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

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 Lia Bronsard, Department of Mathematics & Statistics, McMaster University
Co-supervisors: Dr. Dominik Stantejsky, Department of Mathematics & Statistics, McMaster University; Dr. Lee van Brussel, Department of Mathematics & Statistics, McMaster University

Project Overview

The general idea is to investigate established models in colloidal liquid crystals with the goal of finding defects structures through energy minimization. As this is generally very hard and subject to current research, we propose a simplified setting of a single spherical colloidal particle immersed in the nematic liquid crystal. Therefore, we want to find explicit solutions to the associated Euler-Lagrange equations, namely systems of linear (but coupled) partial differential equations with appropriate boundary conditions.

We expect that the students will be able to successfully employ Fourier transform and Neumann series to derive the fundamental solution and Green's function in order to determine, in a second step, the position and shape of the singular structures. This project will contribute to the understanding of defect structures of vector valued models in soft matter, and includes important terms in the energy which have not been studied so far. 

Supervisor: Professor Tom Chau, Institute of Biomedical Engineering, University of Toronto
Co-supervisors: Erica Floreani, PRISM Lab; Dr. Ledy Holanda, PRISM Lab; Jenny Tou, PRISM Lab

Project Overview

The proposed research project aims to develop deep learning techniques to enhance the quality of EEG brainwave data for brain-computer interfaces (BCIs). Traditional methods for cleaning EEG signals are computationally intensive and fail to effectively suppress certain types of noise. Deep learning techniques are well suited to model complex relationships in data, and can produce pre-trained, deployable models that can easily be run in real-time. This project will focus on the development and assessment of mathematical models and algorithms for real-time cleaning of EEG data using state-of-the-art deep learning techniques. Objectives include the creation of a dataset tailored for developing EEG cleaning algorithms, the replication and validation of previously suggested approaches, the proposal of novel methods, and evaluation of their performance for real-world BCI tasks.

Co-supervisors: Professor Paul Ayers, Department of Chemistry & Chemical Biology, McMaster University; Professor Farnaz Heidar-Zadeh, Department of Chemistry, Queen's University

Project Overview

In the last 30 years, the theory of optimal transport (OT) has emerged as a fertile field of inquiry and a diverse tool for exploring applications within and beyond mathematics, in such diverse fields as economics, meteorology, geometry, statistics, fluid mechanics and engineering. Nowadays, OT theory is also becoming a fundamental tool in Quantum Chemistry, Physics and Machine Learning research.

The general objective of this research project is to solve a specific problem in Quantum Chemistry and Physics (strongly correlated many-electron systems) using Optimal Transport techniques. Towards this goal, we will form a team of students with complementary expertise.

Depending on the student's background, motivation and interests, they could focus on the development of mathematical formalism (pen and paper), modelling of chemical phenomena or the development of scientific computing tools. The concepts/mathematical, computational, machine learning tools will be adapted according to the students' background and the skills the students taking part of this program are willing to acquire. This project is designed to have different difficulty levels and is a way to give the opportunity for students to learn a broad range of techniques in mathematics, computer sciences, chemistry and/or machine learning (in a highly collaborative environment).

Supervisor: Professor Sebastian Jaimungal, Department of Statistical Sciences, University of Toronto

Co-Supervisors: Professor Piotr Zwiernik, Department of Statistical Sciences, University of Toronto; Dr. Ziteng Cheng, Department of Statistical Sciences, University of Toronto

Project Overview

Dynamic programming, traditionally used for sequential decision-making, finds applications in reinforcement learning, economics, finance, forestry, and bioinformatics. Recently, optimization procedures based on expected cost have been enhanced by incorporating more comprehensive risk measures, allowing for direct integration of risk preferences into decision making. This project aims to extend these concepts to more intricate dynamic decision processes that are not necessarily sequential. We propose to broaden Markov risk measures to processes that evolve along a tree or, more generally, a directed acyclic graph. We will demonstrate the practicality of this concept in various fields, such as risk assessment and management in the financial sector.

Supervisor: Professor Masoud Khalkhali, Department of Mathematics, Western University

Co-supervisor: Nathan Pagliaroli, Department of Mathematics, Western University

Project Overview

Large collections of data sets can often be packaged as ensembles of random matrices. A primary goal is to understand the distribution of eigenvalues of these matrices. In this project we apply bootstrapping, a relatively new technique, to understand the distribution of eigenvalues and eigenvectors of specific ensembles of random matrices.

In this project we will use the bootstrap method to find the moments of certain multi-trace and multi-matrix random matrix models of large size N. The large N limit of these moments satisfy an infinite system of nonlinear equations, due to Migdal called loop equations. The loop equations are consequences of Schwinger-Dyson equations. The Schwinger-Dyson equations put constraints on the moments and these constraints help to narrow the search for moments, but this is usually not enough. The process of further narrowing the search space by adding certain extra positivity constraints is called bootstrapping. This idea was recently used in the context of lattice gauge theory. Then it was applied to random matrices. In our project we apply bootstrapping to obtain bounds on moments and eigenvalue distribution of various random matrices. An open problem is to understand the nature of sub leading terms beyond planar contributions.

Supervisor: Professor Annie Lee, Department of Computer Science, Ontario Tech University

Project Overview

Natural language processing (NLP) is an interdisciplinary subfield combining linguistics and computer science to solve text-based problems (called tasks) dealing with natural language. However, popular machine learning approaches (i.e. statistical modeling, deep learning) have dominated the field such as the revolution of Large Language Models (LLMs), a type of foundational model used as scaffold for fundamentally changing the way the NLP tasks are solved . However, machine learning is known to be data and compute-hungry, which only applies to a handful of languages, thus causing a widening gap for the majority of the 7000+ languages spoken in the world. For Low-Resource Languages (LRLs), resources such as text datasets (called corpus) are not available to create LLMS or to solve specific NLP tasks.

In this project, we will examine the various methods for computing language similarity and diversity. The goal is to quantify the distance (as opposed to similarity) between two languages in a manner that would be best for cross-lingual transfer. In this task in particular, we will focus on the a diversity of langauges.

Supervisor: Dr. Mariya Naumova, Rutgers Business School, Rutgers University

Project Overview

The project focusses on constructing a mixed-integer linear optimization model for maintenance and operations of an offshore wind farm based on high-impact risk factors. The participants are expected to contribute to the model development based on the information about the wind farm and its testing under different meteorological scenarios.

Some experience with formulation of mixed-integer linear problems and knowledge of machine learning techniques are desired but not required. If the skills are insufficient, the students will be asked to do some reading prior to the beginning of the project. Moderate programming skills are expected. Students will receive both analytic and computational training. We plan to present the results at the related seminars and conferences. Publication of a paper is possible as well.

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

Project Overview

The goal of this project is to utilize models from statistical mechanics to prove combinatorial formulas for distinguished classes of special functions that play a significant role in several areas of mathematics (e.g., algebraic geometry and representation theory). One extremely successful realization of this goal that has attracted substantial attention in recent years is based on the partition function of a lattice model. Lattice models have been studied thoroughly in condensed matter physics and have fascinating connections with algebraic objects known as quantum groups. From a physical viewpoint, the partition function describes statistical properties of a system in thermodynamic equilibrium. The goal of this project is to examine the existing works on deriving combinatorial formulas from lattice models and then modify these constructions to establish new identities in the context of symmetric function theory.

Supervisor: Professor Asif Zaman, Department of Mathematics, University of Toronto

Co-supervisor: William Verreault, Department of Mathematics, University of Toronto

Project Overview

The distribution of prime numbers has played a foundational role in analytic number theory and its development. A powerful perspective is to ask probabilistic questions about multiplicative functions, whose behaviour is dictated by their value on the primes. Many multiplicative functions, such as the Mobius function, are believed to behave "randomly" like flipping a coin for each prime. A barometer for randomness lies in the study of random multiplicative functions, i.e. multiplicative functions that are defined purely in probabilistic terms. This subject has seen remarkable progress in recent years and this progress has potential applications for analogues with function fields or partitions. This project aims to advance these applications in various potential directions.