Supervisor: Dr. Adam Stinchcombe, University of Toronto
Co-Supervisor: Prof. Mihai Nica, University of Guelph

Project Overview

Boundary integral equations can describe a number of natural phenomena, but their numerical solution can be challenging to implement. In this project we develop solutions to these equations using random walks and ideas from reinforcement learning (a branch of machine learning). This method is particularly adept at handling high dimensional problems and is comparatively easy to implement. Many applications are possible: scattering problems (e.g. the radar cross-section of stealth aircraft), fluid dynamics (e.g. Stokes flow), and elliptic partial differential equations (e.g. from electromagnetics or acoustics). Students will learn some probability theory and some theory behind machine learning. The expected outcome is a high-performance implementation of the method with an accompanying journal article which solves boundary integral equations using random walks and reinforcement learning.

Supervisor: Prof. Annie Lee, University of Toronto

Project Overview

"Taking a step back, the actual reason we work on NLP problems is to build systems that break down barriers. We want to build models that enable people to read news that was not written in their language, ask questions about their health when they don't have access to a doctor, etc." -- "The 4 Biggest Open Problems in NLP", Sebastian Ruder (https://ruder.io/4-biggest-open-problems-in-nlp/)

Natural Language Processing (NLP) has significantly improved due to advancement in deep learning, however deep learning requires large amounts of annotated text for training the machine learning model and is unrealistic for many of the 7000+ languages currently being in use around the world today. Therefore NLP of low-resource languages (LRLs) is still considered a major challenge. Therefore, in recent years, there has been a noticeable increase in NLP research (both by academia and industry) that specifically focused on LRL pairs.

Supervisor: Prof. Azadeh Farza, University of Toronto

Project Overview

Trace Theory denotes a mathematical theory of free partially commutative monoids. Traces (elements in a free partially commutative monoid) are given by a sequence of letters (or atomic actions). Two sequences are assumed to be equal if they can be transformed into each other by equations of type ab = ba, where (a,b) belongs to a predefined relation between letters, called a partial commutation.

The analysis of sequential programs describes a run of a program as a sequence of atomic actions. On an abstract level such a sequence is simply a string in a free monoid over some (finite) alphabet of letters. This abstract viewpoint embeds program analysis into a rich theory of combinatorics on words and a theory of automata and formal languages. Trace theory does the same for concurrent programs.

In this project, we investigate mathematical relaxations of traces, so that the equivalence classes induced by them become larger. This leads to algorithmic benefits when these are used to model behaviours of concurrent systems. The goal to investigate what theoretical results from classic trace theory can be transferred to the new setting, and how these results help in devising algorithms for analysis of concurrent and distributed systems.

Supervisor: Mr. Faisal Habib, Fields Institute
Co-Supervisors: Prof. Arvind Gupta, University of Toronto | Prof. Huaxiong Huang, York University | Prof. Shixin Xu, Duke Kunshan University

Project Overview

We have a dataset of 3D image files. Using machine learning algorithms, we are interested in identifying features from 3D images of blood vessels where brain aneurysms may potentially occur.

Supervisor: Prof. Hans Boden, McMaster University
Co-Supervisors: Prof. Will Rushworth, Syracuse University | Prof. Homayun Karimi, McMaster University

Project Overview

This project will be an investigation of properties of knots that are defined by looking one dimension higher; that is, by viewing the knot and any surfaces that it bounds in 4-dimensional space.

Supervisor: Prof. Jeremie Lefebvre, University of Ottawa

Project Overview

Myelinated nerve fibres, notably found in white matter, orchestrate brain communication between different brain regions. The conduction of action potentials along WM is strongly influenced by myelin, a fatty substance wrapping around axonal membranes. Myelin allows action potentials to both transmit quickly and without attenuation. Most importantly, however, myelin influences axonal conduction delays, that is, the time it takes for action potentials to reach their destination. Tightly calibrated conduction delays are essential for neural communication and memory formation. Memory circuits of the hippocampus are a great example of the importance of myelin– especially in the the formation of certain types of oscillations important for memory encoding. Such oscillations can be impaired in diseases such as epilepsy, resulting in profoundly debilitating consequences on brain function. Mathematical and computational models of such circuits with varying levels of biophysical detail have been generated to understand the mechanisms of hippocampal theta rhythm. However, current approaches do not yet provide satisfactory mechanistic explanations. The goal of this project is to help a team of mathematicians and neuroscientist better understand how such circuits work, and how myelin and other features of these network influence the formation of oscillatory activity required for memory formation.

Supervisor: Prof. Maurizio De Pittà, University of Toronto

Project Overview

We aim to derive simplified equations for diffusion in pipes (or channels) and thin shells built around these pipes. These geometries are chosen because they can be realistically adapted to segment brain cells. At the same time, the type of equations that we want to derive aim to mimic essential features of intracellular molecular signals in a mathematically tractable framework. This is a challenging but fun project that looks for students interested in applied math and multidisciplinary research. The students should have a background in differential geometry and complex variable calculus.

Supervisor: Prof. Mei Zhen, University of Toronto
Co-Supervisors: Dr. Tosif Ahamed, University of Toronto

Project Overview

Random graph theory has emerged as a powerful tool in neuroscience to study the structure and dynamics of network of biological neurons. My lab has pioneered the collection, annotation and analysis of wiring diagrams (connectomes) in the roundworm C. elegans. In this project, students will use random graph theory to analyze and compare connectomes collected across different environmental and across animal development.

Supervisor: Dr. Na Yu, Ryerson University
Co-Supervisor: Dr. You Liang, Ryerson University

Project Overview

In recent years, biomedical hyperspectral imaging (BHSI) has been an emerging technology that provides great potential for many clinical applications, for example, noninvasive disease diagnosis and image-guided surgical procedures. From a mathematical point of view, BHSI is a sequence of three-dimensional (3D) matrices (also known as HyperCubs) where each spectral layer (i.e. two-dimensional matrix) records the radiance observations of biological samples at a specific wavelength going beyond the visible human spectrum from ultraviolet to near-infrared. Simple image methods are not able to extract information of interest (e.g., healthy and malignant tissues) from high-dimensional BHSI data. We propose to develop new and optimal approaches for dimension reduction (especially the size of spectral layers), image classification (e.g., determining material composition in pixels, such as various pathological conditions), and detect targets of interest in BHSI data.

Supervisor: Prof. Yang Xu, University of Toronto

Project Overview

Languages vary substantially, but they share commonalities in how they package meanings into words. This project focuses on colexification, the phenomenon that a single word form encodes multiple related meanings. Recent work shows that colexification relies on similarity relations between concepts, but the nature of these similarity relations is unclear. Students will be developing the mathematical and computational formalisms for solving two subproblems: a) inferring similarity relations among meanings encoded in a word, and b) incorporating this relational knowledge for generative modeling of colexification. Students will work with a large database of colexification across languages and develop interdisciplinary views and skills in computational linguistics, cognitive science, and natural language processing. The theoretical outcome of this project will be to contribute a typology of colexification across the world's languages, and the practical outcome will be to contribute algorithms for reasoning about and generating lexical semantic structures that resemble human languages.