USRA | Undergraduate Student Research Awards

Algebraic geometry, at its most fundamental level, is the study of solution sets of systems of polynomial equations. These solution sets, called varieties, are often very challenging to understand. However, a special class of these objects, toric varieties, can be understood explicitly using tools from combinatorics. This project will involve studying problems in algebraic geometry using tools from toric geometry. Specific problems could include: algebraic hyperbolicity of surfaces in toric threefolds, conics contained in toric varieties, or projections of rational normal curves.  

Requirements: Interested students should have a strong background in linear algebra (e.g. Math 240) and rings and ideals (e.g. Math 340), with added background in commutative algebra/algebraic geometry a plus. Projects are available for up to three students. 

The LWE (learning with errors) problem involves finding a vector s, given that s lies on a list of hyperplanes known only up to some error. The objective of this project is to study and give an independent exposition of a result of Regev [1] which shows that if there exists an efficient algorithm that solves LWE, then there exists an efficient quantum algorithm that approximates the decision version of the shortest vector problem.  This hardness result, and later variants such as RLWE, are often used to motivate approaches to post-quantum cryptography and homomorphic encryption.

[1] On Lattices, Learning with Errors, Random Linear Codes, and Cryptography, Oded Regev, Journal of the ACM 56(6), article 34, 2009.

Given a system of n polynomial equations in n unknowns x1, x2, ..., xn, the determinant of the Dixon matrix is a polynomial in x1 where the variables x2, ..., xn have been eliminated.  We are interested in
polynomial systems of equations which also involve one or more parameters corresponding to physical constants like lengths and masses. In this case the Dixon matrix will be a matrix of polynomials
in x1 and the parameters.

The project is firstly, to study how to construct the the Dixon matrix, secondly, to investigate why it often has a block structure, and thirdly, to investigate how to compute it's determinant. Computing determinants of matrices of polynomials is an interesting problem. We aim to develop a fast method that interpolates the parameters in the determinant from monic images in x1.

The project is suitable for one or two students.

Requirements: Students must have taken a course in linear algebra and also a first course in either abstract algebra or number theory (MATH 340 or MATH 342 at SFU) where you have seen modular arithmetic. Also, students must have taken a second programming course in either C or C++ or Java or Python.  We will use Maple and C for computational experiments.  No prior knowledge of Maple is required.

Given a metric space, its tight span is the set of all minimal one-point extensions of the space. The tight span can be equipped with its own metric, and viewed as its own metric space, so that the original metric space is naturally embedded in its tight span. The theory of tight spans arose in phylogenetics but has become of interest in its own right. In this project the student will fix a finite set of non-negative values and consider metric spaces where distances only take those values. The student will investigate whether the tight span structure can be generalized to this setting.

Requirements: Advanced analysis or discrete math

Many words English words have multiple spellings, for example "humor" and "humour". Historically this was even more common before English spelling was largely standardized. In this project students will use Google Ngram viewer to examine the prevalence of different spellings versus time, and in particular examine the extinction of spellings. Models for the dynamics of spelling prevalence will be fit to the data and assessed.  

Requirements: Dynamical systems or statistics

Clustering is a common used procedure in which a collection of objects is broken up into "natural" groupings. For example, given a finite collection of points in a metric space, you assign points to one of two clusters such that points within clusters tend to be closer together than points in different clusters. Clustering in metric spaces is a well-established field. In this project the student will study clustering in the context of diversities, a generalization of metric spaces where a positive number is assigned to every subset of points, and not just pairs as in a metric space.

Requirements: Discrete math or optimization.

This project investigates spectral methods and radial basis function methods for solving differential equations with a focus on introducing adaptivity. The initial focus will be on singularly perturbed boundary value problems. To numerically resolve boundary layers we will look at various transformation techniques, and we will try to find the optimal parameters in those transformations via an adaptive process. We will also explore the application of spectral methods for differential-difference equations (equations with a delay).

The other issue to be addressed is the treatment of boundary conditions. In particular, we will investigate the use of rectangular differentiation matrices in standard spectral and radial basis function methods. This approach allows for a more systematic treatment of boundary conditions.

This project will combine data analysis and operations research to improve the delivery of health care services to vulnerable people in Vancouver, many of whom have mental health and substance use issues. Data on patient visits, staffing, services offered, and opening hours at a community health centre will be analysed. This data will be linked to emergency department visits at inner-city hospitals. Vancouver Community Health Services at Vancouver Coastal Health will provide data and expert advice for the project.

Key questions to be addressed are:

1. What are the optimal health centre service delivery hours for scheduled and walk-in patients to minimize emergency department visits resulting from gaps in primary care?

2. What are the optimal medical staff schedules to minimize wait times for walk-in patients, subject to staffing, budgetary, and booked patient constraints?

These questions will be addressed using a hybrid optimization approach, which combines integer linear programming with simulation-based optimization. The focus of the research will be on developing a heuristic hybrid optimization approach, which is able to deal with a large number of constraints and multiple objectives. The ultimate goal is a computationally efficient algorithm that is implemented on real data for the community health centre.

Requirements: This project is well-suited to a student interested in operations research, applied optimization, and simulation modelling. Familiarity with integer linear programming, simulation, and Python are important. Good communication skills are essential, because the student will be expected to participate in meetings and consultations at Vancouver Community Health Services.

This is a joint project with Prof. Patricia Brantingham from the SFU School of Criminology. The goal is to develop mathematical models to understand and predict crime patterns in Metro Vancouver. A significant amount of crime data is available from the Vancouver Police Department, so there is an opportunity to test the models against real data. The modelling will likely involve differential equations, both theoretical and numerical.   

Possible topics for a project include:  

* Impact of clustering of bars/pubs in entertainment areas

* Metro stations as hot spot generators

* The impact of road type of crime clusters (like the peaking at intersections along major arteries)

* The time flow of different crime types through the court system in BC based on the seriousness and/ or the complexity of the legal case

* How number and timing of appearances changes in courts depending on the size and location of provincial courts

Requirements: Students are expected to have taken and performed well in MATH 310 and MACM 316. No previous background in Criminology is required.

The idea of counting lattice paths in the plane subject to constraints is a fairly old problem, yet continues to be a fruitful area of modern mathematical research.  In the case of exact enumerative results, the goal is to find and solve a problem by learning about the past results and seeing what kind of results and constraints can be modified to find new, interesting, tractable and/or applicable results.

Requirements: The student would be expected to read some background literature in this area and, together with the adviser, would hone in on a particular problem.  MACM 201 and a third year algebra course is essential.

The project will involve the study of combinatorial problems arising in digital communications.  See http://people.math.sfu.ca/~jed/research.html for background on the general area of research, and http://people.math.sfu.ca/~jed/students.html for examples of previous USRA projects.

Requirements: Students should have completed MACM 201 (Discrete Mathematics II), and preferably have some programming experience. However, the most important attributes are enthusiasm, persistence, and a willingness to learn new skills.

Independently motivated undergraduates in the third or fourth year of their degree are invited to join a research group that uses computational models to understand the fluid mechanics of the weather. There are active projects that investigate a variety of atmospheric phenomena.

One current area of interest involves projects connected with the question, "What is the shape of a cloud"? We have developed a new mathematical model for the motion of cloud edges --- one that has already confirmed the behaviour of "lenticular" clouds caused by airflow over mountains, and somewhat rare phenomenon known as a "holepunch" cloud (search for images!).

Requirements: Some background in a differential equations is essential, as is proficiency in a computational environment such as Matlab.

A result of Flajolet connects continued fraction expansions of polynomials with positive coefficients with weighted Dyck paths, thus giving a nice combinatorial interpretation of continued fractions in terms of lattice paths.  A further result by Viennot connected these weighted Dyck paths to matrices whose minors are all positive.

The student will try to understand these background results, while trying to find new combinatorial instances where these results apply.  

Requirements: This is a rather advanced project, so an appropriate student is expected to have some advanced algebra AND combinatorics (at least third year level for both).

NB: This project is eligible for a VPR USRA ONLY

Graphs are ubiquitous as models for real world networks and data, and an important real-world problem is to find “nice” ways of drawing or representing graphs that help to illuminate their structure.  In this project we will be utilizing some concepts from linear algebra and algebraic graph theory to represent graphs in 3 dimensional space.  Then we will be using a 3D printer to actually print and realize these graphs! 

This project has a mixture of theory and computing and will have up to 3 USRA students in addition to some graduate student involvement.  I will give a series of lectures on the subject to provide background and students will use Maple and some other computing software to work with graphs. 

Be it the spots on butterfly wings, microscale structure in materials or clouds in the sky; pattern formation is ubiquitous in systems governed by differential equations.

This project will look at numerical methods to find patterns in domains with holes for energy minimization problems from material science. We will study the Swift-Hohenberg, Cahn-Hilliard and Ohta-Kawasaki equations in two dimensions.

We will extend the understanding of these equations by using approximations based on Frames.

Requirements: Students are expected to have taken and performed well in MATH 310 and MACM 316. Math 320 or 419 an asset but not necessary

Phylogenetic trees are a central tool in understanding evolution, and are increasingly applied to learning about short-term evolution of infectious diseases as they spread through human communities. These trees are built from genetic sequences of a virus or bacteria. Since data are being gathered over time, we have an opportunity to apply modelling and machine learning methods to study evolution over time, using large phylogenetic trees. This requires a key first step: defining the sets of observations in the trees that machine learning and modelling tools can work with. In this project synthesize a number of these methods into an R package, and/or develop new approaches to this question.  

Requirements: discrete mathematics, familiarity with a programming language. If that language is not R, a willingness to learn R (and R package development) is essential. This project is at the interface of mathematics, evolution and infection, and will require an interest in interdisciplinary work. No previous biological background is required.

NB: This project is eligible for a VPR USRA ONLY

Genetic data from viruses and bacteria contain information about how an infection spreads from person to person, even to the point that we can reconstruct who infected whom with the help of genetic data (with some uncertainty left over). While there are various approaches to do this, none of them can explicitly connect to models that make short-term forecasts. In this project we will make the link from genetic data to prediction using 'force of infection' curves in the R package TransPhylo.

Requirments: ordinary differential equations, introductory statistics, and familiarity with a programming language. If that language is not R, a willingness to learn R is essential. This project links modelling, inference and epidemiology and an enthusiasm for interdisciplinary work is important. No previous biological knowledge is required.

NB: This project is eligible for a VPR USRA ONLY