Supervisors: Professor Hans Boden, McMaster University, Professor Robert Osburn, University College Dublin

Project Description

Quantum knot invariants have their origin in the seminal works of two Fields Medalists, Vaughan Jones in 1984 on von Neumann algebras and Edward Witten in 1988 on topological quantum field theory. The equivalence between vacuum expectation values and Wilson loops and knot polynomial invariants is a major source of inspiration for the development of new knot invariants via quantum groups. Quantum topology has become a driving force in knot theory and low-dimensional topology. Despite its many achievements, the geometric meaning of the Jones polynomial remains elusive and not well understood. This is reflected in the many open problems (volume conjecture, slope conjecture, AJ conjecture,…) and the simple fact that we do not know whether the Jones polynomial detects the unknot!

Khovanov homology, which is defined as the homology of a certain bigraded chain complex associated to a knot diagram, provides a categorification of the Jones polynomial and is known to detect the unknot, thanks to deep results of Kronheimer and Mrowka using gauge theory.

One major problem in the area is to extend these invariants to other 3-manifolds and to higher dimensional knots. Here, virtual and welded knots have an important role to play, and that is because the former gives a purely combinatorial representation of knots in thickened surfaces up to stabilisation, whereas the latter gives a diagrammatic way to study knotted 2-tori and knotted 2-spheres in the 4-sphere.

Inspiration comes not only from the deep and mysterious connections between knots and physics, but also from the strong interaction between knot theory and number theory. A modular form is an analytic object with intrinsic symmetric properties. The study of modular forms has enjoyed long and fruitful interactions with many areas such as number theory, algebraic geometry, combinatorics and physics. Most importantly, they were the key players in Andrew Wiles' spectacular proof in 1994 of Fermat's Last Theorem.

Research will center around the study of specific examples drawn from (classical, virtual, and welded) knots and links. The goal of this project is to combine knot theoretic calculations with techniques from the theory of q-hypergeometric series to explicitly construct new infinite families of quantum knot invariants ("colored Jones polynomials") which have modular properties.

Time permitting, we will also investigate the construction of new mock theta functions from WRT invariants, the stability of the coefficients of the colored Jones polynomial and generalized quantum modular q-hypergeometric series.

Supervisor: Dr. Hadi Salmasian, University of Ottawa

Project Description

Lie groups and Lie algebras are nowadays some of the central objects in mathematics. They play a key role in describing continuous symmetries in geometry, and they arise in many unexpected situations, linking algebra to geometry and combinatorics.

The goal of this project is to study a generalization of the notion of a Lie algebra, which arises in connection with supersymmetry in physics. This generalization is known as a Lie superalgebra. Lie superalgebras share great resemblance to Lie algebras, but what makes them interesting is the points of divergence of their theory from the theory of Lie algebras. For example, one knows that unlike what happens for simple Lie algebras, the category of modules over a simple Lie superalgebra is not semisimple. This means that there are many more interesting examples of representations of simple Lie superalgebras.

One can also go beyond Lie superalgebras and consider algebraic objects with a more refined structure, namely a "grading" by a finite abelian group more general than the cyclic two-element group. Such an object is called a "Lie color algebra". One of the goals of this project is to obtain a better understanding of the representation theory of the latter class of algebras by utilizing an auxiliary algebraic structure known as the "universal enveloping algebra". This problem involves a bit of combinatorial tools such as Young Tableau and the RSK algorithm.

A solid background in 2nd year/3rd year level courses that cover "Linear Algebra" and "Groups, Rings, and Fields" is crucial. Familiarity with document preparation systems such as LaTeX will be an advantage.

In the first stage of the project (roughly the first month) the student will acquire the needed background through learning about basic Lie theory and Lie superalgebras. The next stage is to examine examples, perform computations, formulate plausible conjectures, possibly prove them! The outcome of the project will be a self-contained and well-written report. If some conjectures are proved, the project will be published as a research article. 

Supervisor: Dr. Kevin Cheung, Carleton University

Project Description

Project rationale and expected outcomes

The simplex method for linear programming (LP), developed independently by George B. Dantzig and Leonid V. Kantorovich in the early 20th century, is still very much in use today, especially in integer LP solvers, despite not being known to run in polynomial time.  An attractive property of the simplex method is its relatively lower memory requirement compared to interior-point methods which have good empirical performance and theoretical guarantees on the running time.  As well, the dual simplex method lends itself well to warm starts.

Linear programming was first proved to be polynomial-time solvable by Leonid Khachiyan using the ellipsoid method.  Unfortunately, the method is inefficient in practice.  Nevertheless, it turned out to be an important tool in proving various combinatorial optimization problems to be polynomial-time solvable (see [1] for example) despite having exponentially constraints in the linear programming formulation.  The reason is that the ellipsoid method does not require one to specify all the constraints a priori. It simply requires a polynomial-time algorithm to obtain an inequality that separates a given non-feasible point from the feasible region.

An interesting case is the maximum matching problem.  In a landmark paper, Edmonds [2] gave a polynomial-time algorithm for maximum matching on graphs that are not necessarily bipartite.  Later, he also gave a LP formulation with exponentially many constraints [3].  A question that arose was whether or not the separation problem for this LP formulation was polynomial-time solvable and it was answered in the affirmative by Padberg and Rao [4].  Recently, Chandrasekaran et al. [5] gave a cutting-plane method for maximum perfect matchings that runs in polynomial time.  The proposed research is to revisit this exciting result and obtain a version that leads to practical implementations with lower overhead and transfer the techniques to other problems such as optimization over the matroid polytope and subtour elimination polytope for the Travelling Salesman Problem.

Key tasks and timeline

The project will be in four phases:
1. Literature review (1 - 2 weeks)
2. Devise cutting-plane methods for perfect matching with lower overhead (2 - 3 weeks)
3. Study the techniques on other problems (2 - 3 weeks)
4. Write up 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.

References
1.    M. Grötschel, L. Lovász, A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988.
2.    J. Edmonds: “Paths, trees, and flowers”. Can. J. Math. 17:449—467, 1965.
3.    J. Edmonds: "Maximum matching and a polyhedron with 0,1-vertices". Journal of Research of the National Bureau of Standards Section B. 69:125–130, 1965.
4.    M.W. Padberg, M.R. Rao,: Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7, 67–80, 1982.
5.    K. Chandrasekaran, L. Végh, S. Vempala. “The cutting plane method is polynomial for perfect matchings”.  Math. Oper. Res. 41:23—48, 2016.

Supervisor: Dr. Aziz Guergachi, Ryerson University

Project Description

The physical chemist Emyr Alun Moelwyn-Hughes had once pointed out that "energy among molecules is like money among men: the poor are numerous, the rich few". If there is such a high disparity between the entities that own money/energy and the ones that don't, both in societies and in physical systems such as gases, shouldn't we be concluding that nature is inherently unfair? Some (whether it is people or molecules) seem to be more favoured than others, leading to the emergence of hierarchies and layers within the population at hand. Or, is there some deep wisdom in the way nature handles the distribution of something across a population? In order to gain some insights into these questions, this project will focus on comparing the wealth distribution in different countries around the world to the Maxwell-Boltzmann distribution of energy in gases. Various statistical and mathematical techniques (e.g. Kolmogorov–Smirnov test) will be implemented to carry out these comparisons. The Pareto principle and the Gini index that are traditionally used to describe wealth distributions in societies will also be brought to the comparative analysis. The contexts and assumptions (equilibrium, ideal versus real gases, etc.) under which Maxwell-Boltzmann distribution was developed will be examined, in order to reflect how wealth distribution in societies could differ from energy distribution in gases. Among the questions that will be addressed in this project is how far (or close) the wealth distribution in 'SUSTAINABLE' societies to Maxwell-Boltzmann distribution.

Supervisor: Dr. David Kreindler, University of Toronto

Project Description

The mood disorders - principally major depressive disorder and bipolar disorder - are a significant public health issue affecting one in four people during their lives in total, costing the economy over $6 billion annually. Mood disorders are generally recurrent: approximately half of depression is recurrent; chronic bipolar disorder is typical. Accurately predicting the onset of mood episodes has been a longstanding but so-far-unrealized goal of mood disorders research. This project will focus on advancing a key element of this line of research: characterizing how time series of mood disorder symptoms 'cluster'.

The Sunnybrook Centre for Mobile Computing in Mental Health has collected approximately 150 sets of daily visual analog scale (VAS) symptom severity ratings data, as well as coarser-grained data from standard multiple-choice mood measures, from several prior studies, spanning time scales from 6 weeks to 18 months; data has been collected from experimental subjects including healthy controls and individuals with mood disorders.

Timeline:

Weeks 1-2: Learn about the mood disorders and mood symptoms; review the literature on similarity measures for time series

Weeks 2-4: Adapt existing similarity measures to analyze mood symptom time series data; create code for measures for which code isn't available.

Weeks 3-7: Use software to identify which measures are optimal for which groups of subjects, characterize independent clusters of symptoms, and characterize how clusters vary across groups of subjects

Weeks 8-9: prepare findings for poster presentation and / or publication.

Appropriate students are expected to have advanced knowledge of coding in MATLAB and some knowledge of statistics; excellent organization and communication skills, and the ability to work as part of a team. While this project is based in a teaching hospital, no direct patient contact is involved.

Reference: Serrà, J., Arcos, J.L., 2014. An empirical evaluation of similarity measures for time series classification. Knowl.-Based Syst. 67, 305–314.  https://doi.org/10.1016/j.knosys.2014.04.035

Supervisor: Professor Thomas Uchida, University of Ottawa

Project Description

Background

Mechanical linkages define the motion of industrial manipulators, vehicle suspensions, and deployable structures like folding chairs, artificial satellites, and aircraft landing gear. In many cases, the motion of a mechanism is most naturally described by a system of coupled trigonometric equations. These equations must be solved quickly yet precisely when generating dynamic simulations in real-time applications. For example, many vehicle safety systems run short numerical simulations to predict the vehicle's motion; increasing the speed of these simulations would enable use of more sophisticated stability controllers with the existing electronics, potentially improving vehicle safety at no additional cost.

Typical strategies for solving these systems of equations use approximations or numerical iteration, which sacrifice precision for speed. In previous work, I used Gröbner bases to convert these systems of nonlinear equations into a triangular form, analogous to the familiar echelon form obtained upon applying Gaussian elimination to a system of linear equations. Thus, a Gröbner basis can provide arbitrarily precise solutions to the original system of equations through a process similar to back substitution—that is, without iteration. A Gröbner basis can be generated algorithmically using a computer algebra package like Maple, and triangular systems have been successfully generated for several mechanisms.

Many interesting questions remain. For example, linkage lengths and other system parameters are typically substituted into the equations before computing a Gröbner basis to reduce the number of indeterminates and, therefore, the resources required to compute the basis. It would be advantageous to retain the parameterizations of the original system to avoid computing a new Gröbner basis if the system parameters change. Triangular sets appear to be promising alternatives. A triangular set can be interpreted in this context as a finite set of triangular systems describing all possible mechanism configurations for all choices of system parameters. Because a triangular set partitions the solution into several branches (rather than producing a single branch as in a Gröbner basis), it may be feasible to preserve the parameterizations of the original system without incurring large computational costs.

Research Objective

Explore the use of triangular sets (in particular, the RegularChains package in Maple) for converting systems of coupled trigonometric equations into triangular form for generating computationally efficient simulations of mechanisms.

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 Maple's algorithms to solve real-world problems, and gain experience presenting their progress in oral and written forms. Prior to the midprogram 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) generating triangular systems that describe the motion of mechanical linkages of various complexity, and (2) evaluating differences between triangular systems obtained using Gröbner bases and regular chains. 

Supervisor: Professor Adam Stinchcombe, University of Toronto

Project Description

 Numerically computing solutions to partial differential equations (PDEs) on domains with irregular boundaries is a challenge. In this project we will develop a new method to overcome this challenge that is based on ideas from machine learning, including temporal difference learning and artificial neural networks. This method is particularly adept at handling high dimensional problems and a variety of boundary conditions. Many applications are possible: understanding the electrical activity of neurons and cardiac cells, biologically relevant fluid flow, and inverse problems in elliptic PDEs.

Phases of the project:

1. Learn background mathematics (1-2 weeks). Learn the necessary background about PDEs, Markov processes, the Feynman-Kac formula, artificial neural networks, and temporal difference learning. Students are not expected to be experts in this material beforehand; learning this material is part of the research programme. 

2. Method development (2-3 weeks). Improve the method by testing it on "easy" problems in two dimensions for which traditional methods work or there is a known analytic solution. The main objective of this phase is to quickly iterate ideas to develop the numerical method. The expected outcome is a solid mathematical understanding of the method.

3. High-performance implementation (3-4 weeks). In this phase, we will apply the method to different types of PDEs in dimensions higher than two. The main objective of this phase is to produce a high-performance implementation of the numerical method employing highly parallel GPU computing. We will employ good software design principles and collaborative software development. The expected outcome is a research-quality implementation suitable for applications.

4. Applications (1-2 weeks). Using the high-performance implementation of the improved method, we will study problems from the field of mathematical biology requiring solving PDEs with irregular boundaries. It is expected that the new numerical method will enable insights into the biological phenomenon.  

The main student responsibilities in this project will be to mathematically study the method and implement it in Matlab/Python/C/CUDA. Strong programming skills in at least one of these programming languages is an asset. Students will learn some probability theory and some theory behind machine learning. This project will be supervised by Adam Stinchcombe (Faculty) with assistance from Mihai Nica (NSERC Postdoctoral Fellow) with at least two group meetings a week. 

Supervisor: Dr. Anna Goldenberg, University of Toronto

Project Description

Genetic variants can predispose certain individuals with a high risk of cancer. For example, patients with a germline mutation in their TP53 gene are usually diagnosed with Li-Fraumeni Syndrome (LFS) and are screened for cancer for the rest of their lives, usually receiving annual whole body MRIs (wbMRIs). While deep learning has begun to make a huge impact in MRI-based cancer detection, the paucity of wbMRI data in children who are often screened for LFS, has made the improvement of cancer detection in cancer-predisposed children challenging. To solve this problem computationally, we will train generative adversarial networks (GANs) to generate wbMRI images of early-stage cancers. Having generated sufficient data and trained cancer detection models, we will then test them on real earlystage cancer images from Li-Fraumeni Syndrome patients.

The successful candidate for this position will:
Train a GAN on a subset of 486 wbMRI images in kids from SickKids (2012-2018)
Evaluate cancer detection in wbMRI images from the pediatric LFS population
Work with a team of clinicians and radiologists to ensure the clinical utility of the tool

This tool will be developed with the support of a Terry Fox Research Institute team focused on improved treatment, surveillance, and survival for the LFS population. If successful, this tool will be uniquely positioned to improve cancer screening in children and will be revolutionize cancer screening in the clinic.

Supervisor: Professor Andreas Hilfinger, University of Toronto

Project Description

Biological processes in cells are extremely complex. As a result mathematical models of such systems have to make a large number assumptions based on guesswork. Instead of ignoring or guessing unknown details in complex systems you will utilize our recently established universal balance theorems to characterize stochastic fluctuations in incompletely specified systems. Specifying some features of a system while leaving everything else unspecified has allowed us to translate individual assumptions into rigorous experimental tests despite dealing with infinitely large and complex networks. What we want to understand next, is whether such an approach has enough discriminatory power to identify the correct models instead of merely ruling incorrect ones out, and ultimately to reconstruct entire networks from observed fluctuation data.

You will contribute towards our goal of solving this inverse problem through numerical approaches as well as analytical work based on master equations. You will become an expert in simulating time-continuous Markov processes and will visit a state-of-the-art molecular biology lab to understand how the experimental data is generated. You will be expected to carry out independent research, keep detailed written records of your research, and present your work within our research group and to collaborating scientists.

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.

Supervisor: Dr. Andrew Green, Managing Director and Lead XVA Quant, Scotiabank

Project Description

Deep neural networks provide a new way of rapidly solving high dimensional PDEs. PDEs appear frequently in quantitative finance contexts and are a standard numerical modelling technique used to implement derivative pricing models. However, traditional PDE solvers suffer from the curse of dimensionality and hitherto any model with more than four stochastic factors is usual solved via Monte Carlo methods. Most PDEs that appear in quantitative finance are linear, but some XVA models (see for example Crepey et al, 2017) result in non-linear PDEs or a BSDE representation. The Deep Gelerkin Method (Sirignano and Spiriopoulos, 2018) offer a new way of solving high dimensional PDEs. In a similar fashion the Deep BSDE solver (Weinan E. et al 2017; Henry-Labordere 2017) offers a new way to solve BSDE problems efficiently. This project will implement both the Deep BSDE method and Deep Galerkin method and apply them to problems in XVA and Derivative Valuation.

Supervisor: Dr. Douglas Moseley, University of Toronto

Project Description

Modern medical linear accelerators are equipped with kV x-ray cone-beam computed tomography (CBCT) to perform image-guided radiation therapy (IGRT) for patients receiving cancer treatment. Daily volumetric images taken just before treatment are compared to a planning CT and a 3D adjustment to the patient's position is made. Princess Margaret Cancer Centre has a large digital repository of CBCT images (>400k volumetric images). ImageNet is a large repository of annotated digital images for computer vision research. IGRTnet proposes to construct a CBCT repository available to the scientific community to advance the automatic decision making process around IGRT.

The students will be trained in image registration and fusion for various treatment sites in our skills lab. The next step is to pilot a machine learning project to mimic the expert human observers. The last step is to make this repository available to the community for computer vision research.

Supervisor: Professor Nima Maftoon, University of Waterloo

Project Description

Metastasis involves the dissemination of cancer cell from the primary tumor to the surrounding tissue and distant organs. To metastasize, the cancer cells must detach from the primary tumor, intravasate into the blood stream or lymphatic system, avoid immune protection, extravasate from the blood, finally migrate to distant organs, and proliferate to a secondary tumor. It has been long theorized that blood flow plays a vital role in metastasis and more than two-thirds of metastatic sites could be explained by the blood flow links between the primary and secondary sites. To study circulating tumor cells in metastasis and the location of secondary tumor sites detailed geometry of vascular network is needed. Aim of this project is to develop a computational framework for constructing vascular networks with two applications: (1) for modelling tumor growth and its vascularization, and (2) for geometrical reconstruction of vascular networks in clinical images of vessels with incomplete information due to non-ideal filling of the contrast agent or image resolution. The students will work on developing an algorithm that will use the 3-D geometry of organs as well as skeletons of detectable vessel segments in the clinical image stacks and will reconstruct the missing vessel segments and microvasculature. The team will experiment with a combination of the constrained constructive optimization algorithm, global constructive optimization, simulated annealing and simplified fluid equations to develop the algorithm.

The team will start with reviewing the existing algorithms for in-silico construction of vessel networks and models of angiogenesis. The students will develop an algorithm that uses the patient priors from the incomplete vascular network data to construct the patient specific network. Based on the existing tools, the code is expected to be most easily prototyped in Python. The result of simulations will be validated against an ex-vivo vascular networks dataset obtained experimentally. 

Students are expected to know and be interested in coding, optimization methods, image processing and basics of fluid dynamics. Knowledge of the finite-element and finite volume methods is a plus.