Supervisor Name 
Research Project 

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 and 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 and 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. 