Summer 2013 NSERC USRA (UNDERGRADUATE STUDENT RESEARCH AWARDS) 

DEPARTMENT OF MATHEMATICS - SIMON FRASER UNIVERSITY

These NSERC awards are valued at $4,500 for a minimum of sixteen weeks. The University will supplement this amount from other sources making salaries of at least $5,740 total plus benefits. These awards are provided to encourage undergraduate student interest in research and in graduate studies.

• The SFU NSERC quota is 73
  

**Project Descriptions: Below are brief descriptions of research projects in mathematics from faculty members who wish to supervise qualified undergraduate students. Unless otherwise specified, each project is available to one student only.**

WHO SHOULD APPLY:

Undergraduate students with strong records (B- minimum) in mathematics courses who are Canadian citizens or permanent residents are eligible. Students must have obtained a cumulative average of at least B- over the entire period of undergraduate studies. Please note that this average is a minimum requirement and the cut-off point is often higher.

To hold an award, you must have completed all course requirements of at least the first year of your bachelor's degree program, and must not have started a program of graduate studies. Also, you should be a full-time student at the time of application. Female students or students of aboriginal descent are particularly encouraged to apply.

NB: If you are not found eligible for a USRA award, your application will automatically be submitted for a VPR award

HOW TO APPLY:

Please be sure to review full eligibility criteria, guidelines, and application forms are on the DGC NSERC website.

 To do prior to 01 February, 2013***

• Interested eligible students should contact the supervisor with regards to the project they would like to work on.
• Students must create an NSERC Online Account and complete the application for an Undergraduate Student Research Award Form 202 (Part I) & upload scanned copies of their official transcripts.
• Students must submit to the department, the following:  printed Form 202 (Part I), photocopies of transcripts, their NSERC Online Reference # and 2 reference letters.***

***If you are chosen, you will be notified within 3 business days.***

 To do prior to 07 February, 2013

• Once chosen, the  student must email the selected supervisor and have them complete Form 202 - Part II; the supervisor will need to know the applicant’s reference number to do so.
• The proposed supervisor must complete Form 202 Part II online and submit it online only. The supervisor must submit the form online before the applicant will be able to submit their form.
• The applicant verifies their application and submits online prior to 07 February , 2013
• Reference letters must be requested by 18 January, 2013 and must arrive at the Math Department prior to 01 February, 2013.  Please have reference letters addressed to Dr. Manfred Trummer.

Dr. Manfred Trummer, Chair 
Department of Mathematics 
8888 University Drive 
Simon Fraser University 
Burnaby, B.C. V5A 1S6

**Note: while we do our best to match students up with their preferred projects / supervisors, we cannot guarantee a preferred match due to the competitive nature of the awards.    

If you have questions, please contact

JoAnne Hennessey
Secretary to the Chair
Department of Mathematics
SCK 10503 
Tel: 778.782-4238

Email: mcs@sfu.ca

 PROJECT TOPICS 

Supervisor: Jonathan Jedwab-     jed@sfu.ca

Project: Combinatorial Problems Arising in Digital Communications

The project will involve the study of combinatorial problems arising in digital communications. Students should have completed MACM 201, and preferably have some programming experience. However, the most important attributes are enthusiasm, persistence, and a willingness to learn new skills.

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 Supervisor: Ladislav Stacho -             lstacho@sfu.ca

Project: Optimal Movement of Sensors for the Fault-tolerant Coverage of Line Segment

Consider n points in a line segment representing sensors of an ad hoc network. The sensors are assumed to reallocate themselves as to attain coverage of the line segment (build an impenetrable barrier). In the literature [1,2,3] this problem was studied for various assumptions (sensors have equal or unequal sensing ranges, ...) on sensors in the case when a simple coverage is required (every point of the line segment is covered by at least one sensor). However sensors may become faulty and thus it is important to consider coverage in which every point of the line segment is covered by several sensors, i. e. fault-tolerant coverage.

The aim of this project is to initiate such a study and hopefully to obtain results that would be extensions of corresponding results for simple coverage. The bounds we are interested in are theoretical so students with strong theoretical background should apply.

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Supervisor: Michael Monagan-             mmonagan@sfu.ca

Project: Computational Algebra and Graph Theory Projects X 2

Students: 1 or 2

The Computer Algebra Group at SFU is a group of faculty and students
interested in computer mathematics and systems like Maple and Magma.
We presently have a joint research project with the Maple company
where we are developing algorithms and software for doing mathematics.

We are looking for a mathematics student or a computing student
to work on one or both of the following projects.

Project 1. Computational Algebra
  We are developing a C library for fast computations with
  polynomials over finite fields.  This involves application
  of the Fast Fourier Transform (FFT), coding in C, and a
  basic knowledge of finite fields.
  Familiarity with finite fields and the FFT is helpful but not necessary.
  Students need some programming experience in either C or C++ or Java
  and an interest in mathematical algorithms.

Project 2. Computational Graph Theory
  We will be developing different algorithms and heuristics for
  computing reliability polynomials an Tutte polynomials and related
  polynomials of undirected graphs.  This involves finding identities that
  will allow us to split a graph into subgraphs.
  The algorithms and heuristics will be implemented in Maple.
  Familiarity with Maple is helpful but not essential.
  Some prior programming knowledge is necessary.

 _____________________________________________________________

 Supervisor: John Stockie-   stockie@sfu.ca

Project: Mathematical Modeling of Sap Flow in Trees

There are many interesting questions regarding how sap flows in trees.
For example, scientists still don't understand how exactly the tallest
trees are able to draw sap from the roots to the leaves.  Also, sugar
maple trees have an amazing ability to generate pressure during the
syrup harvest season even though the tree is supposedly dormant.  

The aim of this project is to study a number of mathematical models
for sap flow in a tree stem that will help to answer these and other
questions.  In particular, we'll consider a number of possible models
based on electrical circuits, pore networks, and porous media.  

I'm looking for a student that has taken courses in differential
equations (like MATH 310) or basic numerical methods (like MACM 316) or
(even better) both.

 _____________________________________________________________

 Supervisor: John Stockie and JF Williams-     stockie@sfu.ca   jfw@sfu.ca

Project: Why Don’t Honeybees Freeze in Winter?

Project: Limiting Behaviour of the Neutron Transport Equations

This project concerns the time-dependent 1D neutron transport equations
over half space. This equation describes the distribution function of neutrons which
undergo scattering, emission, and absorption when interacting with the background media.
The goal is to study the limiting behavior of the solution to this equation when
both time and space approach infinity. This investigation is essential to understanding the
boundary layer structure of kinetic equations. Students involved in this project will learn
the generalized-eigenvalue method developed by Case and some spectral analysis related to
this equation.

We are looking for students who have background in complex variable and PDEs.
__________________________________________

Supervisor: David Muraki & Weiran Sun-    muraki@sfu.ca   weirans@sfu.ca

Project: Chaos in the Forced Nonlinear Pendulum

An absolutely classic illustration of chaos in a simple ODE system arises in

the sinusoidal forcing (parametric or direct) of the nonlinear pendulum.  The 

usual textbook visualization of this chaos uses a Poincare map where a so-called 

"stochastic layer" appears in the vicinity of the phase-plane separatrix.  An 

iterated map, known as the "whisker map", has been used as a model to explain 

the essential features within this layer of chaos.  A recent series of lectures 

has uncovered a boundary-layer method for deriving the best whisker-map 

approximation to a given forced ODE system.  This project involves the derivation 

of whisker map parameters and geometric transformations from the pendulum ODE,

and evaluation by numerical computation the accuracy of the optimal approximation.

Students taking MATH 467 in Spring 2013 will have the introductory background

for this type of research.  Proficiency in a computational environment such as 

Matlab is desirable.  

_________________________________________________________________

Supervisor: David Muraki -      muraki@sfu.ca

Project: Computation of Fluid Models for Atmospheric Science

 Undergraduates in the third and fourth year of their degree are invited 

to join a research group that investigates fluid mechanical models for 

atmospheric sciences. There are active projects that investigate a variety 

of atmospheric phenomena.  One such project would involve using a simple 

computational model for fluid flows on a sphere, and comparing the results 

with stratospheric weather data.  Some background in a differential equations 

is essential and proficiency in a computational environment such as Matlab 

is preferable.  

Supervisor: Paul Tupper-      pft3@sfu.ca

Project: Classification of Four-Point Diversities

The study of metric spaces with finitely many points is a well-established field. One question of interest is the classification of all metrics on a given (small) number of points. One form of classification, called combinatorial classification, was performed by Dress for all metrics on 5 or fewer points in 1984. Sturmfels and Yu classified all metric spaces of 6 points in 2004. (There are 339.) This goal of this project is to classify all instances of a related class of mathematical objects: diversities. Diversities generalize metrics in that a real value is assigned to every fine subset of points in the space and not just to pairs. Bryant and Tupper classified all three-point diversities in 2012. The student will classify all four-point diversities. Coursework in discrete mathematics such as graph theory or linear programming is desired.

 ___________________________________________________________

Supervisor: Paul Tupper-      pft3@sfu.ca

Project: Models of Gazing Behaviour

A big area of cognitive research is understanding how people make sense of visual scenes. When a person is presented with a scene, their gaze is attracted to particular visual features. Their behaviour can be described as a series of fixations on features interspersed with `saccades' (jumps). We currently have mathematical models of the timing and location of these fixations and saccades. But we need a mathematical analysis of the models so that we can do a better job of fitting the model parameters to the data. The student will perform such an analysis of the model, which consists of a stochastic partial differential equation. The student will need to have some knowledge of either stochastic processes or partial differential equations. There is opportunity to work in collaboration with scientists at SFU's Cognitive Science Lab.

___________________________________________________________________

Supervisor:  Manfred Trummer  -           trummer@sfu.ca

Project: High-order Method for Differential Equations

This project investigates spectral methods and radial basis function methods for solving differential equations with a focus on introducing adaptivity.  Includes theory, numerical experiments and applications. Requires basic familiarity with Matlab.

__________________________________________________________________

Supervisor:  Karen Yeats-           karen_yeats@sfu.ca

Project: Extra Denominator Reductions

Begin with a graph.  Associate a variable to each edge of the graph, and to each spanning tree associate the monomial of edges not in the tree.  The Kirchhoff polynomial of the graph is the sum over all such monomials.  Beginning with this polynomial we can define a sequence of polynomials by contracting, deleting, and recombining in particular ways.  These sequences terminate, sometimes after 5 steps, sometimes later.  Features of the graph such as triangles give extra steps in the sequence.

Viewing these graphs now as Feynman graphs in a scalar quantum field theory, these polynomial sequences give us denominators after successive integrations and hence give us information about the Feynman integrals.

The project is to study and try to develop a systematic theory of when we get extra steps in the denominator sequences.  A student should have at least one, preferably more, of linear algebra, MACM 201, and programming experience.

________________________________________________________________

Supervisor:  Veselin Jungic & Tom Brown    - vjungic@sfu.ca      tbrown@sfu.ca

Project: Additive Complexity in Infinite Words

Abstract: This project is built around the following open problem: Is there an infinite sequence over four integers that has no two consecutive blocks of the same length and the same sum? It has been shown that infinite words over a finite alphabet with so-called bounded additive complexity must contain long (but finite) sequences of consecutive blocks with the same length and the same sum. This discovery led to the study of the notion of the bounded additive complexity itself which resulted in some surprising connections with other combinatorial properties of infinite words. These discoveries included work of two students in the USRA Program in 2011 and 2012. The student will get familiar with the existing results related to the additive complexity of infinite words and techniques used to obtain those results. The student will also get familiar with van der Waerden’s Theorem and its usage in combinatorics of words.

________________________________________________________________

Supervisor:  Luis Goddyn -         goddyn@sfu.ca

Project: Computer Examination of Matroids whose Circuits form a Hilbert Base

This project would be ideal for a student with both mathematical and  programming skills,
and who is interested in learning a little about graph-like  combinatorial objects called matroids.
This project is concerned with sets of 0,1-vectors that come from matroids. A matroid is either 'good' or 'bad", depending on whether its vector set has a property called `Hilbert base'.

The first goal is to explore which matroids are good. We seek identify which small matroids lie on the 'boundary' between the good and the bad ones. There there is an electronic catalog of several thousand small  matroids, and there is software that can be used to test their goodness.

The second goal is to use the catalog test a couple of hypotheses that  
have been asked, but apparently have never been tested.
  - Is every "minor" of a good matroid is also good?
  - Is it true that "gluing" two matroids together (with k-sums)  
results in another good matroid?

Some programming effort is required to extract the 0,1-vectors from each matroid, and then deploying the Hilbert test in an efficient manner. A modest amount of theory regarding matroids and vectors will be learned by the research assistant. There is significant potential for creative discovery and the  
subsequent preparation of a paper of technical report.

The successful applicant should have some experience with a language like C, C++, Java or Python. Ability to write a scripting language such as Perl, Applescript, or a  shell script etc may be an asset.