Supervisor 
Research project 
Summer 2018

Richard Anstee 
EXTREMAL COMBINATORICS
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns.
We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F.
Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F.
There is an attractive conjecture (A. and Sali) to pursue with some challenging open problems.
A variation is to consider a specially structured family of forbidden configurations.
We will have plenty of problems to work on.
Interested students could contact previous USRA students Santiago Salazar and Jeffrey Dawson.

Summer 2017

Steph Van Willigenburg & Samantha Dahlberg 
NEW GENERALIZATIONS OF THE CHROMATIC POLYNOMIAL
A graph, G, is a set of dots, known as vertices,
connected to each other by a set of lines, known
as edges. A proper colouring of G is a colouring
of the vertices such that no two vertices joined
by an edge are the same colour. If we are given k
colours then the number of ways of colouring G, as
a polynomial in k, is known as the chromatic
polynomial. If we have infinitely many colours
then the chromatic polynomial generalizes to the
chromatic symmetric function.
This project will aim to generalize the chromatic
symmetric function yet further, with the goal of
establishing its basic properties, and trying to
resolve wellknown conjectures regarding the
chromatic symmetric function. One such example is
that it is known that if G is a tree, that is a
graph with no set of edges forming a cycle, then
the chromatic polynomial is the same for every
tree with the same number of vertices. However, it
is conjectured that two trees are nonisomorphic
if and only if they have distinct chromatic
symmetric functions.
This project will be under the supervision of
Stephanie van Willigenburg and Samantha Dahlberg.
A strong background in abstract algebra such as
Math 322 or Math 323, an aptitude for
combinatorics, and programming skills will be an
asset.

Dan Coombs 
ANALYSIS AND MODELING OF SUPERRESOLUTION
MICROSCOPY DATA FOR CELL SURFACE RECEPTORS.
This project, which will be jointly supervised by
one or more professors in a biology department,
will involve computational analysis of multicolour
microscopic images of cell surface receptors,
likely on B cells. This project will require good
programming skills (using Matlab or Python) and
willingness to work closely with the experimental
team.
Students could be from Math, CompSci, (Bio)Physics
or Engineering.

Ailana Fraser

MINIMAL SURFACES
Minimization problems arise naturally in
many branches of mathematics and science.
For example, problems in navigation
involve finding paths of least length
(`geodesics') on the earth's surface.
Minimal surfaces, which are
twodimensional analogs of geodesics, are
minimizers (or simply critical points) of
the area function, and arise naturally in
material science; for example in fluid
interface problems and elasticity
problems. A simple physical example of a
minimal surface is the soap film that
forms after dipping a wire frame into a
soap solution. By the laws of surface
tension this soap film has the property
that it is stable, that is it becomes
larger under slight deformations. The
theory of minimal surfaces (and
submanifolds) has had striking
applications, for example to general
relativity and low dimensional topology.
This project will involve studying
existence of minimal surfaces, properties
of minimal surfaces, and applications.

Richard Anstee

EXTREMAL COMBINATORICS
A problem in extremal combinatorics asks
how many (discrete) objects are possible
subject to some condition. We consider
simple matrices which are matrices of
integer entries with no repeated columns.
We say a simple matrix A has F as a
configuration if a submatrix B of A is
equal to a row and column permutation of
F. We typically restrict our attention to
(0,1)matrices and much has been
determined about the maximum number of
columns an mrowed simple (0,1)matrix can
have subject to condition it has no
configuration F for some given F. We will
consider variations where the simple
matrices are restricted to have entries in
{0,1,2} and have some conjecture to
explore arising from joint work with
Attila Sali.
Interested students could contact my
previous USRA students Foster Tom, Maxwell
Allman, Farzad Fallahi, and Santiago
Salazar.

James Feng

COMPUTER MODELING AND SIMULATION OF COLLECTIVE MIGRATION OF NEURAL CREST CELLSEXTREMAL COMBINATORICS DESCRIPTION
Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos.
For example, the socalled neural crest cells (NCC) migrate in streams along the spine of the embryos of chicks, frogs and
zebrafish. Moreover, NCCs from different sources manage to stay unmixed while migrating side by side. Later, they seem to
be directed to different destinations along the spine, and then toward the front of the body, where they form various tissues
and organs.
There are several mysteries about the collective migration. How do cells interact with each other to maintain cohesion among those from the same source, while keeping a boundary between cell clusters from difference sources? How do the cells decide where to stop or turn into a different route? The intensive efforts by biologists have produced some hypotheses. But as these questions involve the intimate coupling between biochemical signaling and cell mechanics, answering them requires the help of quantitative analysis.
We have developed a mathematical model and numerical techniques for modeling the interaction and collective migration of neural crest cells. For this summer project, the student will generalize our existing tool to study two scenarios of interest: (i) Given an externally imposed gradient of chemoattractant, how does a cluster of NCCs behave? (ii) What chemical and geometric cues may guide different streams of NCCs into different migration paths? Our model involves ODEs describing the dynamics of the signaling molecules and molecular motors, as well as ODEs or PDEs governing the mechanical behaviour of the cells and tissues. The computer program includes MATLAB and Python programming.

Ian Frigaard & Ali Etrati

NUMERICAL SIMULATION OF FLUIDFLUID MIXING AND DISPLACEMENT IN CHANNELS
Displacement of one fluid by another of different properties is a common
process in many industrial applications. Our focus is in displacements of
miscible fluids in inclined ducts, where density and viscosity difference
exists. Depending on the fluid properties, flow rates and inclinations,
different flow regimes and behaviours are observed. For this project we seek
a motivated individual to help in running a large number of numerical
simulations, covering a wide range of parameters in displacement flows. The
simulations are in a 2D channel with high aspect ratio and are written using
PELICANS, an opensource C++ package. The simulations will be sent to
WestGrid and the results will be postprocessed using existing Matlab codes.
The applicant will need to set the correct parameters for each case, submit
the jobs to WestGrid, collect and organize the data, and help with
postprocessing the results. Other data management tasks may also be
included. Experience with Linux, Matlab and knowledge of C++ is an
advantage.
This could be suitable for an Engineering or Science student with
some experience and interest in fluid mechanics, plus physical intuition.

Ian Frigaard & Amir Maleki

VALIDATION OF A MODEL FOR CEMENTING OF OIL/GAS WELLS
Oil and gas wells are cemented to reduce environmental leakage, to
strengthen/support them structurally and to zonally isolate production zones
in the reservoir. A model of this process has been developed at UBC
consisting of 2D hydraulic calculations of viscoplastic fluids in a narrow
channel coupled with a vector concentration equation. The model is
implemented and solved in Matlab using finite difference method. For this
project we aim to validate this model with experimental/field data that we
have available and the main focus will be to aid in this validation. The
work will include running different cementing scenarios, postprocessing the
outcomes and comparing with the experimental data in an intelligent way.
An ideal candidate:
. is proficient in Matlab (reading and compiling Matlab .m files,
troubleshooting Matlab programs)
. has strong mathematical background (multivariable calculus,
ordinary and partial differential equations)
. is familiar with numerical algorithms (Numerical derivatives,
numerical integration, root finding methods)
. can demonstrate strong verbal communication, problem solving and
critical thinking.
This could be suitable for an Engineering or Science student with some
experience and interest in fluid mechanics, plus physical intuition.

Lior Silberman & Avner Segal

DECOMPOSITION OF PRINCIPAL SERIES REPRESENTATIONS VIA ELEMENTARY METHODS
We are looking for a student to perform computations in linear algebra and
finite group actions which will contribute to a project in the representation theory of padic groups.
The only required background is linear algebra and group theory (say at the level of Math 223 and Math 322).
Familiarity with computer algebra systems such as SAGE is desirable but not required.
No knowledge of of more advanced mathematics is needed for the USRA project.
The student will have the opportunity to learn some of the advanced mathematics underlying the project as a whole.

Leah Keshet

SIMULATING THE INTERACTIONS AND MIGRATION OF MULTICELLULAR TISSUES
The behaviour of cells in a tissue depends on their chemical and mechanical interactions,
and on stimuli they receive from their environment.
Our group studies mathematical models for intracellular signaling and its effect on cell shape,
motility, and tissue dynamics. Publicly available software exists for simulating such cellular systems.
This project will consist of adapting such software (CompuCell3D, Morpheus, CHASTE, and others)
to the specific models for signaling studied by members of our group.
This project is suited for CPSC, MATH or PHYS majors who can demonstrate ability to work on large opensource projects
as part of an interdisciplinary team.
Specifically, we are looking for motivated students with experience in testdriven development,
objectoriented programming (C++, Python or MATLAB) and Linux/Unix server administration.
Applicants with scientific computing experience in C++ using template class libraries for numerical
methods, linear algebra, mesh manipulation and multicore processing (e.g. Boost, PETSc, MPI, OpenMP, OpenMesh, etc.)
are strongly encouraged to apply.
Successful accomplishment in this summer work could lead to future research opportunities, including a graduate (MSc.) position in the Mathematical
Biology group of Leah Keshet.

Colin MacDonald

COMPUTING FIRST PASSAGE TIMES AND RANDOM WALKS ON SURFACES
The Brownian motion of particles is a basic fundamental physical
process. Consider the problem of particle moving randomly on a curved
surface, such as a cell wall or material substrate. Starting from a point, what is the average time for such a particle to reach a certain
"trapping region"? This is known as the "mean first passage time".
We can approach such problems using partial differential equations
(PDEs). In this project, we have several goals: (1) to compute
solutions to diffusion PDEs on surfaces using the Closest Point
Method; (2) to implement particle simulations based on closest point
representations of surfaces; and (3) to investigate the role of
curvature in first passage time problems.
Useful skills include proficiency in Matlab/Octave or Python, a
background in numerical algorithms (such as finite differences and
interpolation), and some knowledge of differential equations.
Knowledge/interest in collaborative software development with Git
would be helpful.

Zinovy Reichstein

MINIMAL NUMBER OF GENERATORS FOR AN ETALE ALGEBRA
An elate algebra E over a field F is a finitedimensional algebra of the form E = E1 × · · · × Er,
where each Ei is a finite separable field extension of F . If r = 1, i.e.,
E is a field, the primitive element theorem asserts that E can be generated over F by a single element.
If F is an infinite field,
the primitive element theorem remains valid for etale algebras.
However, it breaks down if F is a finite field.
The project is to find the minimal number of generators in this case,
under various assumptions on E (e.g., assuming that
F  = q and dimF (E) = n is fixed or assuming that q, n and r are fixed).
This problem came up in my joint work with UBC postdoc Uriya First.
Uriya plans to be involved in this USRA project in person while he is still at UBC (his appointment ends on June 10).
He may also be available remotely after he leaves.
We plan to begin with background reading in Galois theory,
the classification and basic properties of finite fields,
the primitive element theorem for fields and etale algebras,
and some related linear algebra (in particular, the Vandermonde determinant and its variants).
Prerequisites are Math 223, 322 and 323 or equivalent.
This general topic and the specific question should be accessible to an undergraduate;
we are hoping to get a definitive answer to our main question (or at least some variants of it)
by the end of the summer.
We also hope that working on this topic will prepare a student for a more indepth study of
abstract algebra and related areas of pure mathematics.

Top

Summer 2016

Richard Anstee 
Title: Extremal Combinatorics
Description: The problem of forbidden Berge
subgraphs will be considered. The relationship
with the problem of Forbidden Configurations will
be explored. We say a (0,1)matrix A has F as a
configuration if a submatrix B of A is equal to a
row and column permutation of F. We say a
(0,1)matrix A has F as a Berge subgraph if a
submatrix B of A has 1's in the positions of the
1's of a row and column permutation of F (B may
also have 1's in other positions).
Let F,m be given. The extremal problems consider
for an mrowed matrix A how many different columns
can you have without having F as a configuration
or perhaps without having F as a Berge subgraph.
The student will consider some recent papers in
this area to seek connections between the two
problems.
Interested students could contact my previous USRA
students Foster Tom, Maxwell Allman and Farzad
Fallahi.

Daniel Coombs 
Analysis and modeling of superresolution
microscopy data for cell surface receptors.
This project, which will be jointly supervised by
Michael Gold in the Cell Biology and Immunology
department, will involve computational analysis of
multicolour microscopic images of cell surface
receptors on B cells. Further details available on
request. This project will require good
programming skills (using Matlab or Python) and
willingness to work closely with the experimental
team.

Eric Cytrynbaum 

James Feng 
An active fluid model for the fruit fly germband
during embryonic development.
A key event in the development of the fruit fly
embryo is the formation of the headtail axis
along the body, partly thanks to the elongation of
an epithelial tissue known as the germband. By
interdigitating cells along the transverse
direction, the germband extends by a factor of two
along the headtail axis. The objective of this
project is to explore this extension process by
mathematical and computation tools.
My group has worked on vertex models that
represent the tissue as an elastic network of
polygonal cells, on which myosin motors exert
contractile force to drive the rearrangement of
the cells. In this USRA project, we will explore a
complementary approach by viewing the tissue as an
effective fluid driven by distributed contractile
forces. More specifically, we model the tissue as
an active viscoelastic fluid whose motion is
driven by distributed body forces and moments that
induces locally elongation along the headtail
axis and contraction in the transverse direction.
The interesting questions include the following:
What magnitude of active forces will generate the
observed amount of axial extension? What are the
roles of the extrinsic forces pulling on the
tissue? Can this model quantitatively predict the
outcome of the germband extension?
The project will involve numerical solution of
PDEs describing the fluid dynamics of the tissue.
Ideally the student has had exposure to fluid
mechanics and numerical methods. Knowledge of cell
biology is not essential but will be a plus. The
student will help build the model and carry out
computations to explore its predictions. See more
background information on my webpage http://www.math.ubc.ca/~jfeng/
under "Research" and "Publications".

Kalle Karu 
Combinatorics of the Bruhat graph
Let S_n be the symmetric group, the group of
permutations of the set {1,2,...,n}. The Bruhat
graph of S_n is a graph with vertices the elements
of S_n, with two vertices connected by an edge if
one permutation is obtained from the other by
switching two numbers. See the picture for
part of the Bruhat graph of S_4 (the picture only
shows 18 out of the 24 elements of S_4).
Bruhat graphs are related to the geometry of flag
varieties in algebraic geometry, to representation
theory of Lie groups, and to other parts of
mathematics. A problem in combinatorics asks to
enumerate all paths in the Bruhat graph. It is
known that this enumeration can be encoded in a
polynomial, called the complete cdindex. The main
open conjecture states that this polynomial has
nonnegative coefficients. The goal of this
project is to work towards proving this
conjecture.
Some elementary group theory will be needed (such
as taught in math 322 or 422). The main problem is
combinatorial and does not need any higher
algebra.

Leah Keshet 
Title: Understanding cell topology and
geometry in tissue simulations
Description: This summer research position
is suitable for a CPSC/MATH USRA student with
experience in Python, Object Oriented Programming
(C++ or Java) and working in a UNIX environment.
The student will be responsible for implementing
algorithms from computational geometry to
automatically find and classify cells according to
their phenotype based on measurements of shape,
size, polarity, movement speed etc. Our group is
interested in modeling cell division, cell
motility, tissue migration and other phenomena
associated with morphogenesis in diverse
biological and physiological systems. The USRA
student will learn to use opensource tools for
extracting rich information from simulations of
biological systems under the guidance of current
group members. Opportunity to interact with both
mathematical modelers and biological researchers
will be provided.
Prior experience with opensource development,
computational geometry, statistics or machine
learning is an asset.

Brian Marcus
A. Rechnitzer 
Capacity of constrained systems.
A constrained system X is a set of ddimensional
arrays defined by a small set of forbidden
patterns. A simple example is the set of arrays of
0's and 1's on the vertices of the ddimensional
lattice such that two adjacent vertices cannot
both be 1. An array is allowed if it does not
contain a forbidden pattern. The capacity of X is
defined as the asymptotic growth rate of the
number of allowed arrays on large ddimensional
cubes. The capacity is closely related to notions
of entropy in information theory and ergodic
theory and free energy and pressure in statistical
physics.
When d = 1, the arrays are simply sequences and
there is a simple formula for capacity given as
the log of the largest eigenvalue of a matrix
associated with the set of forbidden patterns. For
d > 1, there is no general formula for the
capacity of a constrained system and the exact
value of capacity is known only for a handful of
systems. Nevertheless, there are very good methods
of approximating capacity. The methods are based
on ideas in linear algebra, combinatorics and
probability.
The goal of this project is to compare
approximations given by different methods and to
explore some related problems of theoretical
interest. The student(s) will first learn the
basics of capacity and entropy, applications
inside and outside of mathematics, and methods for
computing approximations.
Background in programming and linear algebra is
essential. Background in probability and
combinatorics would be helpful.

Greg Martin 
Statistics of the multiplicative group
For every positive integer n, the quotient ring
Z/nZ is the natural ring whose additive group is
cyclic. The "multiplicative group modulo n" is the
group of invertible elements of this ring, with
the multiplication operation. As it turns out,
many quantities of interest to number theorists
can be interpreted as "statistics" of these
multiplicative groups. For example, the
cardinality of the multiplicative group modulo n
is simply the Euler phi function of n; also, the
number of terms in the invariant factor
composition of this group is closely related to
the number of primes dividing n. Many of these
statistics have known distributions when the
integer n is "chosen at random" (the Euler phi
function has a singular cumulative distribution,
while the ErdosKac theorem tells us that the
number of prime divisors follows an asymptotically
normal distribution). Therefore this family of
groups provides a convenient excuse for examining
several famous number theory results and open
problems.
The goal of this USRA project is to write an
expository article on what is known about these
statistics of the multiplicative group. 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 algebra (say MATH 322, although that
indicates more the level of mathematical maturity
than a catalog of necessary 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.

Dale Rolfsen 
Free group orderings and automorphisms
Free groups F_n are basic objects in the theory of
groups and also arise in topology, analysis and
many other branches of mathematics. Free groups
are orderable, in the sense that the elements can
be linearly ordered in a way that is invariant
under multiplication. In fact there are
uncountably many such orderings of F_n if n>1.
A fundamental question is whether a given
automorphism will respect some ordering of F_n.
This has application to knot theory and other
aspects of low dimensional topology.
The project will involve study of the literature
on orderable groups, the automorphism group
Aut(F_n) and spaces of orderings of F_n and other
groups. It will also involve writing computer
programs which can test whether an automorphism
preserves some order. One goal will be to gain
understanding of (and perhaps solve) the question
of the structure of the space of orderings of F_n,
which is conjectured to be a Cantor set.

Top

Summer 2015

Richard Anstee 
Forbidden Families of Configurations.
The general area of investigation is extremal
combinatorics. The goal is to expand on the M.Sc.
work of Christina Koch and also continue the work
of Maxwell Allman and myself on bounds for a
special family of configurations. 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: Foster Tom and Maxwell Allman.

Daniel Coombs 
Cluster Analysis of Super Resolution
Fluorescence Images.
This project is to work on novel methods for
interpreting data from a very modern microscopic
imaging technique, dSTORM. The images, which are
taken at UBC, are large  each contains ~14 x 106
data points. The points (each representing light
emitted by a single fluorescent molecule attached
to a defined protein in a cell) are, as expected,
distributed in discrete clusters, but the shapes
of the clusters are highly irregular. The goal of
this project will be to learn about existing
methods, understand why they fail for automatic
detection and classification of the observed
clusters, and then develop new approaches to this
problem. The lab produces novel code for
displaying and analysing data acquired from both
light and electron microscopes and we would prefer
an individual who is familiar with the programs we
are using; C++, Matlab and OpenGL. This project
will be joint supervised by Edwin Moore (Cell and
Physiological Sciences) and Dan Coombs
(Mathematics).
Molecularscale simulation of calcium ions within
cardiac tissue.
We want to model calcium ion movement and
interactions with the resident proteins within the
dyadic cleft (which is within cardiac cells).
Input data would be the geometry of the cleft, the
position of the relevant proteins and their
assumed behaviour in response to calcium ions as
well as to other intracellular signaling
molecules. The output of the model would be a
‘calcium spark’, which is the calcium transient
produced by a single dyadic cleft. Since the
volume of the dyadic cleft is measured in
femtolitres, the model will be constructed using
stochastic approaches. Experimental results show
that the positions of the relevant molecules
within the cleft are subject to changes in
response to both physiological and pathological
factors. Changes in the molecules’ positions are
also correlated with changes in both the magnitude
and kinetics of the calcium spark. The goal of
this project is to duplicate the experimental
results and to make testable predictions. This
project will be joint supervised by Edwin Moore
(Cell and Physiological Sciences) and Dan Coombs
(Mathematics).

James J. Feng 
Modeling collective migration of cells during
embryonic development.
Biologists have discovered remarkable patterns of
collective cell migration during early development
of animal embryos. For example, the socalled
neural crest cells (NCC) migrate in streams along
the spine of the embryos of chicks, frogs and
zebrafish. The migration is very rapid, and
resembles metastasis of cancer cells so much that
NCC migration has been used as a model for the
latter. Moreover, NCCs from different sources
manage to stay unmixed while migrating side by
side. Later, they seem to be directed to different
destinations along the spine, and then toward the
front of the body, where they form various tissues
and organs.
There are several mysteries about the collective
migration. How do cells interact with each other
to maintain cohesion among those from the same
source, while keeping a boundary between cell
clusters from difference sources? How do the cells
decide where to stop or turn into a different
route? The intensive efforts by biologists have
produced some hypotheses. But as these questions
involve the intimate coupling between biochemical
signaling and cell mechanics, answering them
requires the help of quantitative analysis.
In collaboration with developmental biologists, we
have been developing mathematical models on
various morphogenetic processes that test the
existing hypotheses and strive for a clear
indepth quantitative understanding. These models
typically involve ODEs describing the dynamics of
the signaling molecules and molecular motors, as
well as ODEs or PDEs governing the mechanical
behaviour of the cells and tissues. This USRA
project will study the signaling pathways
controlling the cellcell communication during
collective cell migration, and explore how the
chemomechanical coupling leads to different
patterns. The student will help build the models
and carry out computations to explore their
predictions. See more background information on my
webpage http://www.math.ubc.ca/~jfeng/ under
"Research".

Leah Keshet 

Leah Keshet
Eric Cytrynbaum 

Lior Silberman 

Top

Summer 2014

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.

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.

Top

Summer 2013

Alejandro Adem 
Symmetries of spaces: groups and topology
A mixture of topology, geometry and group theory
will be used to investigate symmetries of Riemann
surfaces and other familiar examples.

Richard Anstee 
The general area of investigation is extremal
combinatorics. The following is a typical problem
given in matrix notation. Let F be a given kxt
(0,1)matrix and let A be an mxn (0,1)matrix with
no repeated columns and no submatrix F. There is a
conjecture of Anstee, Frankl, Furedi and Pach that
claims that there is a constant c(F) so that n
< c(F) x m^k. The immediate goal is asymptotic
bounds for 2rowed forbidden submatrices
completing the results of Ronnie Chen and Ron
Estrin. The work will focus on specific F in order
to gain insight. A related problem is where we
forbid any row and column permutation of F (so
called problem of Forbidden Configurations).
Interested students could contact my previous USRA
students: Ronnie Chen and Ron Estrin.

James J. Feng 
Title: Modeling morphogenesis of the fruit fly
Biologists have studied the fruit fly Drosophila
extensively as a model system, and have
accumulated a considerable qualitative
understanding. The shape evolution and growth of
the fly's embryo has garnered particular
attention. These are complex processes that are
controlled by biochemical signals upstream, and
actuated through physical forces and mechanical
deformation downstream. Because of this
complexity, it is difficult to draw definite
conclusions based on observations and experiments
alone. In many cases, conflicting hypotheses have
been proposed to rationalize the observations.
In collaboration with developmental biologists, we
have been developing mathematical models on
various morphogenetic processes that test the
existing hypotheses and strive for a clear
indepth quantitative understanding. These models
typically involve ODEs describing the dynamics of
the signaling molecules and molecular motors, as
well as ODEs or PDEs governing the mechanical
behaviour of the cells and tissues. This USRA
project will study the signaling pathways
controlling the dorsal closure process as well as
the chemomechanical coupling during episodes of
cell rearrangement known as intercalation. The
student will help build the models and carry out
computations to explore their predictions. See
more background information on my webpage http://www.math.ubc.ca/~jfeng/
under "Research".

Ian Frigaard 
Title: Displacement and mixing flows
The group has been carrying on research in this
area for the past 5 years, combining a blend of
analytical, computational and experimental
methods. The basic setup is that one dense fluid
is pushed along a pipe, driving a less dense fluid
beneath it. The fluids can have different
viscosities as well as densities, plus the pipe
can be inclined at any angle. The range of flows
observed is very wide and is not yet fully
understood. It ranges from structured laminar
multilayer flows, through inertial driven partial
mixing to full turbulence, driven by buoyancy or
flow rate. The main motivation for our study
is to help understand mixing phenomena that occur
during displacement of fluids in oil wells. This
project will suit someone with some lab experience
who has a practical ability to construct things.
The student will work as part of the team in
modifying the experimental apparatus and
conducting experiments, under supervision. As the
project progresses the student will be exposed to
the types of techniques used to analyse these
flows.

Julia Gordon
sujatha Ramdorai 
Modern computational methods allow us to test out
and compute various aspects learnt in Abstract
Algebra. These are very relevant for Algebraic
Geometry. We shall explore a few such topics, both
the abstract aspects and using mathematical
computational software to do a few explicit
computations.

Christoph Hauert
Wes Maciejewski 
Title: Evolution in Heterogeneous Environments
Description: Typical evolutionary models assume
that the environment is constant/homogeneous over
space and time. In reality, an environment may be
heterogeneous in a number of ways: sites may be
resourcerich or poor, moreconnected or less, and
the size of the population may fluctuate.
Understanding the effects of heterogeneity is a
step towards further understanding the natural
world. The USRA student will investigate how
various aspects of heterogeneity affect the
outcome of an evolutionary process. We will
consider two setups: the invasion of an
advantageous mutant with constant fecundity and
evolutionary game theory. The student will perform
literature reviews, write computer simulations,
and assist in communicating the results of the
research.

Ed Perkins 
Title: Stochastic epidemic models, bond
percolation and critical values.
In one class of stochastic models for the spread
of an infectious disease, carriers of the disease
infect nearby susceptible individuals with
probability p. Infected individuals then recover
and are immune to further infection. There
is a critical value of p, depending on the range R
of infection and underlying lattice, above which
the disease can spread without bound and below
which the disease will die out with probability
one. Bond percolation is a model in
statistical physics in which each bond of range R
is open or closed independently with probability
p. Again there is a critical value of p above
which there may be an infinite connected set of
open bonds (water can flow to infinity), and below
which there can be no such collection. The
two models are closely related. First, a
prospective student has to learn enough
probability to understand these models. In
work with Steve Lalley and Xinghua Zheng, we have
some conjectures on the behaviour of the critical
p's as the range becomes large. Then the
student will run some simulations to test these
conjectures. The student will also write up
a short report on the results. Familiarity with
computer simulations is required as is exposure to
some probability at an undergraduate level.

Lior Silberman 

Michael Ward 
Title: Stability of Localized Patterns for a
ReactionDiffusion System in Biology with
Time Delay
There are numerous examples of morphogen
gradients controlling long range signalling
in developmental and cellular systems. We propose
to investigate one such twocomponent
reactiondiffusion model that includes the
effect of a time delay due to the time needed for
gene expression. I am motivated by a recent
full numerical study of such a system with
GiererMeinhardt kinetics that was undertaken in
S. Lee et al. (Bulletin of Mathematical
Biology, 72(8), (2010), pp. 21392160). I
have recently realized, that by making specific
convenient choices in the powers of the
nonlinearities in the GiererMeinhardt
kinetics, it is possible to undertake a complete
stability theory for a localized state in
this model. This is done by first deriving a
nonlocal eigenvalue problem that represents a
nonself adjoint rank one perturbation of a
Sturm Liouvilletype operator. Secondly, one
can derive a transcendental equation for any
discrete eigenvalues of this problem, which
includes the effect of the delay time for
gene expression. Complex variable methods on this
transcendental equation can then be used to
predict Hopf bifurcations in the model.
Techniques from Math 345, 300, 301, and 316
will be used in the analysis.

Top

Summer 2012

Omer Angel
Lior Silberman 
Title: Value distribution of eigenfunctions on
random graphs
Description: We wish to explore the value
distributions of eigenfunction of random matrices,
especially those associated to random regular
graphs. We hope to obtain both experimental and
analytical results.
The experimental part will involve numerical
linear algebra; experience with Octave, Matlab or
the like would be helpful.

Richard Anstee 
The problem area is Extremal Hypergraph Theory
although I prefer to use the language of matrix
theory. The following is a typical problem. Let F
be a given kxt (0,1)matrix and let A be an mxn
(0,1)matrix with no repeated columns and no
submatrix F. There is a conjecture of Anstee,
Frankl, Furedi and Pach that claims that there is
a constant c(F) so that n < c(F) x m^k. One can
establish that n < m^{2k1e} where e is a
function of k,t and is small. The goal would be to
improve this result. The work will focus on
specific F in order to gain insight. A related
problem is where we forbid any row and column
permutation of F (so called problem of Forbidden
Configurations).
Interested students could contact my previous USRA
students: Connor Meehan and Ronnie Chen.

Bud
Homsy 
Title: Studies of the attraction of floating
particles
Background:
Particles at an interface get
attracted to each other and form aggregates and
clusters. This is commonly referred to as the
“Cheerios effect”, owing to the tendency of pieces
of cereal floating on milk to stick together. It is known that the
attractive force is due to the combination of
surface tension and the curvature of the meniscus
separating the particles. In
spite of this basic fact, prevailing theories of
particle attraction can handle only a few
particles and often make restrictive assumptions.
Our interest in this subject
derives from an interest in understanding how the
“Cheerios effect” influences coating flows. In coating applications,
(such as coating of memory devices and other
recording media, producing photographic film,
etc.), a thin layer of fluid is deposited by a
flow onto a solid substrate.
It has been found recently that particles
lying at the interface of the liquid being coated
can modify the coating properties to a significant
degree. This project will aim to understanding the
forces present at an interface due to the presence
of particles in simple flow topologies.
Project Description:
This project has both
experimental and computational components. The overall objective is
to establish the quantitative relationship between
the particle configuration, the interparticle
spacing, and the force of attraction. On the experimental
side, simple tabletop experiments with well
characterized particles and fluids in controlled
conditions will be carried out.
Fundamental data on the position, velocity
and acceleration of particles as they come
together will provide important tests of theories. On the computational
side, the partial differential equations governing
interface dynamics will be solved numerically
using available Open Source software like Surface
Evolver for groups of multiple particles.

Kalle Karu 
There are several problems that can be chosen
according to student's background knowledge and
interests. The general area is the geometry of
polytopes and fans. Topics include formulas
for lattice point counting, face enumerations,
subdivisions of fans, tropical geometry.
The necessary background is linear algebra,
elementary combinatorics, and preferably some
algebra.

Leah Keshet 
Math Biology

FokShuen Leung 
Title: Mathematics modules in Science One
Description: Science One is a highly selective
firstyear program taught by eight faculty members
from the Mathematics, Biology, Chemistry, and
Physics & Astronomy departments. These
subjects are taught in parallel with occasional
crossover topics. The goal of this project is to
rewrite the mathematics curriculum to be
completely integrated with other subjects. The
student would help design modules with broad
themes  for example, "Decay", "The Normal
Distribution", "Evolutionary Processes"  but
specific mathematical content. A background in
Science One is preferable but not necessary.

Akos Magyar
M. Pramanik 
Title: Problems in density Ramsey theory.
Description: Ramsey theory is a beautiful area at
the interface of analysis, combinatorics and
number theory whose aim is to show that large but
otherwise arbitrary sets of integers or integer
points necessarily contain highly regular
structures. Its basic results are both natural and
easy to formulate, however their proofs can be
surprisingly difficult and varied. A famous
example is Roth's theorem that states that if a
set contains a "positive proportion" of the
integers then it must contain three equally spaced
points. Another is van der Courput's theorem which
establishes the same for the set of primes.
There has been remarkable progress over the past
decade in understanding the underlying principles
behind the different approaches in Ramsey theory,
and by now a number of problems can be formulated
whose solution does not require excessive formal
knowledge. The aim of this project is to introduce
the students to the basic analytic techniques and
possibly try them on some open questions in this
area.

Brian Marcus
A. Rechnitzer 
APPROXIMATING ENTROPY OF MARKOV RANDOM FIELDS
The entropy of a stationary random process
quantifies the degree of randomness of the
process. Entropy is important in information
theory, where it governs optimal rates of data
compression and data transmission, in statistical
physics, where it represents disorder in physical
systems, and dynamical systems where it quantifies
the complexity of a system.
In the case of a onedimensional stationary Markov
chain, there is a simple, exact formula for
entropy in terms of the transition probabilities
of the chain. In two dimensions, the Markov
property generalizes to the notion of a stationary
Markov random field (MRF), which can be viewed as
a random process of arrays on the integer
lattice. While MRF`s can be very concrete,
there is no known formula for the entropy.
However, there have been many methods developed
for approximating entropy of MRF`s. These
methods use linear algebra, probability and
combinatorics.
In this project, the student will first learn the
basics of entropy in theory and practice and the
methods for computing approximations. Then
the student will write computer programs to
compare the performance of the methods.
Background in computer programming, preferably
C++, and linear algebra is required. Background in
probability and combinatorics would be helpful.

Anthony Peirce 
1) NUMERICAL SOLUTION OF A SEMIINFINITE HYDRAULIC
FRACTURE PROPAGATING IN AN ELASTIC MEDIUM:
Hydraulic Fractures (HF) are created in the oil
and gas industry to enhance the production of
hydrocarbons. This has been the subject of some
controversy recently due to the potential negative
impacts of this process also known as "Fracking".
Our goal is to better understand the mechanics of
these fractures that are generated in brittle rock
formations by the injection of a high pressure
viscous fluid.
This project aims to investigate various numerical
schemes to solve the problem of a semiinfinite
Hydraulic Fracture propagating in an elastic
medium at a constant velocity V. This problem is
important from a theoretical point of view as it
forms the fundamental solution for many tip
asymptotic solutions for finite fractures, which
can be obtained via an appropriate rescaling of
the variables. The project will involve partial
differential equations, complex analysis, and
numerical analysis using the MATLAB programming
language, which the USRA will learn during the
course of the project. A strong background in PDEs
(Math257/316, Math 300, and Math 405 is a
recommendation).
2) DEVELOPMENT OF AN EFFICIENT CUDA IMPLEMENTATION
OF AN EXTENDED FINITE ELEMENT (XFEM) ALGORITHM FOR
MODELING HYDRAULIC FRACTURES:
This project involves the development of an
objectoriented set of CUDA routines for the
efficient implementation of an existing Extended
Finite Element Algorithm Developed at UBC (in the
MATLAB language) for modeling propagating
Hydraulic Fractures. We have recently acquired a
GPU server comprising 2 GPU boards with 480 GPU
processors and 6 GB of RAM each as well as 12 Xeon
Multicore CPUs with 96 GB of shared RAM. The
project will be ideal for a combined Honors
student in Computer Science and Mathematics. Being
a development of C++, CUDA is a language developed
to exploit the multithreading capabilities of the
CGU architecture processors. The USRA will learn
both the numerics behind the XFEM algorithm as
well as the CUDA programming language. (A strong
background in numerics Math405  or equivalent
courses in CS and C++ programming are
recommended).

Top

Summer 2011

Richard Anstee 
The problem area is Extremal Set
Theory although I prefer to use the
language of matrix theory. Let F be a given kxt
(0,1)matrix and let A
be
an mxn (0,1)matrix with no repeated columns and
no submatrix F. There
is a
conjecture of Anstee, Frankl, Furedi and Pach that
claims that n is
O(m^k).
One can establish that n is O(m^{2k1e}) where e
is a function of k,t
and
is small. The goal would be to improve this
result. The work will focus
on
specific F in order to gain insight.

Neil Balmforth 
1) Modelling Mudflow
Mudflows and debris flows are significant
geological hazards
(USGS, http://vulcan.wr.usgs.gov/Glossary/Lahars/images.html).
Mud
itself
is
a
``viscoplastic
fluid'',
meaning
that
the material
will remain solid unless the forces acting upon it
exceed a threshold,
the yield stress; once this threshold is reached,
the material
flows like a fluid. The goal of this project is to
analyse a
theoretical
model for the sudden release of a pile of mud (the
dambreak problem)
and find its final shape, extracting predictions
for the
``yield surfaces'' that separate the regions that
flowed from those
that remained rigid. The project will build an
experiment
to compare with the theory, using a transparent
analogue laboratory
fluid
(Carbopol). Markers placed in the fluid will be
tracked to
determine the flowing and stagnant regions, and
gain insight into
the flow history.
Refs: Viscoplastic dambreaks and the Bostwick
Consistometer,
(www.math.ubc.ca/~njb/Research/bosto.pdf)
2) Dynamics of granular toys
Piles of granular materials display fluidlike
behaviour when they
avalanche under gravity, yet behave like solids
once they come
to rest. This project will explore one or more of
the following
granular ``toys'', each of which is meant to
illustrate and study
the mechanics of flowing granular media.
a) the granular pencil on an inclined plane:
the threshold for the onset or cessation of
avalanching depends
on how superposed layers of grains flow over one
another.
Idealized to the extreme, one can explore how a
grain rolls
irregularly over a layer of static grains, or
equivalently a polygon
(pencil) rolls down an incline.
b) rocking and rolling granular drums:
cylinders either empty or completely full of
grains rolls
like solid objects down an incline. If the cyliner
is partially full,
on the other hand, the dynamics is far more
complicated, becoming
unsteady
and erratic as grain avalanche within the rolling
drum.
c) laboratory bulldozers:
granular currents can be established by placing a
stationary
wall above a moving conveyor belt; the bulldozing
action of the wall
can be steady or unsteady, depending on the belt
speed and whether the
granular medium avalanches or flows as a whole.
The goal of the project is to build simple
laboratory experiments
and theoretical models for each of these toys.
Refs: Granular dambreaks
(www.math.ubc.ca/~njb/Research/gran.pdf)
3) Shear flow instability and vortex pairing
As illustrated by the meandering of the Gulf
Stream, sheared fluid flow
is often unstable, with jets and shear layers
rolling up into arrays
of vortices.
(http://earthobservatory.nasa.gov/IOTD/view.php?id=5432
http://weathervortex.com/wakes.htm)
It is commonly believed that these vortices suffer
a secondary instability once they are formed
wherein
vortices pair up, interact and merge together. The
purpose of
this project is to demonstrate that this pairing
does NOT always occur,
and periodic arrays of vortices can sometimes be
stable.
The project will derive a reduced model for shear
instability
and then exploit it to build vortex equilibria and
test their stability
towards perturbations that seek to pair and merge
the vortices.
The problem has a counterpart in plasma dynamics,
and will involve
some numerical work. However, the numerical scheme
that will be used
is straightforward.
Refs: Dynamics of Vorticity Defects in Shear.
(www.math.ubc.ca/~njb/Research/defect.pdf)
Onset of meandering in a barotropic jet
(www.math.ubc.ca/~njb/Research/jet1.pdf)

Gordon Slade
Martin Barlow 
Title: Branching random walk.
The project will study random networks arising
from branching random
walks. We are interested in the electrical
resistance of the network,
and time it takes a random walk in the network to
move a distance R
from the origin.

Michael Bennett
Greg Martin 
Title: Computational problems
related to Egyptian fractions
Background: Egyptian fractions, that is, sums of
reciprocals of
positive integers, arise in a variety of contexts,
ranging from
recreational number theory to the socalled optic
equation. Recent work
in this area has typically used sophisticated
sieve methods; these
results have usually described "average behaviour"
of egyptianfraction
representations of rational numbers, without
necessarily explicitly
determining extremal cases.
Project Description: The project is primarily
computational in nature,
based upon understanding and implementing an
algorithm designed to find
egyptianfraction representations of a given
positive rational number,
with specified bounds upon the number and size of
terms. From this
algorithm, one should hope to explicitly quantify
results of Martin,
related to old questions of Pal Erdos.

George Bluman 
Scattering
of
Waves.

Jim Bryan 
Title: Quivers: geometry,
algebra, combinatorics, and physics.

James J. Feng 
Title: Particlebased computer
simulation of flow
We wish to explore how blobs of soft material
interact with each other
when subject to forcing, and how they yield to
flow like a fluid. For
this, the student will integrate and adapt
computer programs to
implement different models of internal forcing.
The computation will be
based on the socalled smoothed particle
hydrodynamics, and will, we
hope, yield an elegant mesoscopic model for the
flow of soft materials.

S. Gustafson
TaiPeng Tsai 
Title: Numerical study of
soliton dynamics
Note: This project has both computational and
theoretical elements.
It would be suitable as either an individual or
"team" project.
Background:
There are many partial differential equations of
physical importance,
such as nonlinear Schroedinger and Kortewegde
Vriestype equations,
whose solutions have a "dispersive wave" nature,
but which possess
solitary wave ("soliton") solutions whose spatial
profiles are constant
in time. There is a large literature on the
analytic and numerical
study of the stability properties of these
solitons. There are
comparatively few results on the interaction of
solitons, except for
those few special equations which are ``completely
integrable", and for
some recent literature on collisions of 2 solitons
whose sizes are
almost the same, or else have very different
magnitudes.
Project:
Set the domain to be the circle, for simplicity of
numerical
computation. We propose to consider dispersive
equations which posses a
branch of solitary waves, one part of which is
stable, the other part
unstable. We plan to first research and develop
numerical methods to
study the dynamics of solutions near this branch,
particularly near the
stabilityinstability transition, where we expect
to observe some rich
phenomena. We will then propose some conjectures,
gather numerical
evidence to support them, and try to prove them.

Bud
Homsy
Harish Dixit 
Title: Studies of the attraction
of floating particles
Background:
Particles at an interface get
attracted to
each other and form aggregates and clusters. This
is commonly referred
to as the “Cheerios effect”, owing to the tendency
of pieces of cereal
floating on milk to stick together.
It is
known that the attractive force is due to the
combination of surface
tension and the curvature of the meniscus
separating the particles. In
spite of this basic fact, prevailing
theories of particle attraction can handle only a
few particles and
often make restrictive assumptions.
Our interest in this subject
derives from an
interest in understanding how the “Cheerios
effect” influences coating
flows. In coating
applications, (such as
coating of memory devices and other recording
media, producing
photographic film, etc.), a thin layer of fluid is
deposited by a flow
onto a solid substrate. It
has been found
recently that particles lying at the interface of
the liquid being
coated can modify the coating properties to a
significant degree. This
project will aim to understanding the forces
present at an interface
due to the presence of particles in simple flow
topologies.
Project Description:
This project has both
experimental and
computational components. The
overall
objective is to establish the quantitative
relationship between the
particle configuration, the interparticle spacing,
and the force of
attraction. On the
experimental side,
simple tabletop experiments with well
characterized particles and
fluids in controlled conditions will be carried
out. Fundamental data
on the position, velocity and
acceleration of particles as they come together
will provide important
tests of theories. On
the computational
side, the partial differential equations governing
interface dynamics
will be solved numerically using available Open
Source software like
Surface Evolver for groups of multiple particles.

Kalle Karu 
Title: Vector bundles on toric
varieties
Toric varieties are algebraic varieties
constructed from combinatorial
data, such as polytopes and polyhedral cones.
Similarly, vector bundles
on a toric variety are described by combinatorial
and linear algebra
data. A main open problem in the field is to
determine if a toric
variety has any nontrivial vector bundles. The
goal of this
project is to study examples of toric varieties
and look for nontrivial
vector bundles on them. The main tools are
combinatorics (of convex
polytopes and cones), linear algebra, and possibly
computer algebra.

Anthony Peirce 
Title:
Novel
Approximation
Schemes
for
Hydraulic
Fractures
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 CO_{2}
sequestration and the enhancement of fracture
networks to capture
geothermal energy.
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 piecewise cubic Hermite
approximation to the
crackopeningdisplacement which is used along
with a collocation
technique on a uniform mesh. We will also consider
various
approximation strategies for a propagating
fracture, which does not
coincide with the prescribed uniform mesh. The
Hermite 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

M. Pramanik 
Title: Patterns in sparse sets
A set is sparse or thin if its "size"
(appropriately construed) is
small in relation to the ambient space that it
belongs to. For example,
the standard Cantor middlethird set is sparse on
the real line (in
what sense?). The primes are sparse on the
integers, but not as sparse
as say the perfect squares. It seems,
superficially at least, that the
sparser a set is, there is less of it to actually
get a handle on,
which is why it is all the more striking when it
turns out that some of
these sets have lots of geometric patterns and
structures in them. A
thriving area of research touching upon many
branches of mathematics
centers on understanding the properties that
ensure such structures.
This project is concerned with identifying
patterns in some specific
examples of sparse sets. The theoretical component
of the project will
involve analyzing some very concrete sets, such as
the Cantor set or
its higher dimensional counterpart, the Sierpinski
gasket. There will
also be a computational aspect of the project,
where we will formulate
numerical experiments to address some
longstanding open questions in
geometric measure theory.
Experience with a mathematical computational
system (Maple, Mathematica
or Matlab) will be required for the experimental
part of the project.

D. Schoetzau 
Title:
Exactly divergencefree discretizations for
buoyancy driven flow
problems
Description: We plan to develop, implement and
numerically test novel
finite element methods for the numerical
simulation of buoyancy
driven flow problems, in which the incompressible
(Navier)Stokes
equations are coupled with a diffusion equation.
This project
is part of a bigger research effort devoted to the
understanding and
advancing of exactly divergencefree finite
element methods in the
context of multiphysics flow problems.
More specifically, starting from an existing
(Navier)Stokes solver
and an existing elliptic solver (with codes
available in MATLAB), we
plan to develop suitable discretizations of the
coupling terms, and
to devise a solver for the fully coupled problem
by employing
suitable linearization and iteration techniques.
We wish to perform
numerical benchmark and accuracy tests, and to
simulate stationary
nonisothermal flows.
This project has a strong computational component
and involves computer
programming in MATLAB. Some background in
computational PDEs would be a
helpful, but not absolutely necessary.

Zinovy Reichstein 
Cubic
hypersurfaces and a conjecture of Cassels and
SwinnertonDyer

