Syllabus for MA 876 Topics in Partial Differential Equations

Syllabus for MA 876 Topics in Partial Differential Equations, Spring 2021

Course Information

Lecturer: Ryan Goh

Email: rgoh@bu.edu
Office: MCS 243
Office Hours: (on Zoom) W 10-11a, Th 3-5p and by appointment, zoom links for office hours will be posted on the Blackboard course page
Web page: http://math.bu.edu/people/rgoh/

Lectures:

Time: Tuesday, Thursday, 9:30-10:45am;
Location: BRB 121
Textbook:There will be no textbook for this class. References will be provided and updated as needed.

Additional References
Here are some other references which may provide an alternative viewpoint and presentation of the course material and which you might find useful.
  • Cross, M., Greenside, H. Pattern Formation and Dynamics in Nonequilibrium Systems
  • Hoyle, R., Pattern Formation: An introduction to Methods
  • Murray, J., Mathematical Biology Vol. II
  • Haragus, M., Iooss, G. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems
  • Ball, P., Nature's Patterns: A tapestry in three parts
  • Collet, P., Eckmann, J.P., Instabilities and Fronts in Extended Systems

    Course Webpage: http://math.bu.edu/people/rgoh/teaching/ma876-ss21/course-page.html

    Course Description:

    Broadly, this course will focus on the mathematics of pattern formation and coherent structures. It will investigate how tools from dynamical systems and functional analysis can be used to rigorously characterize the formation, stability, and dynamics of such regular spatial structures in prototypical and historically important PDE models. Examples include traveling fronts, localized pulses, periodic lattices, and spiral waves, and arise in diverse areas of application, such as solitons and convective rolls in fluids, stripes and spots on animal coats and vegetation patches, waves in cardiac dynamics, and phase separation in bi-metallic alloys. Theoretically, this course will highlight and discuss the use of the following types of tools and how they are used to study patterns and coherent structures:
  • Spatial Dynamics, center manifolds, normal forms,
  • Heteroclinic and homoclinic bifurcation theory
  • Functional analytic viewpoint
    Depending on time we may also consider the Fourier-Bloch theory which is used to establish spectral stability of patterns, as well as the modulational approach
    A coarse schedule of the course is as follows Introduction and history: A little bit about history and interest of pattern formation
  • Discuss center manifolds, normal forms, and symmetry, with applications to patterns
  • Getting solutions from other solutions:
  • Spatial Dynamics
  • Heteroclinic and homoclinic bifurcations, Lin's Method, Silnikov variables
  • Essential instabilities and bifurcation of fronts and pulses
  • Defect solutions
  • Functional analytic perspective: Fredholm operators, Lyapunov-Schmidt reduction, applications to patterns
  • Stability: localized perturbations of periodic patterns
  • Dynamics: the modulational approach

    Homework

    There will be no homework is this course. Occasionally I will give out suggested or interesting problems for you to think about and attempt. You are more than welcome to submit solutions for comment or discuss with me!

    Class presentation

    Each registered student will be required to give two lectures (i.e. a week's worth of lectures) on a course related topic. You could present on a paper, a chapter in a related reference, work out some interesting exercises which add up to an interesting result, or compose an interesting numerical project which sheds light on some of the phenomena we'll discuss. Some potential topics you could consider are listed below. We will determine the lecture schedule after the course begins. You must meet with me at least once by February 16 to discuss your presentation and its suitability (though you are more than welcome to meet with me more!) and must let me know by February 18 what you plan to speak about.
    Some potential topics (in no particular order), feel free to suggest another! :
    Localized patterns and homoclinic snaking
  • Modulations and patterns, Rigorous validation of approximation equations
  • Numerical continuation of patterns
  • Patterns in discrete systems or lattice systems
  • Patterns and noise
  • Non-local PDE and Patterns
  • Stability of a specific type of pattern we discuss in class
  • Stability of traveling waves
  • Approaches to nonlinear stability: semi-groups, pointwise methods, diffusive stability, etc...
  • Stability of Multi-pulse solutions
  • Hexagon forming Fronts
  • Target patterns and spirals
  • Grain Boundaries in patterns
  • Classification of defects in patterns
  • Existence of attactors in pattern forming equations
  • Geometric singular perturbation and coherent structures
  • Patterns in a specific application you're interested in: Math. Bio, water waves, the Marangoni problem, phylotaxis, etc...
  • Heterogeneities and patterns
  • Patterns away from onset: use of calculus of variations, comparison principle arguments, analytic continuation/global bifurcation,

    Grades

    Your course grade will be based on your lectures.

    Learn From Anywhere and Classroom policies

  • Zoom links for lectures and office hours can be found in the course page on Blackboard .
  • All class sessions will be recorded for the benefit of registered students who are unable to attend live sessions (either in person or remotely) due to time zone differences, illness or other special circumstances. Recorded sessions will be made available to registered students ONLY via their password-protected Blackboard account. Students may not share such sessions with anyone not registered in the course and may certainly not repost them in a public platform. Students have the right to opt-out of being part of the class recording. Please contact me to discuss options for attending the course in such cases.
  • I will also post my lecture notes to the blackboard website a day or two after each class.
  • Please let me know ASAP if a you run into issues with connecting to course materials or resources. I will try to make office hours accessible to students in all time zones. Please contact me if you are unable to make any of the sessions and I will try to work something out with you.
  • You are welcome to switch your class participation modality between remote and in-person for any reason. Please let me know if you decide to do so. At the current class size, we do not need to do rotations, and all registered students are able to attend in person each day if desired.
  • We will use the LFA app for those who are attending the course in person. Please use this app to update your weekly plans on attending lecture.
  • We will observe all University mandated COVID-19 policies on social distancing and mask wearing.

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