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MA6M5 Geometric Measure Theory

Lecturer: Filip Rindler

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 one hour lectures

Assessment: 100% Oral Exam

Formal registration prerequisites: None

Content:

Geometric measure theory is the study of geometric objects with the tools of measure theory. It occupies a central place in modern Geometric Analysis, where it has led to the resolution of many conjectures such as Plateau's problem and intriguing questions about soap bubbles. It is also extremely useful as a toolkit of methods that have enabled many new discoveries in other fields of Mathematics, spanning from Mathematical Material Science, over the Theory of PDEs and the Calculus of Variations, all the way to Number Theory. This course will give an introduction to this important area.

 Outline:

  • Motivation: Plateau's Problem
  • Hausdorff Measures
  • Area & Coarea Formula
  • Rectifiability
  • Forms and Stokes' Theorem
  • Currents
  • Integral Currents
  • Deformation Theorem
  • Resolution of Plateau's Problem

 Books:

  • Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free- Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, 2000.
  • H. Federer, Geometric Measure Theory, Grundlehren der mathematischen Wis- senschaften, vol. 153, Springer, 1969.
  • S. G. Krantz and H. R. Parks, Geometric Integration Theory, Birkhauser, 2008.
  • P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995.
  • F. Morgan, Geometric Measure Theory, 5th ed., Elsevier, 2016.
  • L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, vol. 3, Australian National University, Canberra, 1983.