USRA | Undergraduate Student Research Awards

The hidden subgroup problem involves finding a subgroup, given evaluations of a function which is distinctly constant on its cosets. The abelian case can be solved by a quantum computer and results in a quantum algorithm to factor integers efficiently. The objective of this project is to study the explicit representation theory involved in generalizations to non-abelian groups.

A Cayley graph is a visualization of a group paired with a finite set that generates the group. The vertices are the elements of the group, and an edge connects two group elements if one is equal to the product of the other and a generator from the set.  For example: A finite, multiplicative  cyclic group generated by x and its inverse has a cycle as its Cayley graph. Because of the high number of symmetries coming from the group, they are often very beautiful graphs. At the same time, they permit one to use graph theoretic tools to study groups. In this project we consider the problem of enumerating walks along the edges of Cayley graphs, and try to determine formulas for the number of walks of a given length that start, and end at the vertex that represents 1. This is related to the problem of trying to identify whether a given product of arbitrary generators is equal to 1 or not, the so-called “word problem” of group theory.

I am looking for motivated, independent students that have taken courses in group theory, and combinatorics. (The project involves generating functions).  They should have some programming experience — we will be building Jupyter notebooks, and prior experience will be an asset. Interested students should contact me directly to discuss their suitability for this project, and the potential of this topic for an undergraduate honour’s thesis.

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: degenerations to toric varieties, 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. 

This is a joint project with Prof. Patricia Brantingham from the SFU School of Criminology. The goal is to advance our mathematical model used in prior research to understand and predict crime patterns in cities. A significant amount of crime data is available from the Vancouver Police Department and other police departments. There is an opportunity to test the models against real data in multiple cities. The modelling will involve difference and/or differential equations, with agent based simulations on street networks.

The topics for a project include:

• How crime attractors and crime generators are formed
• Metro stations as specific generators of crime
• The impact of road capacity on crime clustering (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
• The impact of city growth and concentrations of shopping and entertainment facilities on crime concentrations

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

This project will use mathematical modelling and data analytics to improve access to care for emergency hospital admissions in British Columbia. Patients admitted to hospitals often wait an unacceptably long period of time in the emergency department before they receive a hospital bed. This has significant consequences both for the quality of health care and the efficient functioning of the emergency department. The British Columbia Ministry of Health will provide data for this project, which is part of a wider collaboration to improve health care in BC.

This project will involve developing a queue network model for hospital admissions. Discrete event simulation will be used to analyze this model. Statistical methods will be used to calibrate and validate the model using data. The student will be encouraged to investigate innovative approaches to combining machine learning with simulation. The ultimate goal of the project is to determine the number and type of hospital beds required to meet quality of care targets for emergency department admissions.

Requirements: This project is well-suited to a student interested in operations research, data science, and simulation modelling. A course in operations research or probability models, such as Math 208W or Math 348, would be beneficial. Programming experience in R and/or python is essential.

In this project we are interested in investigating the behaviour of solids immerse in a gas. We will apply coupled equations of the gas and solid to model their motion. In particular, kinetic equations will be used to describe the evolution of the gas and equations derived from Newton's Second Law will be for the solid. Their interaction is through the moving boundary conditions for the gas together with the formulation of the drag force generated on the solid due to exchange of momentum. We are interested in studying existence, uniqueness and long-time behaviour of such a system. In particular, we want to rigorously justify the decay rate that has been observed in numerics.

Arithmetic geometry studies the interplay between geometry and number theory (arithmetic) -- the solutions of polynomial equations in several variables that take values in the rational numbers, for instance. It is a field that has time and again exposed deep and intricate connections within mathematics. Due to the nature of the objects studied, it is also a field that is particularly suited to computational exploration.

Exact projects will be developed in collaboration with interested candidates.

Requirements: Prerequisites require a strong background in mathematics, and particularly in algebra: MATH 340, 341, 342; MATH 440, 441 etc.

This project consists of adapting code for computing periods of Riemann surfaces [1] to use a certified numerical integrator [2]. The project can possibly be extended to compute integrals over non-closed paths.

Implementation would happen in the Sage computer algebra system, and would require a solid knowledge of the Python programming language. The student should be prepared to work with numerical algorithms in a rigorous mathematical setting. Experience with combining complex scientific software solutions is a plus. Interested students should contact the supervisor for more information and to check if the project is right for them.

[1] Nils Bruin, Jeroen Sijsling, Alexandre Zotine, Numerical computation of endomorphism rings, ANTS XIII: Proceedings of the Thirteenth Algorithmic Number Theory Symposium, The Open Book Series 2-1 (2019), 155-171.

[2] Frederic Johannsen, "ARB" library. See http://fredrikj.net/blog/2017/11/new-rigorous-numerical-integration-in-arb/

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.

The Ohta-Kawasaki energy is a model from material science whose minimization leads to a fourth-order partial differential equation. Solving it on periodic domains gives a rich class of distinct solutions such as stripes, spots, spheres of different packings and double gyroids.

This project will look at numerical methods to find patterns in irregular domains which are not amenable to simple descriptions. We will use approximations based on Frames to find non-periodic solutions using a Fourier series description.

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.

Amalgamation is a concept from model theory that describes when two structures sharing a common substructure can be embedded into a third structure while respecting the substructure. A class of objects of which any two objects can be amalgamated when they have a common substructure is said to have the amalgamation property. When a class of structures has the amalgamation property it can often be shown to have a Fraisse limit, which is an object with many interesting theoretical properties. Examples of Fraisse limits include the rationals (the Fraisse limit of the class of all finite linear orders) and the Rado graph (the Fraisse limit of the finite graphs). In this project you will investigate which classes of finite metric spaces have the amalgamation property.

Recently a study by Finkelstein, Stevens, al-Gharbi claimed, using records of comments on various political websites, that psychologist Jordan Peterson is a "gateway drug" to extremist alt right content. The student taking on this project will examine this study and others like it and develop a mathematical model of ideological development and content consumption. Such a model can then be used to study other examples of gateways in content consumption, not necessarily in the political context.

# Students: up to 2

As part of this USRA, students will be involved in an interdisciplinary collaboration joint with Prof. James Wakeling in BPK.  The goals of the USRA project are to design and implement computational methods to study the impact of localized activation of muscle fibres, as well as understand why fast- and slow-twitch fibres are differentially recruited in motion. As part of the project, the students will learn about the foundations of skeletal muscle mechanics, the finite element method, and coding in C++. The computational studies will build upon existing codes. A background in PDE and computing is essential.

# Students: 1

In Math 314, students encounter the use of trigonometric Fourier series to solve simple PDE with specific boundary conditions.  Trigonometric functions arise naturally as the eigenfunctions of the Laplacian with those boundary conditions. In this USRA, we'll investigate the behaviour of a different set of eigenfunctions - those of the Steklov eigenvalue problem for the Laplacian - and compare this with some standard spectral methods. The concepts we'll use include some from PDE, numerical analysis and linear algebra. A background in analysis, PDE and computing is needed.

This project investigates spectral methods and radial basis function methods for solving differential equations. The focus is on designing new methods for differential-difference equations (equations with a delay), in particular spectral methods.

Requirements: Students should have completed Macm 416 and Math 310. Familiarity with delay equations is desirable.

Models for Atmospheric Science 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!).

Some background in a differential equations is essential, as is proficiency in a computational environment such as Matlab.  Students enrolled in the Spring 2020 fluid dynamics course are particularly qualified.

In this project we will explore possible hypocoercivity of a kinetic equation with internal state (KEIS) used to model bacteria motion. Although a rather general framework has established by Dolbeault-Mouhot- Schmeiser to show hypocoercivity for transport equations without smoothing effect, such framework does not apply to the equation that we are studying. The aim of this project to identify hidden structures of the KEIS which can lead to hypocoercivity.

Description: A voltage graph is a directed multigraph with possible loops whose edges are marked by elements of a group. The covering graph of a voltage graph is a special product of the voltage graph with the group. The proposed project is going to study under which conditions on the voltage graph and the assignment on its edges the resulting covering graph has certain properties. In particular, under which conditions the covering graph is Hamiltonian.

A successful candidate has taken courses in group theory, and graph theory. Some programming experience is a plus.