Summer 2017 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.
SUPERVISOR: Dr Nilima Nigam
PROJECT: Optimizing the lowest 'sloshing' eigenvalues
Consider a simply connected planar domain D, with boundary B. The boundary length is fixed. We are interested in real constants p and functions u which are harmonic in D, have zero Neumann data on a subset B_N of B, and satisfy the Steklov condition
du/dn = p u
on B\B_N. If B is a Lipschitz curve, we know that the 'sloshing' eigenvalues p form a countable set of non-negative numbers whose only accumulation point is at infinity.
In this project, we would like to explore the dependence of the first non-zero sloshing eigenvalue on the length of B_N as well as the number of connected components of B_N. The work will use a combination of techniques from PDEs and spectral methods, and computation (in Matlab) will be involved.
SUPERVISOR: Dr Cedric Chauve
PROJECT: Graph-theoretic methods for improving clustering genes in families
In this project, we want to develop a combinatorial method to improve the clustering of genes into families of genes that are evolutionary related. This method will be applied to a data set of 18 mosquito genomes I have been working on for a few years now (http://science.sciencemag.org/content/347/6217/1258522.long).
A gene family is a set of genes that are assumed to have evolved from a single ancestral gene. We have, from the large-scale sequencing that was done in the initial work on these mosquito genomes, the sequence of most of the genes in these genomes. Moreover we already have an initial clustering in gene families and evolutionary trees for all these gene families.
However, there is clear evidence that this clustering is sometimes wrong, with some families that are split into several subfamilies (often two).
The goal is to develop a (hopefully simple) combinatorial optimization method that will analyze the evolutionary trees and location of genes along the genomes to modify the initial clustering and correct likely errors.
This project is well suited for a student with the following skills and interests:
- interest for applying mathematics, especially combinatorial algorithms, to big data genomics
- good skills in discrete mathematics and discrete algorithms
- good skills in programming in at least one of python, C/C++ or Java
This project will be well balanced between theoretical work, programming, actual analysis of real data with existing programs and the programs we will develop.
It will provide to the student an opportunity to develop valuable skills in the field of genomics and big data.
This project will be done in collaboration with a graduate student who is working on these mosquito genomes.
PROJECT: Fast simulation of anomalous diffusion processes
The goal of this project is to implement a fast Matlab solver for integro-differential equations involving the Fractional Laplace operator. These equations are able to model anomalous diffusion processes and have been recently used in many applications such as material science, finance, bioengineering, continuum mechanics, graph theory, and machine learning.
In this case, classical numerical schemes such as the finite element method may be very inefficient. Indeed, they require the storage of large-sized densely populated matrices that waste memory and computational resources. In order to speed up the numerical computations, the idea is to employ some recent advanced techniques based on computational harmonic analysis and compressed sensing. These methods are able to exploit the sparsity of the solution with respect to a suitable set of basis functions and to compress the discretization using “short and fat” matrices, which are less demanding in terms of memory and more computationally tractable.
Required: Proficiency with Matlab and familiarity with basic concepts of functional and numerical analysis. Previous knowledge of the finite element method is useful, but not mandatory.
PROJECT: Optimal sampling strategies for compressive imaging
Compressive imaging deals with the reconstruction of images from small numbers of measurements. Its applications include medical imaging (MRI, CT), lensless imaging, electron and fluorescence microscopy, and infrared imaging.
A fundamental question in compressive imaging is the design of good (or optimal) sampling strategies. For example, in Fourier-based imaging, this corresponds to the selection of frequencies in k-space. This project will explore the design and analysis of optimal sampling strategies based on the mathematics of compressed sensing and nonlinear approximation. In particular, the aim is to understand the conditions which ensure optimal nonlinear approximation of images via compressive sensing for popular sparse regularization techniques such as wavelets and their various generalizations.
This project will combine techniques from approximation theory, wavelets and compressed sensing. Experience with Matlab is useful.
SUPERVISOR: Dr Ben Adcock
PROJECT: Numerical methods for parameter assessment from time-dependent Magnetic Resonance signals
Description: Data from MRI scans is typically used to produce images that are interpreted for diagnosis. However, there is substantial amounts of information in the data that is not extracted during the image formation process. Discarded information such as temperature change, blood flow velocity, perfusion, diffusion, structure and physiology can provide practitioners with much broader clinical insight into the patient.
The purpose of this project is to develop numerical methods for extracting this information. A major difficulty in extracting this information is that it requires the solution of a nonlinear and ill-posed inverse problem, which means that standard numerical methods typically produce poor reconstructions. Nevertheless, progress has recently been made by exploiting the sparsity of the underlying quantities so as to regularize the underlying problems. Aiming to enhance reconstruction fidelity, this project will explore the further development of these techniques. Three particular areas of focus will be (i) the extension from single-coil to multiple-coil (i.e. parallel) MRI, (ii) the development of new gridding strategies for dealing with nonuniformly-acquired data, and (iii) the implementation of novel joint sparsity-promoting regularizers.
Linear algebra, numerical analysis and Matlab are essential. Basic optimization and Fourier transforms are useful.
PROJECT: Tight Spans for Metric Structures
Given a metric space, its tight span is the set of all minimal one-point extensions of the space. The tight span can be equipped with its own metric, and viewed as its own metric space, so that the original metric space is naturally embedded in its tight span. The theory of tight spans arose in phylogenetics but has become of interest in its own right. In this project the student will work to see how much of the theory of tight spans can be generalized to metric structures, which are basically metric spaces with extra structure (one example being normed spaces). We see already how this generalization works in some cases, but we'd like to push it further.
PROJECT: LONG TIME BEHAVIOUR OF AGGREGATION EQUATIONS
Aggregation equations are widely used in math biology to model the collective behaviour of a group of agents/particles (such as birds flocking or fish swarming). Various aggregation models have been designed to take into account of many types of interactions such as repulsion, attraction, and alignment. One particularly interesting question is the long time behaviour of the system. In this project we will investigate the confinement of the system given an initial configuration and study whether or how flocking forms when the system evolves for a long enough time.
SUPERVISOR: Dr Mrni Mishna
PROJECT: Random generation of maps
Maps are planar graphs embedded in the plane. This project will develop methods and algorithms for the random generation of maps. The goal is to generate very large maps.
The algorithms will be implemented, and the second phase will be to study properties of these graph classes by analyzing randomly generated maps.
SUPERVISOR: Dr Michael Monagan
PROJECT: Sparse Polynomial Interpolation
We are looking for one student, either a mathematics major, or a computing major, or a joint mathematics and computing major, who likes mathematics and programming to work on the following project.
We are developing a library of algorithms for interpolating sparse polynomials in many variables. A sparse polynomial is a polynomial with few non-zero terms. The algorithms involve elementary number theory, linear algebra, polynomial algebra, and using the Fast Fourier Transform (FFT).
Programming is in Maple and C and the the parallel extension of C known as Cilk which is easy to use. The project also involves the analysis of failure probability of the GCD algorithm. There are several sources of failure each of which needs analysis.
The student should have taken a first course in either algebra or number theory (MATH 340 or MATH 342 at SFU) and have programming experience in either C or C++ or Java. Knowledge of Maple, the FFT, and experience with parallel programming is not required.
SUPERVISOR: Dr David Muraki
PROJECT: Computation of Fluid Models for Atmospheric Science
Independently motivated undergraduates in the third or fourth year of their degree are invited to join research efforts that use computational models to understand the fluid mechanics of the weather. There are active projects that investigate a variety of atmospheric phenomena.
One such project investigates the unusual phenomenon of a "holepunch" cloud --- a circular hole in a thin cloud layer initiated by an aircraft fly-through. Some background in a differential equations is essential, as is proficiency in a computational environment such as Matlab.
PROJECT: Gromov-Hausdorff Distances for Diversities
Diversities are a generalization of metric spaces in which, rather than distances being assigned to just pairs of points, any finite set of points is assigned a positive number. An on-going project is to determine which concepts from metric space theory can be fruitfully extended to diversities. One such example is the Gromov-Hausdorff (G-H) distance. The G-H distance is a way of assigning a distance between compact metric spaces themselves. The goal of this project is to find an analogue of the G-H distances that provides distances between compact diversities.
PROJECT: Parallelizing C++ code to run finite element models of muscle contraction
In a collaborative venture between Profs. Nilima Nigam (Math) and James Wakeling (BPK) we are developing and using a finite element model to understand mechanisms within the muscle that define and control muscle contractions. The current model is written in C++ using a series of finite element modeling libraries (Deal.ii). We are looking for a student to parallelize the code to utilize multiple processors simultaneously.
This research opportunity will provide the student with experience working within a multidisciplinary team, and to develop skills in biomechanics and mathematics and computing. The student will take a lead role in the parallelization part of this venture.
You will be responsible for such tasks as
- Interacting with other team members and developers of the Finite Element Model.
- Developing the necessary code modification to parallelize the process.
- Validating the accuracy, efficiency and speed improvements of this code.
- Preparation of project reports
Desired skills and qualifications:
- Minimum of 2 years of student experience
- Demonstrated experience of C++ coding (please provide examples of projects and codes written)
- Motivated and reliable
- Ability to work both independently and as part of a team
Dr Wakeling's website: http://www.sfu.ca/bpk/people/faculty_directory/wakeling.html
Note: This position is eligible to count as a co-op work term and would take place during Fall 2017 or Spring 2018
SUPERVISOR: Dr Ralf Wittenberg
PROJECT: Dynamics and Statistics of Spatiotemporally Chaotic PDEs
The Kuramoto-Sivashinsky equation, a 4th-order scalar, nonlinear partial differential equation (PDE), is a prototypical example of a class of PDEs whose solutions exhibit complex and chaotic dynamics in space and time, and for which the underlying energy transfer mechanisms across different spatial scales remain incompletely understood.
One goal of this project is to update and extend the available statistics for such PDEs, to provide reliable descriptions of their chaotic attractors at much larger spatial scales than were previously computationally accessible. Depending on progress and the interests of the student, there is also potential for analysis of the dynamics in appropriate function spaces.
A background in differential equations and Fourier series is highly desirable for this project, as is some proficiency with Matlab and an exposure to spectral methods.