Geometric measure theory (MATS2110)
Lecture: Tuesdays and Thursdays, 14:15-16:00, MaD 380
First lecture: Tuesday, March 18, 2025.
Exceptional teaching time: Wednesday April 30 (14:15-16:00, MaD 381) instead of May 1.
Exercises: Tuesdays, 12:15-14:00, MaD 355
First exercise session: Tuesday, March 25, 2025.
See also the course page on SISU
General information about this course: pdf
Topics for the presentations: pdf
Geometric measure theory provides tools to investigate geometric properties of sets that are not necessarily regular enough to be studied by differential geometric methods, but which appear naturally for instance in calculus of variations, harmonic analysis, theory of PDEs, or dynamical systems. An overview of some of these applications is given in this survey by Tatiana Toro.
Program (may be updated during the course)
1. Review of Hausdorff measure and dimension [O, §2.1, 2.2, Lemma 3.1], [M, p.8-11, p.54-59, §4.10]
2. Density theorems; Frostman's lemma and applications [O, §3, §5.1]
(notes 1, notes 2)
3. Ahlfors regularity; uniformly distributed measures; Haar measures [M, §3]
(notes 1, notes 2)
4. Riesz energies of measures, projection theorems by Marstrand, Kaufman, and Mattila [O, §4]
(notes 1, notes 2)
5. Rectifiable and purely unrectifiable sets, Besicovitch's projection theorem [O, §5.2-5.4]
(notes 1, notes 2)
6. Kakeya sets [O, §7.1-7.2]
(notes 1, notes 2)
7. Proof of Besicovitch's projection theorem [O, §5.5]
(notes 1, notes 2)
Literature:
Notes for each lecture of the course will be uploaded below. These are largely based on the following references:
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[M1] Mattila, Pertti: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge, 1995
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[O] Orponen, Tuomas: Geometric Measure Theory, lecture notes, University of Helsinki, 2018
Additional references:
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[BP] Bishop, Christopher J. and Peres, Yuval: Fractals in probability and analysis. Cambridge University Press, 2017
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[F] Falconer, Kenneth: The geometry of fractal sets, Cambridge University Press, 1986
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[M2] Mattila, Pertti: Fourier analysis and Hausdorff dimension, Cambridge University Press, 2015
Exercises:
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7
Evaluation:
Grade scale 0-5
The final grade is the rounded arithmetic mean of the grade for the exercises and the grade for the presentation (talk and essay).