Proposed research projects for Summer 2011

Supervisor Name
Research Project
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^{2k-1-e}) 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, 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,

2) Dynamics of granular toys

Piles of granular materials display fluid-like 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

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.
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.
Onset of meandering in a barotropic jet
Martin Barlow
Gordon Slade
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 so-called optic equation. Recent work in this area has typically used sophisticated sieve methods; these results have usually described "average behaviour" of egyptian-fraction 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 egyptian-fraction 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 Feng
Title: Particle-based 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 so-called smoothed particle hydrodynamics, and will, we
hope, yield an elegant mesoscopic model for the flow of soft materials.
Stephen Gustafson
Tai-Peng 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.


There are many partial differential equations of physical importance, such as nonlinear Schroedinger and Korteweg-de Vries-type 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.


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 stability-instability 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 and
Harish Dixit
Title: Studies of the attraction of floating particles


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 table-top 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 CO2 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 crack-opening-displacement 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:
Malabika 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 middle-third 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 long-standing 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.
Dominik Schoetzau
Title: Exactly divergence-free 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 divergence-free finite element methods in the context of multi-physics 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
non-isothermal 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 Swinnerton-Dyer