Supervisor: Professor Camelia Karimianpour, University of Toronto
Co-supervisors: Dr. Daniel Johnstone and Dr. Malors Espinosa Lara, University of Toronto 

Project Overview

Doing research in the area of number theory and the Langlands program requires a deep and computation-friendly understanding of Galois theory for nonarchimedean local fields. This project will focus on exploring this topic by tackling the problem of finding a complete classification of cubic extensions of local fields. Such a classification is of interest to representation theorists as it gives insight into the representation theory of reductive groups over local fields. In addition to establishing such a classification, we will also discuss the fundamentals of the representation theory of local fields and how understanding the Galois theory of field extensions leads to an understanding of tori in reductive groups and thus to a way in which to parameterize their irreducible representations. 

Supervisor: Professor Annie Lee, University of Toronto

Project Overview

What good does improved interpretability do for us? It has been found that in many machine learning modeling contexts, even though sometimes interpretable methods come at an expense of accuracy, the penalty is limited, giving more interpretable models a leg up especially in scientific research, such as in medical contexts. Through extensive literature research, we hope to provide insights for researchers to holistically evaluate interpretability human-centric (understandability) and differentiate those that are passive and active. 

Supervisor: Professor Ejaz Ahmed, Brock University

Project Overview

The project focusses on estimation of model parameters and prediction based on high dimensional data (HDD), when variables are larger than the sample size. The techniques are available to analyze HDD and they mostly rely on penalized regularization methods to do variable selection and estimation simultaneously. These methods do well when the model has a few strong signals and with sparse signals. In this project we also consider when. some of the features may have weak-moderate influence on the response of interest.  Basic to moderate statistical and mathematical skills; coding in R and Python; standard writing skills, and team player are most significant requirements for the projects.  Students will be involved in data cleaning, building basic models, doing simulations, and applying the proposed strategies to real datasets.  The most important aspect of the proposal is that it will provide answers to the question of "what are the tools and tricks, pitfalls, applications, challenges and opportunities in HDD". Students will receive both analytic and computational training. My team) plan to present the results in various machine learning and computer science communities in addition to the statistical community to better facilitate interaction between the two groups.

Co-supervisors: Professor Ada Chan, York University, and Dr. Sooyeong Kim, York University

Project Overview

Markov chain models of dynamic processes on graphs havebeen exploited in a number of successful applications, of which the most notable one is perhaps Google's PageRank algorithm. Many graph invariants are derived from the properties of the random walk on the graph, which is an example of a Markov chain. These random-walk-based invariants can be effectively used to describe how well-connected the graph is, determine measures of clustering, and analyze the behaviour of spreading processes on the graph. Many such invariants can be determined from the eigenvalues of the corresponding transition matrix, utilizing techniques from spectral graph theory. Analysing the effect of graph manipulations (such as adding/removing edges, joining two graphs together, and so on) on these measures of connectivity provides insight into the control of stochastic processes on a network, and into the synthesis of robust networks.

Supervisor: Dr. Masoud Khalkhali, Western University

Co-Supervisor: Dr. Asghar Ghorbanpour, Western University

CANCELLED

Project Overview

In this project we aim to compute the eigenvalues and eigenfunctions of discrete Laplacians on discrete hyperbolic surfaces. In general there are no closed formulas but recent bootstrapping techniques   have been  successful in solving this problem for two dimensional hyperbolic surfaces.  We shall investigate this technique in the case of k-regular graphs which can be understood as discrete finite analogues of hyperbolic two dimensional surfaces. 

Supervisor: Professor Asif Zaman

Co-supervisor: William Verreault, Université Laval

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.

Supervisor: Dr. You Liang, Toronto Metropolitan University

Co-Supervisor: Dr. Na Yu, Toronto Metropolitan University

Project Overview

The project aims to develop innovative and non-invasive machine learning algorithms to detect the boundary between tissues, such as muscle, sclera, choroid, and retina from biomedical hyperspectral images (BHSI). This technique enables healthcare providers to understand and diagnose eye and brain diseases more quickly and accurately, especially Spaceflight-Associated Neuro-ocular Syndrome and Amyotrophic Lateral Sclerosis. BHSI were generated using a near-infrared hyperspectral microscope, by the experimental research team from the University of Toronto/St. Michael's Hospital. Algorithms and data analysis results will be validated with clinical specialists and experiments.

Students should have

1. taken fundamental math and statistics courses such as calculus, linear algebra, mathematical optimization, probability, and statistics

2. strong programming skills using Matlab, R, or Python 

Supervisor: Professor Huaxiong Huang, York University

Co-supervisor: Faisal Habib

Project Overview

Heart disease is one of the leading courses of death in the world, with 8.9 million deaths in 2015, representing 45% of all non-communicable disease deaths. The number of death due to heart diseases increases in a faster pace than that of population growth, reached an estimated 17.9 million in 2017. Since most adverse medical events such as heart attack happen outside of clinical settings, new technologies using remote sensing and monitoring offer tremendous value as they can provide information needed to provide care at home. In addition, such collected information may be used for early event detection and prediction.

Students need strong programming skills to succeed in this project. Students in computer science are encouraged to apply.