SUMMER 2012 NSERC USRA PROJECT DESCRIPTIONS

Joint Supervisors: Adam Oberman and Paul Tupper

Project: nonlocal diffusions and mathematical finance (2 students)

A classic Nobel winning problem by Merton is the design of optimal portfolios.

The solution of this problem is based on the concept of stochastic control: optimizing your expected utility in a random environment.

Newer models take into account nonlocal effects, which model the fact that stock prices sometimes make unexpected large jumps.

This project will involve learning some backgrounds and performing simulations.

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Project: Flocking behaviour and nonlinear equations (2 students)

The behaviour of flocking birds and insects has been used as a model for a variety of topics, such as reputation management on the internet, political opinion forming, and controlling drone planes.   This project will involve learning background material on ODE and PDEs for modeling flocking, and performing simulations.

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Surpervisor: Jason Bell

Project: Power series satisfying a homogeneous linear differential equation 

This project looks at the following question: If a power series with integer coefficients and finite radius of convergence is the solution to a non-trivial homogeneous linear differential equation, does it follow that it is the diagonal of a multivariate rational function?  At the moment this is known only in the case that the power series is algebraic and for some other special classes.  The background needed is some number theory and algebra.  Applicants should be able to do some programming and should also be familiar with LaTeX.

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Supervisor: Jonathan Jedwab

Project: Combinatorial Problems Arising in Digital Communications

The project will involve the study of combinatorial problems arising in digital communications. See here for background on the general area of research, and here for examples of previous USRA projects. 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.

Supervisor: Paul Tupper

Project: What is the diversity analogue of uniform spaces?

Topologies and metric spaces are two fundamental notions of space in
mathematics. Every metric space has a standard topology associated
with it. However, it turns out that intermediate between topologies
and metric spaces there are uniform spaces (also known as
uniformities) which do not have a metric but still have the concept of
uniform convergence.

On the other hand, my colleague David Bryant and I have developed a
generalization of metric spaces called diversities. In a diversity,
any finite set of points has associated with it a measure of how
"diverse" it is. For pairs of points, this reduces to a metric.

We want to know what is the analogue to uniform spaces for
diversities. The student will work to answer this question by
familiarizing themselves with the concepts of topologies, metric
spaces, uniform spaces, and diversities, and the relations between
them. Category theory may play an important role in discovering the
correct analogue, but no prior knowledge of this is required.

The only prerequisite for this project is good performance in Math 320
or equivalent.

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Supervisor: Manfred Trummer

Project: High-order methods for differential equations

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

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SUpervisors: Peter Borwein and Vahid Dabbaghian

Project: Application of Fuzzy Logic in Complex Social Systems

Human decision-making behavior is a fuzzy system. Therefore, fuzzy models, and in particular Fuzzy Cellular Automata (FCA), can be applied for better understanding the dynamic of urban transformations, such as infectious diseases, crimes, and other social disorders, among individuals in a high risk society. The models will be developed in the software package Matlab, so we expect that students have some computer programming skills.

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Supervisor: Karen Yeats

Project: c2 invariants of Feynman graphs.

The c2 invariant of a scalar Feynman graph is an arithmetic invariant which gives information about the Feynman integral but is much easier to compute.  There are many basic things which we do not know about this invariant.  This project is to compute some examples and so develop some reasonable conjectures.

The student should have a taste for discrete problems and some background either in discrete math or in number theory.  No background in quantum field theory is necessary.  The project will entail some programming.

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Professor Michael Monagan 

Project: Computational Algebra Projects 

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 MITACS project, a joint research project with the 
Maple company where we are developing algorithms and software for 
doing mathematics. For example, the Maple company asked us in 2004 
to develop a package for graph theory and now they would like us to 
develop one for computational group theory. 

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

Project 1. Developing a Maple package for computations in finite groups. 

Students need to have taken a first course in group theory, 
e.g. MATH 341 Groups at SFU. Students need basic programming 
skills and an interest in finite groups. 

Project 2. Extending Maple's graph theory package. 

 Students need to have taken a first course in graph theory,
 e.g. MATH 345 Graph Theory at SFU, or a computing course on
 graph theory algorithms, e.g. CMPT 405 Design and Analysis of
 Computing Algorithms at SFU.  Students need basic programming
 skills and an interest in graph theory algorithms.

Both projects will be done in Maple. Familiarity with Maple would 
be helpful but it is not essential. However, both projects will 
involve a significant amount of programming and experimentation. 
To see what summer NSERC students have done in previous years, 
please take a look at the posters at 

http://www.cecm.sfu.ca/research/posters.shtml 

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Project Title: Computing Flow Polynomials

Supervisors: Bojan Mohar and Jessica McDonald

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Supervisor: Mary-Catherine Kropinski

Title: Integral Equation Methods and Fluid Dynamics (2 students or 1 student who chooses 1 project) 

Integral equation methods have become increasingly popular in computational simulations involving the solution to partial differential equations (PDEs). One important advantage over finite element or finite difference methods for solving PDEs is dimension reduction: unknowns are distributed about the boundary instead of the entire domain. There are two projects involving integral equation methods and fluid dynamics: 

Project 1: Vortex Motion on the Sphere 
This project involves computing the motion of vortices on the earth's surface in the presence of multiple islands. This requires coupling the solution to an integral equation with a set of ordinary differential equations for the motion of the vortices. The work will focus on accurate time integration methods for this problem. 

Project 2: Integral Equation Formulations in Fluid Dynamics 
This project involves reformulating the Navier Stokes equations as integral equations. In addition to computing, this work will involve analysis. 

Both of these projects will require basic familiarity with matlab. 

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Joint: Adam Oberman and Steve Ruuth