Geometric Measure Theory 7.5hp

Educational level: Third cycle

Subject area: Mathematics

Grade scale: Pass/Fail

Course description: Geometric measure theory uses measure theory to analyze geometric problems. Any geometric shape in space, for instance a minimal surface or a fractal set, can be viewed as the support of a measure. An understanding of the measure will therefore increase the understanding of the set. The great advantage of considering the set as being the support of a measure is that measures are very general objects with good analytical properties (such as compactness properties) even if the set is bad. In this course we will cover the basics of measure theory.

Learning outcomes

After completing this course the student should:

● Have a good understanding of Geometric Measure Theory (GMT). In particular the regularity theory as developed by Almgren.

● Be able to independently read and understand advanced mathematics.

● Be able to discuss and synthesize mathematics.

● Be able to situate GMT in the larger field of regularity theory for PDE and understand its strengths and weaknesses.

● Have a good understanding of basic measure theory.

● Have a good understanding of analysis on rough manifolds.

Course main content

We will follow Leon Simon's Lectures on Geometric Measure Theory, the best understanding of the material will come from pursuing the table of contents.

● Basic measure theory (Radon and Hausdorff measures, densities of measures.)

● Analysis on manifolds (gradients, sobolev inequalities etc.)

● BV functions, varifolds and rectifiable sets.

● Regularity theory for minimal surfaces.

Eligibility and prerequisites.

This course is open for dedicated PhD students. It is desirable to have a good grounding in mathematical analysis (such as SF2713 Foundations of Analysis) and measure theory (such as SF2743 Adfvanced Real Analysis). Some basic understanding of PDE and Sobolev spaces is also desirable.

Literature

Leon Simon “Lectures on Geometric Measure Theory” Proceedings of the centre for mathematical analysis, the australian mathematical society, Volume 3, 1983

Examination

• Presentation of a topic during a seminar.

• Active participation in seminars.

• Oral exam at the end of the course.

Offered by

SCI/Mathematics

Examiner and contact person.

John Andersson