Geometric measure theory (MATS2110)
Lecture: Mondays and Tuesdays, 12:15-14:00, MaD 380
Exercises: Wednesdays, 12:15-14:00, MaD 355
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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.
Program (selected chapters of the lecture notes as indicated)
1. Hausdorff measure and dimension, density theorems [Chapters 1-2, Appendix A]
2. Mass distribution principle, Frostman's lemma, Riesz energies of measures [Sections 5.1, 6.1, Chapter 3]
3. Haar measures, uniformly distributed measures [Sections 4.2-4.4]
4. Projection theorems by Marstrand, Kaufman, and Mattila [Sections 5.2-5.3]
5. Fourier transforms of measures [Sections 7.1-7.2, 7.4-7.6]
6. Rectifiable and purely unrectifiable sets [Sections 6.1-6.4]
7. Besicovitch's projection theorem [Sections 6.5-6.6, statements in Section 8.1]
Literature:
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[M1] Mattila, Pertti: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge, 1995
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:
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).