AMSI Summer School 2010 Course Materials
Soap Films - Minimal Surfaces and Partial Differential Equations (SPF)All enquiries should be directed to: |
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Synopsis
Minimal surfaces exhibit some intriguing behaviour that, to some extend, we are familiar with from playing with soap films and soap bubbles. Having minimal surfaces as the Leitmotiv, and depending on students' background and various interests we will cover topics in
- Geometry of surfaces – parametrisation, first and second fundamental forms, notions of curvature;
- Calculus of variations – minimising "energy"/ surface area, first and second variation of functionals;
- Introduction to elliptic partial differential equations of second order (including Sobolev spaces and basic solution techniques from functional analysis);
- Fun results on minimal surfaces.
Contact Hours
7 hours of lectures per week, with consultation as requested/required. For information on timetabling please visit the timetable web page.
For Credit (FC) and Not For Credit (NFC)
Are you taking this subject for credit in your home institution? If so, I need to know. I’ll take a roll in the first class. You can change your mind afterwards, but students taking a subject for credit must confirm their intention to do so by the start of the third week of the school.
Prerequisites
We'll assume familiarity with
- the fundamental concepts of analysis in Euclidean Space (infs and sups, open and closed sets, continuity, completeness and compactness, differentiability);
- basic notions of linear algebra (vector spaces, inner products, quadratic forms, eigenvalues);
- basic notions of multivariable calculus (level sets, graphs of functions, parametric representation of curves and surfaces, divergence theorem);
- some basic notions of complex analysis, metric spaces – would be helpful, but we can talk about them;
- some introductory knowledge of differential geometry on curves and surfaces would be fantastic, but not assumed.
Obviously, the stronger your background, the happier the lecturer! I will circulate a questionnaire in the first lecture asking you to indicate at which level you classify your knowledge of the various topics.
Assessment
I’m open to negotiation, but the proposal is:
- Problems assigned during lectures (50%) – this component can include a short project, depending on class size and interest;
- Take-home exam (50%)
Resources
Lecture notes
My lecturing style is to write on the board from my handwritten notes. For the last few years I have experimented with scanning volunteer students' notes after the class (they get free editing in exchange...), and everybody seemed happy. Last semester we also photographed the boards, but had some non-standard arrangements for linking them, as files were getting large.
Textbook
For the first week I will be using my own notes – some introduction to differential geometry and calculus of variations.
Week two and three will be following material chosen from "Partial Differential Equations" by Lawrence C. Evans, AMS, Graduate Studies (GMS 19). This book is currently not available at La Trobe, but we are trying to have a copy by the time the summer school starts. It seems a good idea if the book is available at your institution, to arrange to bring a copy along, but please negotiate to share with your friends.
The last week will be devoted to selected topics on minimal surfaces and I plan to follow (in a rather free way) some bits and pieces from "Minimal Surfaces", by U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab, Springer, Comprehensive Studies in Mathematics 296.
Course downloads
- Background Notes
- There are currently no background notes available for this course
- Lecture Notes
- There are currently no lecture notes available for this course