Math 577 - Section 02 - Spring 2003
Dr. Joceline Lega
- Office: Mathematics 511
- Phone: 621-4350
- MATH 101
- Tuesdays and Thursdays, 9:30 am - 10:45
- Office Hours:
Tuesdays, 2-3 pm & 4-6 pm.
I will not use a text for this course.
- Course description:
Patterns (sand ripples, stripes or spots on animal coats, ...) are
ubiquitous in nature. A striking feature of pattern-forming systems is
the similarity in the patterns they exhibit in spite of the
differences in the physical or biological nature of these systems. A
century of experimentation and a few decades of modeling/analysis have
brought some answers to this puzzling phenomenon. In particular, the
near-threshold theory of pattern formation, which involves multiple
scales analysis and which has its roots in bifurcation theory for
dynamical systems, has now become a standard tool in the analysis and
modeling of nonlinear phenomena.
This course will discuss near- and far-from-threshold pattern
formation. Focus will be placed on theoretical issues, but both experiments
in and numerical simulations of pattern-forming systems will be discussed.
The main topics are listed below.
- Pattern formation in nature and in laboratory experiments.
- Patterns near threshold: bifurcation theory, multiple scales
analysis, envelope equations.
- Applications to pattern formation in biological systems.
- Patterns far from threshold and the phase-diffusion equation.
- Order-parameter equations as partial differential equations:
analytical solutions, linear stability analysis, complex dynamics.
- (time-permitting) On the validity of order-parameter equations.
Prerequisites for this course include good undergraduate knowledge of
partial and ordinary differential equations, linear algebra, and numerical
analysis. Students will also be expected to have some knowledge of one
of these topics at the graduate level.
Students will work in teams on a long-term project.
Each team will turn in progress reports on its project throughout the
semester, will give an oral presentation at the end of the semester and will
turn in a final written report, in the form of a research paper, which will
be due May 6th 2003 (the last day this course meets). Progress reports
will not be graded (but I will of course give you feedback on your work).
Project topics will preferably come
from mathematical biology. Possible topics are listed below:
Homework problems will be assigned regularly. Students will present homework
solutions in class.
Click here to see homework assignments
- Grading policy:
The project will count for 80% (60% for the written paper and 20% for the oral
presentation) and the homework for 20% of the semester grade. I will ask
each team to rate the participation of its members and this will
also be taken into account in establishing individual final grades.
- Interesting websites:
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