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 Lp-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

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

  2. P. Mattila: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, Cambridge University Press (1995).
  3. C. J. Bishop and Y. Peres, Fractal Sets in Probability and Analysis, Cambridge University Press (2015).
  4. 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

GroupDayTimeRoomInstructor
1.Wednesday 10-12 C122  

Course feedback

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

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