Proposed research projects for Summer 2014

Note to students: this list will be continuously updated with new projects until the application deadline in February.
Supervisor Name
Research Project
Richard Anstee
The general area of investigation is extremal combinatorics. The work will involve exploring problems of forbidden families of configurations. The goal is to expand on the M.Sc. work of Christina Koch and also continue the work of Lu and myself on bounds for a special family of configurations which made use of Ramsey Theory. The following is a typical problem given in matrix language. Let m be given and let F be a given kxt (0,1)-matrix. Let A be an mxn (0,1)-matrix with no repeated columns and no submatrix that is a row and column permutation of F. We seek bounds on n in terms of m,F (so called problem of Forbidden Configurations). It is perhaps surprising that n < cm^k, for some c, but we can do even better for many F.

Interested students could contact my previous USRA students: Ron Estrin and Foster Tom.
George Bluman,
Zinovy Reichstein
Title: The mathematics of parallel parking

The student will study a mathematical model of parallel parking. This project will combine the theory of Lie algebras (in particular, computations in enveloping algebras) with methods of applied mathematics aimed at practical applications.  We are looking for a mathematics undergraduate with a strong physics background. During the project the student will learn about Lie algebras and the computation of global groups resulting from their commutators, including applications to a model for parallel parking.  This model has free modelling parameters which require investigation by the student.  The student will also do a literature survey and consequent study of existing mathematical models for parallel parking.
Daniel Coombs
Title: An age-structured model of mosquito biopesticides and malaria control

Description: Bio-pesticides are natural pathogens that are called into service to control pest populations. An important class of biopesticides are fungal pathogens of mosquitos. These fungi can replace or be used in conjunction with chemical pesticides in controlling the incidence of malaria. Additionally, genetic engineering of the fungi has been performed and variants have been produced that are shown to control malaria parasites within the mosquito. This raises the possibility of reducing the burden of malaria by curing the mosquitos. However, there are some potential issues regarding the effectiveness of these biopesticides that are related to the precise timing of the malaria infection process, the fungal pathogen infection process, and the life-stages of the mosquito. In this project, a new age-structured model of these effects will be developed and analyzed with the goal of understanding (a) the possible efficacy of existing biopesticides of this type, and (b) how future biopesticides might be optimally designed to control the impact of malaria.
Ian Frigaard,
Kamran Alba
Project 1: Mixing and displacement in pipe flows

We seek a motivated individual to help in modifying an existing experimental apparatus in order to be able to conduct experiments involving two-fluid displacement flows in an inclined pipe. The applicant will need to understand the flow experiments to be run, help in design of new components and other modifications, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. Once modified, the person will assist in various operations associated with the flow loop: mixing and fluid preparation, operating the flow loop, running careful experiment, image processing of the data. Machining and instrumentation experience, data acquisition, etc. are considered as advantages. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.

Displacement of one fluid by another is a common process in industrial applications, where the fluids are not always Newtonian and where a range of fluid properties and densities are used. Here we focus on pipe flow displacements in inclined pipes, where there is also a significant density difference. Depending on the fluid properties and flow rates the fluids either mix, or displace with a clean interface, or stratify during the displacement. We seek to understand these flows mostly experimentally also partly computationally though numerical simulations.

Project 2: Gas migration in viscoplastic fluids

We seek a motivated individual to help run experiments in a small-scale apparatus. The apparatus is made of an acrylic container and involves viscoplastic fluids, ultra-sensitive pneumatic components and high speed flow imaging. The applicant will need to understand the flow experiments to be run and may need to help in design of new components, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. The person will assist in various operations associated with the experiment: mixing and fluid preparation, running careful experiment, image processing of the data and rheometry measurements of the non-Newtonian fluids. Experience with pneumatic, machining and instrumentation, data acquisition etc. is an advantage. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.

Through fundamentally studying the topic of gas bubble movement in a viscoplastic bed with intend to resolve the gas-migration problem in cemented oil & gas wells. After drilling oil & gas wells, the annulus section between the production casing and rock formation is cemented using cement slurry. The slurry is then left to set and solidify. In this stage of the process gas may enter the cemented annulus creating channels that provide an undesirable flow path of the reservoir fluids including hydrocarbons into the wellbore and near-surface environment. Our aim is to study this problem mostly experimentally (also partly computationally and analytically) in order to finally design the cement slurry fluid such that it minimizes the gas intrusion from formation into the wellbore. This will in return, decreases the environmental impacts and increases the well productivity.

The positions are likely to appeal to applied mathematics or engineering physics students. The mathematical content of the positions lies in understanding physical fundamentals, in data analysis and some computation.
Stephen Gustafson
Title: Soliton Stability: Analysis and Numerics

Background: Many partial differential equations of physical importance -- examples include nonlinear Schroedinger and Korteweg-de Vries equations -- describe dispersive waves, but nevertheless possess "soliton" solutions whose spatial profiles are constant in time. Some of these solitons are well-known to be stable (hence physically observable) against certain mild perturbations, but with the exception of one or two very special "completely integrable" cases, the response of a soliton to rougher (high frequency) noise is not understood.

Project: This project has both computational and theoretical elements (and could in principle be suitable as a "team" project). Building on existing numerical schemes and codes, we aim to compute the response of one-dimensional periodic solitons to various classes of noise, and propose some conjectures based on these computations. On the analysis side, we will review the few proofs available in the literature of stability against rough perturbations, and propose analytical approaches to our conjectures.
Kalle Karu
The general area of the research is lattice polytopes, cones over polytopes, and fans obtained by gluing such cones. There are several problems that can be chosen according to student's background knowledge and interests. The problems are about subdivisions of polytopes (triangulations), counting lattice points inside a polytope, and polytopes with special properties, such as nonsingular polytopes, for example. Techniques used in the study involve combinatorics, linear algebra, and some abstract algebra (group and ring theory).
Leah Keshet Project 1: Simulation of interacting cells in epithelium.
Project 2: Analysis and simulation of molecular motors on microtubule tracks.
Leah Keshet,
Eric Cytrynbaum
Title: Modernizing the Life-Science Calculus Experience at UBC (Math 102/103).

Fok-Shuen Leung
Title: Visualizing calculus from the completeness of R to Taylor's Remainder Theorem.

Description: Consider the content of a "standard" pair of first-year courses in differential and integral calculus. What is the logical structure of this content, and how can it be visualized meaningfully? The goal of this project is to create an online setting that will provide this visualization. This will likely take the form of a navigable directed graph in which clicking on an edge will yield an explicit proof of implication.

Programming experience is strongly preferable but not necessary. The candidate should have good mathematical writing skills and enjoy reworking proofs.
Greg Martin
ABC Triples.

The ABC conjecture is an extremely ambitious assertion in number theory, roughly saying that three numbers that are additively related cannot all have unusual factorizations. More precisely, consider triples of positive integers a, b, c with a+b=c, and let R be the product of all the distinct primes that divide a, b, or c. For any positive number epsilon, the ABC conjecture asserts that there are only finitely many such triples (a,b,c) such that R < c^{1-epsilon}. For example, taking a=1 and b=2^m, the ABC conjecture says that numbers of the form 2^m+1 are "almost squarefree".

It might be considered annoying that we need the epsilon in that statement; however, the "naive ABC conjecture" - namely the assertion that R must be at least c - is certainly false. We know several constructions that give infinitely many counterexamples to this naive statement. Some of the simplest such constructions, however, are "folklore examples" and quite hard (or impossible) to find in the literature. People have also found it interesting to search for numerical examples of triples (a,b,c) where R is smaller than c (even though no finite set of such examples can prove or disprove the full ABC conjecture). Whether they come from theoretical constructions or numerical searches, triples (a,b,c) with a+b=c where R < c are called "ABC triples".

The goal of this USRA project is to write an expository article on what is known about ABC triples. Of course this goal provides a concrete body of knowledge to assimilate and master; this knowledge is quite accessible for strong undergraduate students. Ultimately we would hope to publish a joint paper in the American Mathematical Monthly; this would require a very well-written article that is both mathematically rigorous and also inviting to the non-specialist reader. The successful candidate for this project should have a solid understanding of number theory (say MATH 312) and also solid skills in analysis (possibly MATH 320, although that indicates more the level of mathematical maturity than any specific content); some of the required techniques can be learned during the project. At least as importantly, the candidate should have excellent mathematical writing skills: composing proofs should be seen not just as a challenge to overcome but also as an opportunity to enlighten and even entertain potential readers.
Anthony Peirce
Novel Approximation Schemes to model Hydraulic Fracture


Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. Examples of HF occur in nature as well as in industrial applications. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy. They have recently received considerable attention in the media due to the intense hydraulic fracturing of horizontal wells in order to release the natural gas embedded in shale-like rocks – a procedure referred to as “fracking.”

We plan to investigate the numerical solution of an integral equation that governs the propagation of a fracture in an elastic medium under conditions of plane strain. We will consider a collocation scheme to solve this integral equation. Of particular interest is the robustness of the solution to mesh refinement. The objective of this project is to devise autonomous mesh refinement strategies that will be able to achieve uniformly convergent schemes that are much more efficient than can be achieved using a uniform mesh. The new adaptive scheme scheme will then be used to solve the dynamic model for a hydraulic fracture propagating in an elastic medium. It is proposed that a computer code will be developed in MATLAB and the numerical solutions will be checked against existing asymptotic solutions.

For more information please check my web site:
Lior Silberman
Project 1. Computation of eigenfunctions on polygonal domains

Abstract: Motivated by the Polymath7 project and the collocation method we will investigate a finite-element method for computing approximate eigenfunctions on plane domains.  Some programming background required.

Project 2. Topology of modular links

Abstract: We will use computer calculations to investigate topological invariants of the spaces obtained by removing closed geodesics from the unit cotangent bundle of the modular surface.  Programming experience will be required (familiarity with python an advantage); background in group theory, algebra and topology will be useful.  For pictures of what the spaces look like see here.

Projects from last year: 2013