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SMSTC Semester 1 2018-19, Tuesdays & Wednesday 09:00-10:30am
Geometric Measure Theory (20 hours)
The purpose of this course is to equip students with some essential techniques from geometric measure theory and allow them to start reading research papers. This field studies the geometry of sets and measures in Euclidean and metric spaces, and is motivated by problems that come from areas like complex analysis, harmonic analysis in non-Euclidean spaces, and partial differential equations. One can also think of this subject as the study of geometric problems with the assistance of measure theory. Some typical example problems that we will discuss are as follows:
1. In measure theory we learn that there are actually a rich family of sets of Lebesgue measure zero, but how can we compare their sizes in a meaningful
way? If we construct a Cantor set by removing middle thirds, and another by removing middle fourths, intuitively the latter one should be bigger than
the former, but Lebesgue measure is not sensitive enough to distinguish their sizes since it returns zero for both. This leads us to the notions of Hausdorff measures and dimensions to help distinguish the sizes of small sets.
2. What happens to the dimension of a set under transformations like orthogonal projections and Lipschitz mappings? For example, if we look at the
orthogonal projections (or shadows) of a line segment, then they are usually of positive length except if we project parallel to the segment, in which case
we get a point. In that situation, the dimension has dropped to zero, but that case is quite exceptional. How true is this for general sets? That is,
in how many directions can the dimension of a set drop when I perform an orthogonal projection?
3. What does the geometry of a measure behave like as I zoom in on it? To answer this question, we will study tangent measures and their applications.
These are the set of measures obtained by zooming in on the support of a measure and taking weak limits of the rescaled measures. For example, the
tangent measures of surface measure on a manifold will just be Lebesgue measure supported on the tangent plane, though there are measures whose
tangent measures at a point can be very large.
4. Given a set in Euclidean space, when can I parametrize it, or when is it contained in a countable union of nice surfaces? This leads us to the study of
rectifiable sets. An n-rectifiable set is like an n-dimensional manifold with a weak differentiable structure, in the sense that it is well approximated by
tangent planes at almost every point. These arise very naturally in several topics, for example, in classifying which sets in the complex plane support
bounded analytic functions on their compliments. We will discuss several classifications of these sets.
Prerequisites: Only real analysis is required, although complex analysis or a parallel course in harmonic analysis would improve the experience.
Course Material: We will use Mattila’s book ”Geometry of Sets and Measures in Euclidean Space” as a guide, and any additional material will be covered in supplementary notes. In addition, there are some recommended sources for further reading:
”Geometry of Fractal Sets,” by Falconer.
”Measure theory and fine properties of sets,” by Evans and Gariepy.
”Fourier Analysis and Hausdorff dimension” by Mattila.
There will be weekly exercises on the material. Students who want to sit in on the class for no credit are welcome.
All information about this course (homework, notes, office hours, etc) will be posted on my webpage, http://www.maths.ed.ac.uk/~jazzam/.