Supervisor 
Research project 
Summer 2019

Leah Keshet 
MATHEMATICAL MODELING AND COMPUTATIONAL BIOLOGY
At least one USRA position is offered for talented
undergraduate(s) in the group of Leah Keshet to
combine mathematical modeling and computational
biology. Interested students will be working with
other group members on several projects related to
cell motility and the collective behaviour of
cells in a tissue. Students will use computational
software (for example: Cellular Potts Model),
contribute to writing code (guided by a
postdoctoral fellow) and to running and testing
codes for a variety of conditions. Students with a
computational background and familiarity with
Python, MatLab, C++, and similar programming
languages, as well as an interest in applied
mathematics will be seriously considered.

Ian Frigaard 
FLUIDS PROJECT
Required skills: Computation using OpenFOAM
By 2013, more than 550,000 oil & gas well had
been drilled in Canada. Before hydrocarbons can be
produced every well undergoes primary cementing.
This operation consists on sealing the annular
section between the steel pipe that stabilizes the
well (named casing), and the rock formation. The
seal should increase production and prevent
subsurface fluids from percolating to surface.
Nevertheless, gas leakage to surface is common. A
public perception is that 1020% of wellbores
leak, which has both health & safety
consequences and environmental/ecological impact.
This project will study this process from a fluid
mechanics perspective.
In the field, after the well is drilled, the
casing is lowered into the open well. In this
point, the space inside and outside the casing is
occupied by the drilling mud that keeps the
hydrostatic balance between the hole and the
formation. Then, cement slurry is pumped downwards
inside the casing, reaches the casing’s bottom,
and flows up into the annular section displacing
the drilling mud upwards. A good seal will not
leave residual mud anywhere.
In the lab, we use two flow loops to simulate the
field process. We have carefully designed and
built the loops to achieve dynamic similarity. We
can control the key parameters of the process,
such as flow rate, eccentricity, rheology, and
fluid’s densities. The data acquisition is through
imaging with high sensitivity cameras and
automated instrumentation. The objective is to
capture experimental data relevant to theoretical
predictions of the fluidfluid displacement flows
under a wide variety of scenarios.
We are seeking 2 students, who will perform some
combination of experimental work and associated
computations. Experimentally, the students will
assist in all operations associated with the
experiment: fluid preparation, running experiment,
image processing of the data, rheometry
measurements of the fluids and data analysis. The
students will need to understand the physical
background to the experiments and may need to help
in design of new components, undertake bits of
machining/manufacturing, and implement changes to
the current apparatus. Computationally, the
students will run and analyse selected simulations
using OpenFOAM.
Interested students should contact Alondra
Renteria alondra.renteria.ruiz@gmail.com and Ali
Etrati etrati@alumni.ubc.ca with resume and
transcript. The intention is that the successful
students be funded under the NSERC USRA program,
so grades should be good!

Liam Watson 
CURVES ASSOCIATED WITH TANGLES
Khovanov homology, in its original form devised by
Khovanov about 20 years ago, is a combinatorially
defined homology theory for knots and links.
BarNatan extended this theory to tangles using a
certain cobordism category, more precisely a
category whose objects are crossingless tangles
and whose morphisms are embedded surfaces that
interpolate between such tangles. This made
calculations of Khovanov homology much more
efficient. Very recently,
KotelskiyWatsonZibrowius observed that if one
specialises to tangles with four ends, a certain
version of BarNatan's theory can be interpreted
in terms of immersed curves on the 3punctured
disc. This is particularly interesting in the
light of similar interpretations of Heegaard Floer
and instanton homology.
This project has one main goal: compute the
immersed curves associated with as many 4ended
tangles as possible. A computer program to find
the immersed curves from BarNatan's invariant has
already been written by Zibrowius. What is missing
is a program that computes BarNatan's invariant
in this setting. One option would be to adapt one
of the already existing implementations of
Khovanov homology. Alternatively, it might be
easier to just write an implementation of the
algorithm from scratch.
Prerequisites for this project are an excellent
grade in a first course in algebraic topology
(covering classification of surfaces, simplicial
or singular homology and the Euler characteristic)
as well as some experience in programming, ideally
python and/or C++. Java would be required if we
decide to adapt existing implementations of
BarNatan's theory.

Yaniv Plan 
STABILITY OF MATRIX COMPLETION WITH CONVEX
OPTIMIZATION
The matrix completion problem is as follows: Given
a subset of entries of a lowrank matrix, the goal
is to fill
in the missing entries by leveraging the lowrank
structure. There are competing convex optimization
programs for this, but
while one has the strongest theoretical backing
(nuclearnorm minimization), another seems to work
better in practice (maxnorm
minimization). We study the difference both
theoretically and through numerical simulations. A
key will be to understand the
underlying geometry of the corresponding convex
sets.

Ben Williams 
SPACES OF GENERATORS FOR MATRIX ALGEBRAS WITH
INVOLUTION
The transpose operation makes the ring of n x n
complex matrices into an algebrawithinvolution.
That is, the ring is an algebra over the complex
numbers, and is equipped with an additive self map
A > A* that reverses the order of
multiplication, and so that A**=A. We will say
that an rtuple (A1, A2, ... , Ar) of n x n
matrices _generates_ this algebra if no strict
subalgebrawithinvolution contains every element
of the rtuple. Let X(n,r) denote the space of all
such generating rtuples. This project will study
the topology of the space X(n,r) and of a related
space Y(n,r) where the rtuples are considered
only up to an involutionpreserving change of
basis. It will study both particularly in the case
of small values of n (and r) where it is possible
to understand the spaces completely.
The project will require a strong understanding of
linear algebra, knowledge of homology will be
highly desirable. Some commutative algebra or
algebraic geometry will also be helpful.

Lior Silberman 
PHASE RETRIEVAL THROUGH OPTIMAL TRANSPORT
In some imaging applications (Xray
crystallography, for example) the resulting image
essentially captures the magnitude (amplitude) of
the Fourier Transform of a function of interest.
However, the Fourier transform is complexvalued,
so information (the phase) is lost.
The problem can be rephrased as an optimization
problem  of selecting a candidate function whose
Fourier transform best matches the observed
signal. We will investigate whether techniques of
regularized optimal transport can be brought to
bear on this problem by quantifying distances and
gradients in the space of candidate functions.
Background in real analysis and some programming
experience are required.

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. We will pursue
some 4rowed choices for F that when forbidden in
A should yield a quadratic bound on the number of
columns of A. There are many variations of this
problem including considering specially structured
families of forbidden configurations. We will have
plenty of problems to work on.
Interested students could contact my previous USRA
students Santiago Salazar and Jeffrey Dawson and
Cindy Tan.

Priscilla Greenwood 
The Mathematics of Stochastic Neuron Models
A student might work on a problem from the
following small book which is available on line.
The background needed is some knowledge of simple
systems of ode's and some probability. There are
problems at many levels.
[BOOK] Stochastic neuron models PE Greenwood, LM
Ward  2016  Springer In this book we describe a
large number of open problems in the theory of
stochastic neural systems, with the aim of
enticing probabilists to work on them. These
include problems arising from stochastic models of
individual neurons as well as those arising from
stochastic models of the activities of small and
large networks of interconnected neurons. We
sketch the necessary neuroscience background to
these problems so that probabilists can grasp the
context in which they arise. We focus on the
mathematics of the models and theories.

Andrew Rechnitzer &
Mike Bennett 
Elliptic curves and continued fractions
Elliptic curves are a fundamental object in number
theory. The problem of cataloguing elliptic curves
(up to various symmetries) is a difficult
computational problem and continues to attract
significant attention. Recently Bennett developed
a connection between elliptic curves over Q and
solutions of a family of Diophantine equations
called Thue equations. These equations are very
difficult to solve rigorously, however by
exploiting properties of continued fractions, one
can solve them heuristically. In this project we
will investigate the link between solutions of
Thue equations and continued fractions over other
fields  such as the Gaussian integers.

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Summer 2018

Leah Keshet 
COMPUTATIONAL AND MATHEMATICAL BIOLOGY OF CELLS
The group of Leah Keshet offers placement for USRA
students with computational experience to work on
problems in mathematical biology. Our group is
interested in modeling cell shape, cell motility,
tissue migration and other phenomena associated
with developmental and cell biology. We currently
work on understanding the behavior of normal and
malignant cells, as well as intracellular
chemicals that influence cell shape, adhesion,
contractility, and motility. .
The student will be in charge of refining and
running publicly availble software (Compucell3D,
Morpheus) or helping with development and testing
of new software, to simulate collective cell
behaviour corresponding to mathematical models.
Students with engineeringmath or CSmath double
major are especially encouraged to apply.
Particulaly promising students who succeed at the
USRA research will have an opportunity of being
strongly considered for future MSc graduate
studies with the Keshet group.

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.

Bernie Shizgal 
PSEUDOSPECTRAL METHODS OF SOLUTION OF THE
SCHROEDINGER AND FOKKERPLANCK EQUATIONS
The time dependent solution of a large class of
FokkerPlanck equations for the translational
distribution functions of electrons and/or
reactive species can be obtained numerically with
an efficient pseudospectral method defined with
nonclassical basis polynomials. The solutions are
expressed in terms of the eigenfunctions and
eigenvalues of the linear FokkerPlanck equation.
The FokkerPlanck eigenvalue problem is
isospectral with the Schroedinger equation so that
the pseudospactral methods developed can be
applied to both eigenvalue problems. There are
large number of quantum problems of current
interest that involve the Yukawa, Krazner,
Pseudoharmonic, WoodSaxon and Hulthen
potentials.
There are well defined projects for up to four
undergraduate summer students. A publication with
two undergraduate students was recently published;
Shizgal, Ho and Zang, “The computation of radial
integrals with nonclassical quadratures for
quantum chemistry and other applications”, J.
Math. Chem. 55, 413 (2017). The pseudospectral
methods have been described in “Spectral Methods
in Chemistry and Physics”, Springer 2015 
http://www.springer.com/gp/book/9789401794534

James Feng 
COMPUTER SIMULATION OF COLLECTIVE MIGRATION OF
NEURAL CREST CELLS IN AN OBSTACLE COURSE
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. Recently, our group built a model to
explain NCC collective migration in terms of the
chemical signaling inside the cells and between
the neighbors.
The project for summer 2018 extends the existing
model to explore an intriguing phenomenon: how NCC
cells negotiate obstacle courses, as they are
shown to be able to do in experiments. The student
will generalize our current ODEbased model to
account for chemoattraction and filopodial
sensing. The project will require MATLAB and
Python programming.

James Feng & Joerg Rottler

HOW THE BACULOVIRUS GETS INTO CELL NUCLEUS:
MOLECULAR DYNAMICS SIMULATIONS
We seek a summer student interested in theoretical
molecular biophysics. The goal of the project is
to understand how baculovirus can break the
protein gel filling the pores of the nuclear pore
complex and hence enter the cell nucleus. This
will be done using molecular simulations.
A background in biophysics and experience in
programming/scientific computing (python, C, Linux
OS etc) are required. This is a joint project
between Prof. Joerg Rottler (Physics) and Prof.
James Feng (Mathematics)

Mike Bennett 
ELLIPTIC CURVES OVER NUMBER FIELDS
The proposed research centres on the problem of
tabulating elliptic curves over quadratic fields,
to test various conjectures (Birch,
SwinnertonDyer, Modularity, etc.) and to
complement work done in the LMFDB (the London
Database of Lfunction calculations). The work
would have a definite computational flavour, but
could otherwise be tailored to fit the skillset
of the student, involving classical invariant
theory (as used to great effect recently by
Bhargava), algebraic number theory and complex
analysis. It is likely that code would be
produced, in one or more of the standard
computational algebra packages (Sage, Magma, Pari,
etc).

Ian Frigaard 
CEMENTING OF OIL & GAS WELLS
Oil & gas wells are sealed by a process called
primary cementing, in which sequences of fluids
are pumped and displace one another along an
annular flow path. The fluids are nonNewtonian,
have different densities and other rheologies. At
UBC we are studying these flows by both
experimental means and by modelling/simulation.
For this project we seek a motivated individual to
help in running fluid flow experiments within our
custombuilt flow loops, participating in some
construction, design and calibration tasks,
possibly running some numerical simulations etc..
The applicant needs to have practical skills, a
good physical sense of fluid flows and proficiency
in a lab setting. This could be suitable for an
Engineering or Science student with some
experience and interest in fluid mechanics, plus
physical intuition.

Lior Silberman 
PROPERTY (T) FOR MAPPING CLASS GROUPS
We will investigate numerically the spectral
behaviour of groups defined through topology. In a
recent breakthrough KalubaNowakOzawa recently
showed that computational techniques can show an
eigenvalue gap in the actions of a discrete group,
the outer automorphism group of the free group F_5
(more precisely, the group Out(F_5) has "Kazhdan
Property (T)").
A related family of groups, "mapping class
groups", are connected to the geometry and
topology of surfaces, and it is not known whether
they have Property (T) or not. We will investigate
this question numerically.
Background in group theory and linear algebra (at
the level of Math 223,322) is required.

Lior Silberman 
PHASE RETRIEVAL THROUGH OPTIMAL TRANSPORT
In some imaging applications (Xray
crystallography, for example) the resulting image
essentially captures the magnitude (amplitude) of
the Fourier Transform of a function of interest.
However, the Fourier transform is complexvalued,
so information (the phase) is lost.
The problem can be rephrased as an optimization
problem  of selecting a candidate function whose
Fourier transform best matches the observed
signal. We will investigate whether techniques of
regularized optimal transport can be brought to
bear on this problem by quantifying distances and
gradients in the space of candidate functions.
Background in real analysis and some programming
expeirence are required.

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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 CELLS
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.

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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.

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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

