2023-2024
Room: E210 (Max Planck Institute, second floor)
Schedule: every Monday 15:30-17:00 (start on 23rd October)
Link to homepage of the course: MPI Webpage
Lecture notes: link
The aim of the course is to present some classic results in geometric measure theory and to introduce some of the basic tools used in the field.
The topics will tentatively include the following (and can be modulated depending on the audience): Hausdorff measure and dimension; Rademacher's theorem and rectifiable sets; tangent measures and Marstrand's density theorem; the Fourier transform of measures, Frostman's lemma, and Marstrand's projection theorem; Besicovitch's projection theorem and Kakeya sets.
23/10/23 (First Lecture). Plan of the course. Review of measure theory (definition of measure, weak convergence, compactness). Definition of Hausdorff measure and dimension.
30/10/23 (Second lecture). Recalling the definition of Hausdorff measure. Mass distribution principle. Dimension of the Cantor set. Frostman's lemma. Application to dimension of cartesian product.
06/11/23 (Third lecture). Riesz energies and relation betweeen finiteness of the energy and dimension. Proof of Marstrand's theorem, Part 1.
13/11/23 (Fourth lecture). Fourier transform of measures and relation to Riesz energies. Proof of Marstrand's theorem, Part II. Covering theorems. Hausdorff upper and lower densities. Example of Cantor set with zero lower density everywhere.
20/11/23 (Fifth lecture). Rademacher's theorem: proof by blowup. Definition of rectifiable and purely unrectifiable sets. Decomposition theorem in rectifiable and unrectifiable part.
27/11/23 (Sixth lecture). Tangent cones and rectifiability criteria. Approximate tangent cones and rectifiability criteria.
04/12/23 (Seventh lecture). Rectifiable sets and approximate tangent planes. A set with two null projections in the plane is purely unrectifiable. Criteria for rectifiability (only statements): Marstrand-Mattila (existence of density for sets); Preiss (existence of density for measures); Marstrand's density theorem; Besicovitch theorem and conjecture (lower density $\ge\frac12$ implies $1$-rectifiability.)
11/12/23 (Eighth lecture). Construction of a Kakeya set by duality. Proof that a Kakeya set has box dimension 2. Proof that a Kakeya set has Hausdorff dimension 2.
08/01/24 (Ninth lecture). Recalling the Kakeya set construction by duality. Projections of the 4-corner Cantor set.
15/01/24 (Tenth lecture). Introduction to Marstrand's theorem. Tangent measures and uniform measures. Locality of tangents. Tangents of measures with density are uniform.
22/01/24 (Eleventh lecture). Conclusion of Marstrand's theorem.