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 agestructured model of mosquito
biopesticides and malaria control Description: Biopesticides 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 lifestages of the mosquito. In this project, a new agestructured 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 twofluid 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 smallscale apparatus. The apparatus is made of an
acrylic container and involves viscoplastic fluids,
ultrasensitive 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
nonNewtonian 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
gasmigration 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 nearsurface
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 Kortewegde Vries equations  describe dispersive waves, but nevertheless possess "soliton" solutions whose spatial profiles are constant in time. Some of these solitons are wellknown 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 onedimensional 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 LifeScience Calculus
Experience at UBC (Math 102/103). Description. 
FokShuen
Leung 
Title: Visualizing calculus from the
completeness of R to Taylor's Remainder Theorem. Description: Consider the content of a "standard" pair of firstyear 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^{1epsilon}. 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 wellwritten article
that is both mathematically rigorous and also inviting to
the nonspecialist 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 Propagation 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 shalelike 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: http://www.math.ubc.ca/~peirce 
Lior
Silberman 
Project 1. Computation of eigenfunctions on
polygonal domains Abstract: Motivated by the Polymath7 project and the collocation method we will investigate a finiteelement 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. 