Introduction to Geometric Measure Theory - Spring 2017
Lecturer Prof. S. Wenger

Start date
Mondays 13:15 - 15:00 and Fridays 10:15-12:00, Science de la terre 2.301

20 February 2017
Geometric Measure Theory (GMT) studies geometric properties of singular surfaces (of any dimension) through the use of measure theory. The following problem lies at the origins of GMT. Does every k-dimensional surface without boundary in Euclidean n-space bound a (k+1)-dimensional surface of minimal volume? One of the principal achievements of GMT has been to develop a sufficiently rich and powerful theory of surfaces in Euclidean space which can be used to solve this problem and many related geoemtric variational problems. GMT is a very active area of current research and has found applications in many areas of mathematics, far beyond the problem of area minimization.

In this course, we give a gentle introduction to Geometric Measure Theory. The only prerequisites are the courses Analysis 1 - 4. A course in (abstract) measure theory is useful but not required.

Outline of course (might change slightly as we go along):
A) Review of measure theory:
  1. Outer measures
  2. Hausdorff measure and dimension
  3. Riesz representation theorem
  4. Weak compactness of Radon measures and Banach-Alaoglu theorem
B) Lipschitz maps:
  1. Lipschitz extensions
  2. Rademacher's theorem
  3. Area and co-area formulas
  4. Rectifiable sets
C) Differential forms
  1. m-vectors and m-covectors
  2. Differential forms and Stokes' theorem
D) Theory of currents:
  1. Definition and basic examples
  2. Homotopy formula and push-forward
  3. Integer rectifiable and integral currents
  4. Slicing
  5. Proofs of Closure and Boundary Rectifiability Theorems
  6. MBV functions and proof of the slice-rectifiability theorem

Preliminary notes for the course, taken by C. Guo and to be updated regularly during the semester.

  • L. Simon: Lectures on Geometric Measure Theory. Australian National University.
  • S. Krantz, H. Parks: Geometric Integration Theory, Birkhaeuser.
  • F. Morgan: Geometric Measure Theory: A beginner's guide, Elsevier/Academic Press.
  • L. Evans, R. Gariepy: Measure theory and fine properties of functions, CRC Press.
  • P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press.
  • L. Ambrosio, P. Tilli: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications.
  • H. Federer: Geometric Measure Theory, Springer Verlag.
  • L. Ambrosio, B. Kirchheim: Currents in metric spaces, Acta Math. 185 (2000), 1 - 80.