||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
||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 2-rowed 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
Interested students could contact my previous USRA students: Ronnie Chen and Ron Estrin.
||Title: Modeling morphogenesis of the fruit
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 in-depth 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".
||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 multi-layer 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.
|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.|
|Title: Evolution in Heterogeneous
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 resource-rich or poor, more-connected 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.
||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.
||Title: Stability of Localized Patterns for a
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 two-component reaction-diffusion 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 Gierer-Meinhardt kinetics that was undertaken in
S. Lee et al. (Bulletin of Mathematical Biology, 72(8), (2010),
pp. 2139-2160). I have recently realized, that by making specific
convenient choices in the powers of the nonlinearities in the
Gierer-Meinhardt 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
non-self adjoint rank one perturbation of a Sturm Liouville-type
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.