Fall Semester 2007

Geometric Measure Theory


Lecture: Mon 13.15-15, ML F40
Wed 12.15-14, ML H41.1
Exercises: Fri 13.15-15, HG G5, see Exercises
Testat condition: See Exercises
Year: Masters, Doctoral
Language: English
Prerequisites: Mass und Integral (4. Semester Math) - required
Differential Geometry and/or PDE - helpful
Lecturer: Tom Ilmanen, HG G63.2, 044-632-5443
or write to Hedi Oehler, oehler@math.ethz.ch
Assistant: Jörg Hättenschweiler, johaette /at/ math.ethz.ch

Description:

Geometric measure theory studies detailed properties of irregular sets and functions in Rn. Some central notions are

  • Hausdorff measure
  • rectifiable and unrectifiable sets
  • covering theorems
  • varifolds and currents
  • first variation

    Applications include minimal surfaces with singularities, and singularities of nonlinear PDE. The class will be strongly oriented towards solving exercises.

    Literature:

    [1] L. Simon, Lectures in Geometric Measure Theory, 1984.
    [2] R. Schaetzle, Geometrische Masstheorie, script, Tübingen, 2006/07.
    [3] Fanghua Lin and Xiaoing Yang, Geometric Measure Theory - an Introduction, International Press, 2002.

    The following are good background reading:

    [4] M. Struwe, Analysis III (Mass und Integral), script, ETH, 2002.
    [5] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Adv. Mathematics, CRC Press, 1992.
    [6] F. Morgan, Geometric measure theory: A beginner's guide, Academic Press, 2000.
    [7] K. Falconer, Fractal geometry: Mathematical foundations and applications, John Wiley & Sons, 2003. (Either of the predecessor volumes is also good.)

    T. Ilmanen, Dept Math, ETH, Sep 2007