# Geometric measure theory and singular integrals, spring 2017

**Teachers:** Henri Martikainen and Tuomas Orponen

**Scope:** 10 cr

**Type:** Advanced studies

**Teaching: **

Weeks 3-9 and 11-18, Tuesday 14-16 in room C122 and Friday 10-12 in room C124. Two hours of exercise classes per week.

Easter holiday 13.-19.4.

**Topics: **The course investigates the connection between the geometry of planar sets and the boundedness of the Cauchy transform (a singular integral operator in the plane). Specific topics may include:

- Basic concepts of the theory of singular integral operators; in particular the Cauchy transform.
- Curvature and rectifiability of sets and measures in the plane.
- Boundedness of the Cauchy transform on curves.
- Analytic capacity and rectifiability.
- Characterising rectifiability via P. Jones' beta-numbers; connection to curvature.

**Prerequisites: **Basic knowledge on measure theory, Lebesgue integration and *L ^{p}*

*-*spaces as covered e.g. in the courses "Mitta ja integraali" and "Reaalianalyysi I".

## News

- The first exercise set is now available. These are for the exercise session on Wednesday, 1 February.
- The list of topics covered on the lectures of the first part of the course: LOG.
- The second set of exercises is now available. These are for the exercise session on Wednesday, 15 February.
- Draft lecture notes for the second half of the course are now available.
- The third set of exercises is now available. These are for the exercise session on Wednesday, 1 March.
- The first part of the course is over:
**no lecture on Friday, 3 March.**The second part of the course by Tuomas starts on Tuesday, 14 March. - The fourth set of exercises is now available; session on March 29.
- Ella's presentation on Tuesday, Mar 21. A normal lecture on Wednesday, Mar 22.
- The fifth set of exercises is now available; session on April 12.

## Topics for presentations

Below are a few suggestions for the presentations. If you're interested in one of them, contact either one of the lecturers (e.g. after a lecture), and we'll discuss the details and the schedule. The schedule is tentative.

- Geometry of 1-rectifiable sets (reserved by Ville Marttila, 15.3.)
- Analytic capacity (as in the books of Tolsa and Mattila, Chapter 1 and Chapter 19) (Reserved by Janne Siipola, 26.4.)
- Tangent measures (as in the book of Mattila, Chapter 14) (Reserved by Hans Groeniger, 5.4.)
- The traveling salesman theorem (as in the book of Bishop-Peres, Chapter 8). (Reserved by Ella Tamir, 22.3.).
- A short, complex-analytic proof of the L2-boundedness of the Cauchy transform on curves, as in a paper of Coifman-Jones-Semmes (Two elementary proofs of the L2 boundedness of the Cauchy transform on Lipschitz curves, J. AMS Vol. 2, No. 3 (1989), 553–564 (reserved by Juuso Nyyssönen, 3.5.)
- Lecture notes for the second half of the course

- Course material

X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón–Zygmund theory, Progress in Mathematics, Vol. 307, Birkhäuser Verlag, Basel, 2014.

- P. Mattila: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, Cambridge University Press (1995).
- C. J. Bishop and Y. Peres, Fractal Sets in Probability and Analysis, Cambridge University Press (2015).
- Lecture notes for the second half of the course

## Registration

No registration required; come to the first lecture.

## Exercises

You should complete at least 60% of the exercises.

### Assignments

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Wednesday | 10-12 | C122 |

## Course feedback

Course feedback can be given at any point during the course. Click here.