models of moral value variation across cultures and explore the extent to which these models can build fairer representations of cultural moral values in large language models. Candidates should have a strong background in mathematical and statistical modeling, computer programming, and experiences with quantitative data analysis and large language models.

Supervisor: Professor Augusto Gerolin, Department of Mathematics and Statistics, University of Ottawa
Co-Supervisors: Professor Khabat Heshami, Department of Physics, University of Ottawa & National Research Council of Canada; Professor Michael Schuurman, Department of Chemistry and Biomolecular Sciences, University of Ottawa & National Research Council of Canada

Project Location: Fields Institute (Toronto, ON)

Project Overview

Channeling the power of quantum mechanics will play an increasingly important role in the technological landscape of the future, including creating synthetic quantum materials with specific superconducting or magnetic traits, identifying molecules with beneficial pharmaceutical properties and transforming machine learning methods with quantum technologies.

Experimentally investigating these quantum materials, molecules and devices is prohibitively costly and time intensive, so theoretical studies and computational modeling are critical to mitigating costs and accelerating discovery. 

This project will contribute to the development of a mathematical framework and novel computational algorithms to enhance optimization techniques in quantum computation and quantum chemistry. On the learning side, students will gain proficiency in mathematical methods and computational algorithms based in (quantum) optimal transport theory.

Optimal Transport is a field of mathematics and economics that has recently emerged as a fundamental tool in machine learning. In optimal transport, the goal is to compute the minimum distance (or cost) between two probability distribution functions, as well as the optimal transport plan that achieves this minimum. Quantum Optimal Transport extends this theory to quantum states by minimizing a cost between two density matrices and identifying the quantum channel that achieves this minimum.

Students participating in this project should be motivated to explore either the theoretical (pen-and-paper) or computational aspects, or both. Ideal candidates should have a solid foundation in undergraduate-level probability theory, optimization and linear algebra, or possess strong programming skills. Prior experience with mathematical aspects (or language) of quantum mechanics is beneficial, but not required.

Undergraduate students from mathematics, physics, and computer science backgrounds are equally encouraged to apply. Our aim is to tailor goals based on each student’s background and motivation while fostering an interdisciplinary and collaborative team. Students will engage in complementary yet overlapping tasks, encouraging diverse perspectives and skill development.

Supervisor: Dr. Afroditi Talidou, Cumming School of Medicine, University of Calgary

Project Location: Fields Institute (Toronto, ON) - the project supervisor will be fully remote

Project Overview

Neuronal axons in the brain are classified as either myelinated or unmyelinated. Myelin, a compacted cell membrane wrapping axons in discrete sheaths, facilitates the rapid and efficient propagation of electrical signals. Changes in myelin sheath geometry, such as alterations in length or thickness, can result from neural activity or injury. This research project aims to investigate how structural changes in myelination influence the function of individual neurons or neural networks. Depending on the student's background, the focus could be on developing a mathematical model or creating computational tools for parameter exploration within an established model. The students are expected to have some knowledge of differential equations (ordinary or partial) and their numerical solutions.

Supervisor: Professor Maia Fraser, Department of Mathematics and Statistics, University of Ottawa
Co-Supervisor: Professor C.-M. Michael Wong, Department of Mathematics and Statistics, University of Ottawa

Project Location: University of Ottawa (Ottawa, ON)

Project Overview

Computer formalization of mathematics, the step-by-step writing of mathematical proofs using a proof assistant, is gaining increasing attention in various areas of mathematics and is also poised to play a crucial role in AI-based tools for research mathematics. This research project will explore certain topics in low-dimensional topology, possibly touching on contact geometry, and in parallel seek to formalize some of the material using Lean, currently the most widely used proof assistant for mathematics. Depending on the students' background, the emphasis could be more on formalization or more on novel mathematical results - or ideally both! The students are expected to have some knowledge of topology and basic differential geometry, such as curves and surfaces. Knowledge of manifold theory is a plus but not required.

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: Professor Krzysztof Kapulkin, Department of Mathematics, Western University
Co-Supervisor: Mr. Daniel Carranza, Department of Mathematics, Johns Hopkins University

Project Location: Western University (London, ON)

Project Overview

The goal of the project is to formalize a selection of theorems in topological combinatorics in the proof assistant Lean. The project combines three distinct areas: 1. topology (specifically homotopy theory) 2. combinatorics (specifically graph theory), and 3. formal theorem proving. Formal theorem proving, using computer proof assistants, is an emerging area in mathematics with several prominent mathematicians leading teams formalizing major results. Our goal is to formalize a selection of theorems in topological combinatorics, e.g., Lovasz's proof of Kneser's conjecture and the topological realization of discrete homotopy groups. In the course of this project, students will gain familiarity working with computer proof assistants and knowledge of selected topics in topological combinatorics. The specific formalization goals can be selected based on each student's background and focus on either more combinatorial or more homotopy-theoretic aspects.

Supervisor: Professor Masoud Khalkhali, Department of Mathematics, Western University
Co-Supervisor: Professor Asghar Ghorbanpour, Department of Mathematics, Western University

Project Location: Western University (London, ON)

Project Overview

Quantum error correction (QEC) offers techniques and theories to protect information in quantum computing systems. Since their inception in 1985, there have been significant advancements in the development and construction of various classes of codes. Notable milestones include the CSS codes (1996) and the stabilizer codes introduced in the late 1990s, which provided a unifying mathematical framework for quantum error correction. These advances paved the way for the emergence of more sophisticated QEC methods, such as Kitaev's toric codes (1997), which are foundational in the field of quantum topological codes. This Project aims to study topological QEC's via methods of random matrix theory.

Random matrix theory (RMT) has proven to be a powerful tool in understanding complex systems, including quantum error correction. The statistical properties of random states, such as those in the Laguerre ensemble, provide insights into the behavior of quantum systems under noise and decoherence. In particular, random matrix models have been used to analyze the stability of QEC codes and to study problems like Page's problem, which relates to singular value distributions and their implications for error resilience.

This project will investigate the application of RMT to the study of topological quantum codes, focusing on statistical models that mimic realistic noise environments and their implications for code performance.