## Geometric Measure Theory

**Objectives** and Course Description

Geometric Measure Theory is the study of the geometry of sets by the means of measures supported on such sets, or more generally, the geometric study of measures. GMT gives a way of measuring the size and dimensionality of fractals at a finer level than Lebesgue measure. Within the theory, there are rough fractals, which have fractional Hausdorff dimension and which are not well approximated by smooth objects. At the other end of the spectrum, there are “smooth” objects, which in this course means rectifiable sets. We will develop tools to study these sets and for recognising them through their projections. At the end, time permitting, we will touch upon a recent geometric construction of a “tangent space”, or decomposability bundle, for a measure. The tools are applicable to the study of singularities of geometric and analytic objects, such as minimal surfaces or solutions to partial differential equations.

**HWs:**

**LEcture times**

**Lectures: **Sylvester Eriksson-Bique

Tuesday + Thursday 17.1-2.3.2023

14:15-16:00 MaD 245

**(NOTE: Thursday 23.2. we are in MaD 259)**

**Office hours: **Sylvester Eriksson-Bique

Tentatively, Thursdays 16-17, MaD 316.

Available upon request at other times.**Demo sessions:** Jiayin Liu

Friday 27.1-10.3.2023

14:15-16:00 MaD 381**Exam: **

The preferred way of completing the course is to do a written project and presentation. If this does not work for you, a take home exam will be organized on the week of 13th-17th of March.

## Syllabus

This is the first time I give this course and the first time it is given with the proposed topics. Thus the following schedule is only a rough estimate, and we may move a bit slower or faster. Contact me if the pace is not ok for you.

**Week 1: **Notions of dimension, Hausdorff measures and linear relaxations and duality.

Chapter 1-2 in notes.

**Week 2:** Duality continued, Frostman’s lemma, Density of Hausdorff measures.

Chapter 2 in notes.

**Week 3:** Rademacher’s theorem, co-area formula and inequality, Riesz energies

Chapter 3-4.

**Week 4:** Uniformly distributed measures, Rectifiability and Besicovitch projection theorem.

Chapter 5-6.

**Week 5:** Tangent measures

Chapter 7.

**Week 6: **Fourier Transform of measures.

Chapter 8.

**Week 7: **Kakeya Sets and Decomposability bundles. (Time permitting)

Chapter 9 and Chapter 7 starred material.

## Literature

- Lecture notes will be written during the course and the progress is available here:

(This link will expire around April 2023. If I forget to update this link, then please contact me by email for a copy of the notes.)

- Lecture notes of Tuomas Orponen and Katrin Fässler from previous years. These notes are very close to the present course, but there will be some differences in presentation. You may choose either to follow for most of the topics.

Link to lecture notes of Orponen and Fässler.

- Mattila, Pertti:
*Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability*, Cambridge, 1995

**Additional references**:

- Bishop, Christopher J. and Peres, Yuval:
*Fractals in probability and analysis*. Cambridge University Press, 2017 - Falconer, Kenneth:
*The geometry of fractal sets*, Cambridge University Press, 1986 - Mattila, Pertti:
*Fourier analysis and Hausdorff dimension*, Cambridge University Press, 2015

## Grading

The course can be completed in one of two ways:

- HW (40 %) and a 30 minute oral presentation, which includes a roughly 5 page written report (60%).
- HW (60 %) and a take home exam (40%)

*Out of the above two options, the first is highly recommended.*

You **must** complete either a take home exam or a presentation to receive credit for the course.

If for a justified reason you are unable to complete some of the requirements, please contact me, and we may find a compensatory way.

The course is 5 cr, and graded 0-5. As a base level, I will use the following rubric, which however can be adjusted in a way that increases grades.

90 %, 5

80 %, 4

70 %, 3

60 %, 2

50 %, 1

<50 %, 0/Fail

An excellent presentation can give a bonus 10 % to the score (i.e up your grade by one).

## HW

The HW is worth 10+bonus points, and consist of a written portion (5pt), a group portion (5pt) and a bonus section (2pt).

Each student should complete the written portion on their own, and think ahead of time about the group portion. You should attempt to solve the group portion, and write at least a sketch of your ideas – but you can complete the solution in the sessions. In the demo sessions, the group problems are discussed and worked into presentations.

The last two weeks will have shorter written HW (10 pt for written + 2 pt bonus), since we will use the discussions for presentations.

HW should be returned in the demo sessions and to get the credit for the group portion, you should participate in the discussions. If you are unable to participate, you may submit written solutions to all problems which are graded on completeness and correctness.

At the discretion of the grader, only some problems are graded on correctness and the rest on completeness.

Generally, no late assignments will be accepted. If you are sick, or otherwise unable to complete an assignment, please contact the instructor in advance, and discuss alternate options.

The lowest HW score is dropped and ignored in the grading.

Additionally, there are bonus questions (2pt) each week, which will be discussed depending on time. Submitting written solutions to them will give you extra credit that compensates for lost points.

## Inclusivity

The course should be an encouraging and open space for discussion, learning and the exchange of ideas. Every participant should be able to join fully and comfortably. If you find challenges with this, or if any course policy is difficult for you, please discuss these with the instructor. Also bear in mind, that the university has resources and support in the event of harassment or of personal difficulties. I can assist you in identifying such resources.

If you need a religious exemption, or other compelling exemption, please contact me at least 1 week prior to the requests effect. I may not be able to accommodate requests that arrive late.

## Project ideas

You should decide on a project by the end of Week 3 and report the project to the professor. I will follow up on this.

You should feel free to contact me for help or discuss the project in office hours. Generally speaking, it should include at least one result, one proof, some reflection/citations, one example and one definition. You can choose the topic to be anything which at least peripherally connects to the course.

The lecture notes contain many suggested projects. Here are some as well. The topics with a star are more challenging, with two stars very challenging and those with an L are a bit lighter. A lot of the topics can be made easier or harder depending on interest and time of the student. The starred ones may need to be longer than 5 pages, and you may only be able to give a summary in 30 minutes. In those cases a good survey is good, while providing some detail. You can also work in teams of two-three students.

1. Hausdorff dimension of 𝐴×𝐵- examples when it does not hold, and conditions on factors which ensure that it holds. (L)

2. Sets of finite perimeter. E.g. Federer’s characterization or existence of total variation measure.

3. Mass Transference Principle of Beresnevitch and Velani.

4.Hausdorff dimension and Brownian motion. (L)

5. Singularities of Sobolev functions.

6. Singularities of Sobolev functions. [7] (L)

7. Hausdorff content and Sobolev capacity. [7] (L)

8. Kirchheim’s metric differentiation theorem. (*)

9. Besicovitch’s projection theorem in higher dimensions.

10.Weak tangent planes and characterizations of rectifiability in terms ofthem. (*)

11. The basics of integral currents. (*/L-depends on how much is covered)

12. Jones traveling salesman theorem.

13. Dorronsoro’s theorem in R^d (partly or in pairs). (*)

14. Iterated function systems and their Hausdorff dimension. (L)

15. Packing dimension. (L)

16.Hausdorff dimensions of slicing sets by planes. (L/*-can follow mostly Mattila)

17. Porous sets and Hausdorff measures. (L)

18.Volberg and Konyagin construction of a doubling measure on metrically doubling spaces. (*)

19. Marstrand’s theorem. (*)

20. Wasserstein metric on measures and Kantorovich duality. (L)

21. Proof of the co-area formula. (L/* – quite technical)

22.Suslin sets – the very basics of them. Say the theorem that Suslin sets are measurable, and/or that a set is Borel if and only if it is Suslin andit’s complement is Suslin. (* – technical and difficult to choose what is essential)

23. The Method of stationary phase. (L/* – mostly because technical)

24. Rainwater lemma for measures. (L)

25. Weaver derivations – definition mostly and examples. (*-open ended)

26.Choose notion of dimension, Topological dimension, Assouad dimension,Assouad-Nagata dimension, Lipschitz dimension – expand on the lecturenotes. (**)

27. Big-pieces of bi-Lipschitz maps property. (*)

28.Asymptotically doubling measures and Lebesgue differentiation theoremfor them. (L)

29. Reifenberg’s theorem and Reifenberg flat sets. Sketch of proof (**).

30.Four corner cantor set and the unboundedness of the Cauchy transform.(**)

31. Essential infimum and supremum of measures. (L)

32.Non-bilipschitz equivalent Cantor sets with the same Hausdorff dimension.(*-Can assume and simply state the Ergodic theorem. Will give a referencefor this.)

33.Set cover problem in Computer Science, its linear relaxation and log(𝑛)-approximation algorithm. (L – can mention NP-completeness, but not wise to try to prove.)

34.Quasiconvexity of R^𝑑\𝐸, and for which compact sets 𝐸? Conditions in terms of Hausdorff measure? Conditions in terms of 𝑑−1-rectifiability? Combine concepts of this course and literature. (*)

35.Cut measures and 𝐿^1-embeddability. Connection to sets of finite perimeter.(**,Can present the basic result, and mention the connection to the sparsestcut problem.)