Supervisor: Ada Chan, York University
Co-supervisor: Hanmeng Zhan, York University

Project Overview

Quantum walk is a natural generalization of classical random walks on graphs. Like its classical counterpart, quantum walk is an important tool for the design of quantum algorithms. For example, Grover's search algorithm, which provides a quadratic speedup over classical search algorithms, can be viewed as a quantum walk.

We focus on continuous-time quantum walks in this project. Continuous-time quantum walks have been used to simulate quantum gates. More importantly, Childs has shown in his seminal paper that continuous-time quantum walk can be regarded as a universal model for quantum computation.

Perfect state transfer and fractional revival are two interesting phenomena on continuous-time quantum walks concerning the transfer of information over quantum networks. Fractional revival is also useful in generating entanglement. The behaviour of the continuous-time quantum walk on a graph depends on the spectral properties of the graph. We propose to apply spectral graph theory to study fraction revival in continuous-time quantum walks.

Detailed Description

The continuous-time quantum walk on a graph is determined by the Hamiltonian associated to the graph. Typically, the Hamiltonian of the walk is either the adjacency matrix or the Laplacian matrix of the graph.

In particular, most known results for fractional revival concern the quantum walks using adjacency matrices. We propose to investigate fractional revival on quantum walks with Laplacian matrices. From a spectral angle, if a graph has perfect state transfer or Laplacian fractional revival, the involved vertices must be strongly cospectral. This property has led to a non-existence result on Lapcian perfect state transfer in trees. We plan to look further into strong cospectrality, in the hope of finding examples of Laplacian fractional revival, or showing that it does not happen on certain families of graphs. We will also extend some known results on adjacency strong cospectrality, such as equivalent formulations, to the Laplacian counterpart.

The project requires a solid background in second year linear algebra and an introductory level of graph theory. Experience in SAGE and LaTeX will be an advantage.

Students will start the project by learning necessary background in spectral graph theory and continuous-time quantum walks. They will be doing computation in SAGE to explore the behaviour of Laplacian continuous-time quantum walks on specific families of graphs, and formulate conjectures based on their computational output. Students will spend the remaining time proving their conjectures and writing up their results.

Supervisor: Alec Jacobson, University of Toronto

Project Overview

An elastic object will resonate or vibrate along modes at fixed frequencies. These are called the vibration modes or harmonics. Algebraically these motions are orthogonal and form a basis for any motion of the object. Computationally, we can approximate these harmonic motions by solving an eigenvalue problem on the matrix that represents the elastic energy of the shape. In this analysis, the shape is assumed to be perfectly elastic and the motions are small so that the shape does not crack, break, or fracture.

In this project, we will generalize the notion of vibration modes to brittle materials. The goal is to find a sequence of motions at fixed frequencies that fracture the shape into parts. Analogous to the classic vibration modes, these motions should be orthogonal and form a basis. Our methodology will leverage advances in the field of sparse optimization and compressed sensing. 

Detailed Description

After an introduction to the related literature in the first week of the summer, the students will begin by implementing standard harmonic analysis in two-dimensions. Through a series of experiments we will introduce a sparsity inducing norm on the continuity of the motion. We will then generalize the notion of orthogonality and demonstrate that our computed motions properly span the full space. The final weeks will be split between experiments with other applications and preparing a report on the summer's work. If successful, our "fracture harmonics" will be useful in the study of brittle materials and in the simulation of those materials. We will see immediate applications in the Computer Graphics industries such as Visual Effects and Computer Games.

The students undertaking this project will not only work on novel research with the intent to publish an academic paper, but will also have the opportunity to become familiar with the computer graphics and geometry processing scientific literature. This project will gather topics in numerical methods, sparse linear algebra, partial differential equations and computational geometry. In addition, the students will be invited to join the Dynamic Graphics Project (dgp) at the University of Toronto, Department of Computer Science, where we host weekly seminars and group research discussions with graduate students.

Supervisor: Andreas Hilfinger, University of Toronto

Project Overview

Biological systems are extremely complex, yet our observations of them are often very limited. How to then reliably infer model properties from experimental data is a key theoretical question and the subject of ongoing mathematical work. We will attempt to answer aspects of this question with regards to models of diabetes and how constrained they are given typically observed blood concentrations within individual patients. In particular, we will express model solutions of one (measurable) variable in terms of another (measurable) variable without solving for the whole model at once. Additionally, we will use real experimental data to quantify the day-to-day variability of blood glucose dynamics under identical conditions, which will have important implications for individualized medical care in diabetes patients.

Detailed Description

Physiological processes involve many variables of which we can only observe a small number at the same time. Furthermore, the data typically exhibit significant "noise" which additionally obscures the underlying dynamics.

How to reconstruct the whole from partial observations is an ongoing area of research in the analysis of stochastic processes. In this project we will use general mathematical theorems to find classes of possible models that are consistent with the given experimental data.

The type of question we will ask is "how does observing only two connected variables constrain the class of allowed models?". We will answer aspects of this question, by making quantitative predictions of one variable in terms of the other instead of finding solutions in terms of model parameters. For example, we will quantify how different models of red blood cell formation predict the concentration of glycated hemoglobin (HbA1c) in terms of time-varying glucose levels. Analyzing how the same long-term average blood glucose levels can give rise to different HbA1c levels will quantify non-linear effects in the control of red blood cell levels. In contrast to statistical models, such mechanistic models of the underlying physiological dynamics are causal and therefore allow us to make predictions about the effect of medical interventions. Furthermore, we will quantify the day-to-day variability in glucose levels under (daily repeating) identical conditions. This will allow us to determine whether model parameters could be used to characterize individual diabetes patients in an effort to personalize medical interventions.

Your undergraduate research experience will have three formal milestones.
1) Interim presentation after two weeks to describe your project goals and how you plan to achieve them.
2) Final presentation to summarize your work.
3) Final written report of your research findings. 

Experience with stochastic processes would be beneficial but is not a necessity. For our numerical approaches a solid background in differential equations and computer programming is required (which programming language is not important). In general, your key pre-requisite is not a mathematical one but simply an intense curiosity about how the world works!

Supervisor: Angele Hamel, Wilfrid Laurier University

Project Overview

Chromatic symmetric functions sit at the intersection of graph theory and enumeration. These symmetric functions generalize chromatic polynomials, objects in graph theory that count the number of colorings of a particular graph. By contrast the chromatic symmetric function of a graph not only counts the colorings, it counts the number of vertices of each color. One fundamental question in this domain concerns expressing a chromatic symmetric function in terms of the elementary symmetric function basis and deriving beautiful formulas for these coefficients. A second key question concerns whether two graphs can have the same chromatic symmetric function. The primary student tasks will be to generate examples of chromatic symmetric functions, to explore the structure of graphs through examples, to familiarize oneself with the proof techniques--either combinatorial or algebraic--related to coefficients and uniqueness, to formulate conjectures, and to prove them. The objective would be a journal publication.

Detailed Description

Graph coloring is a vibrant area of combinatorial research, and the search for a proof of the famous Four Color Theorem drove much of the development of graph theory throughout the 20th century. One key tool in this area is the chromatic polynomial, which counts the number of colorings of a particular graph. The chromatic symmetric function—the focus of this project—is a generalization of this polynomial, and it acts like a super chromatic polynomial as it not only counts the colorings, it counts the number of vertices of each color. This facilitates deeper knowledge of the structure of the graph, and allows exploitation of the machinery of classical symmetric function theory.

Symmetric functions are a long-standing part of algebraic combinatorics, and students may be able to use tools from either algebra or combinatorics. One fundamental question in this domain concerns expressing a chromatic symmetric function in terms of the elementary symmetric function basis and deriving beautiful formulas for these coefficients. A second key question concerns whether two graphs can have the same chromatic symmetric function. For most graphs, the chromatic symmetric function does not uniquely determine it, but for certain types of graphs it does, e.g. for some types of trees. For this project, we will look at formulas for coefficients and at uniqueness questions for particular graph classes, exploiting the underlying graph structure. But which graphs to explore?

A number of graph classes such as trees and cycles, have already been investigated by other researchers, and there has been particular focus in the literature on both clawfree graphs and on tree graphs, owing to clawfree and tree conjectures in Richard Stanley's defining 1995 paper. It is a standard approach in graph theory to describe graphs in terms of the induced subgraphs they do not contain, or are "free" from, and this considerable graph theory literature on H-free graphs, where H is some set of induced subgraphs, can be capitalized on as well.

The primary tasks and student responsibilities will be to generate examples of chromatic symmetric functions and related graphs, to explore the structure of graphs through examples, to familiarize oneself with the proof techniques--either combinatorial or algebraic--related to coefficients and uniqueness, to formulate conjectures, and to prove them. The students will also use packages in Sage to test examples. The outcome to aim for would be a journal publication.

Supervisor: Asif Zaman, 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. However, there is a strong need for experimental data on distributions related to random multiplicative functions. This would facilitate further questions in the subject and perhaps precipitate new conjectures in the area.  

Detailed Description

Random multiplicative functions are often at the heart of many conjectures for deterministic multiplicative functions. Chief among these deterministic ones is the lambda Liouville function which outputs +1 if an integer has an even number of prime factors and -1 if it has an odd number of prime factors. Based on the ideas of flipping a random coin, it is widely believed that the partial sums of the Liouville function exhibit squareroot cancellation. Indeed, this is equivalent to the Riemann hypothesis. 

A random multiplicative function that models the Liouville function is well known to exhibit squareroot cancellation. The Central Limit Theorem might suggest that this is best possible. However, Harper (2017) proved that these sums have better than squareroot cancellation and he further established their precise order of magnitude . His ideas draw from connections to areas of probability concerning critical multiplicative chaos, branching random walks, and freeze transitions. Now, the distribution of partial sums of random multiplicative still remains unknown. Extensive computational data is sorely needed to shed more light on this fundamental question.

The overarching goal of this project is to address this need. First, you will analyze recent literature and proof techniques for multiplicative functions to design vital computational investigations.  Next, you will be trained in numerical methods to produce large data sets for your proposed investigations. The goal is to give strong evidence for existing theorems or suggest possible conjectures. Finally, you will summarize the outcomes of your research in an appropriate academic forum so the community can use this work to guide future research. 

Supervisor: Huaxiong Huang, York University,
Co-supervisor: Shixin Xu, Duke Kunshan University

Project Overview

I direct the Fields CQAM lab on Health Analytics and Modeling: https://www.cqam.ca/health-analytics-and-modelling. We have several ongoing projects with industrial partners such as Nurologix, Rostrum, and also the Diabetes Action Canada and the National Research Council. The students from FUSRP can choose to work on a number of projects, including using machine learning to predict stresses (physical or psychological) and identify risk factors in diabetes, or develop mathematical modeling for prediction lung functions using noninvasive means, and finally model reduction techniques for problems from industrial and other applications.

Detailed Description

Depending on background and skills, students will be given a range of possible projects to work on.
1. Using machine learning techniques for identify high risk diabetic patients for developing complications using electronic medical records and other data;
2. Using machine learning for lie detection, using ECG and other physiological data;
3. Using computational models to predict lung functions using breathing data;
4. Using machine learning and model reduction techniques to improve computational efficiency.

Our lab is also developing new collaborations and it is possible for students to work on new problems as well. Students choosing to work with me and my postdocs are expected to have basic programing skills and interested in working on problems with real world applications.

Supervisor: Kevin Cheung, Carleton University

Project Overview

Symmetry is an important and useful property in mathematics. Often, symmetry can be exploited to reduce complexity of proofs and computations. Yet, symmetry-breaking computations sometimes require writing ad hoc code, leading to results that are highly dependent on the correctness of the implementation. The proposed project aims to develop frameworks for reliable combinatorial search routines that exploit symmetry and verification of computed search results. The expected outcomes are proofs of concepts for reliable combinatorial search and verifiable computed results for selected test cases.

Detailed Description

Project rationale and expected outcomes

Symmetry is an important and useful property in mathematics. Often, symmetry can be exploited to reduce complexity of proofs and computations in areas such as combinatorial search (e.g. graph isomorphism), Boolean satisfiability, constraint programming, integer programming, representation theory etc. Yet, symmetry-breaking computations sometimes require writing ad hoc code, leading to results that are highly dependent on the correctness of the implementation. In some cases, verifying computations that involve symmetry-breaking routines can only be carried out by independent replication, a tedious and time-consuming task that is not frequently performed.

The goal of the proposed project is to explore ways to improve reliability of computations involving symmetries. Two specific approaches can be taken: 1. Using formal methods for the development of combinatorial search routines. 2, Designing frameworks for verifiable results for computations involving symmetry breaking. To limit the scope of the study, the focus will be limited to graph search and binary integer programming. The expected outcomes are proofs of concepts for reliable combinatorial search and verifiable computed results for selected test cases.

Key tasks and timeline

The project will be in four phases:

1. Literature review (1 - 2 weeks)
2. Design of frameworks for reliable search under symmetry and verification (1 - 2 weeks)
3. Implementation (3 - 4 weeks)
4. Write up of results obtained (1 - 2 weeks)

Student responsibilities

Students assigned to this project are expected to conduct scientific research activities which include consulting relevant literature, discussing and/or presenting existing literature in a group setting, proposing questions for further inquiry, implementing and testing ideas and hypotheses, summarizing and reporting observed and derived results.

Supervisor: Milad Lankarany, University of Toronto

Project Overview

The prodigious capacity of our brain to process information relies on efficient information processing mechanisms. Understanding mechanisms of neuronal information processing enables the development of bio-inspired intelligent systems and facilitates the development of implantable microsystems (neural prosthesis) that help patients with neurological disorders to experience a healthy life again. Inferring these mechanisms from recorded brain signals is extremely challenging because the neural systems' activities are complex, nonlinear, and multi-scale. These challenges are referred to as lack of observability. Relying solely on experimental recordings of the brain is not sufficient to tackle these challenges, mathematical models and engineering methods need to be incorporated.

The proposed Research Project focuses on the creation of multi-scale mathematical models and the development of advanced engineering algorithms, which link these models to the experimental recordings of the brain, to conquer the lack of observability in the neural systems and to enhance the controllability of the brain's functions.

Students will work on the development of mathematical model representing neural activities of different regions of the brain. Using the experimental data available in the lab, students not only learn about modeling biological neurons but also gain a great hands-on experiences with different software to analyze the brain data.

Detailed Description

In our lab, we have developed biophysically-realistic models to recreate activities observed by biological neurons and neuronal networks. As well, we have developed several Bayesian-based algorithms to infer hidden parameters of these mathematical models from noisy, sparse, and incomplete recorded data. One line of research will be taken by students is to use mathematical techniques like stability analysis to understand the dynamics of the models. The other line of research will be on the development of Parameter Estimation algorithms to fit the model to experimental data. Through our collaborations with Neuro-surgeons at The Toronto Western Hospital, we access to different types of human brain data corresponding to different neurological disorders such as Parkinson's disease and epilepsy.

Supervisor: Thomas Uchida, University of Ottawa

Project Overview

Mechanical linkages define the motion of industrial robots, vehicle suspensions, and deployable structures like reclining chairs, artificial satellites, aircraft landing gear, and umbrellas. If we specify the geometry of a mechanism, we can perform simulations or build a physical model to determine how it moves. The inverse problem---mechanism synthesis---is far more challenging. In this case, we do not know the geometry of the mechanism and may not even know how complex it is. We are only given the desired path of one or more points and we must determine what mechanism produces this motion. Many beautiful mechanisms have been designed to achieve specific goals, but there are few general principles. In this project, we will explore the mechanism synthesis problem using optimization and algebraic approaches, and will design new mechanisms using our strategies.

Detailed Description

Background

Mechanical linkages define the motion of industrial robots, vehicle suspensions, and deployable structures like reclining chairs, artificial satellites, aircraft landing gear, and umbrellas. If we specify the geometry of a mechanism, we can perform simulations or build a physical model to determine how it moves. The inverse problem---mechanism synthesis---is far more challenging. In this case, we do not know the geometry of the mechanism and may not even know how complex it is. We are only given the desired path of one or more points and we must determine what mechanism produces this motion. Many beautiful mechanisms have been designed to achieve specific goals, but there are few general principles.

Research Objective

Explore the mechanism synthesis problem using optimization and algebraic approaches, and design new mechanisms using our strategies.

Specific Aims

Students will gain exposure to a research program at the intersection of computational mechanics and computer algebra. Students will read and discuss key journal papers in the area, learn about and use popular software (Python and Maple) to solve real-world problems, and gain experience presenting their progress in oral and written forms. Prior to the mid-program presentations, students will develop an understanding of the problem and form testable hypotheses; thereafter, students will seek answers to their research questions.

Expected Outcomes

At the conclusion of the program, students will have contributed to this research program by (1) developing methods for describing mechanisms algebraically, and (2) designing mechanisms using numerical optimization and symbolic computing.