Fractal sets in analysis, spring 2015

Lecturer

Pertti Mattila

Scope

10 sp.

Type

Advanced studies

Prerequisites

Basic knowledge of measure and integration theory such as the courses "Mitta ja integraali" and "Reaalianalyysi I". "Reaalianalyysi II"  and basic knowledge of Fourier analysis are useful but not necessary.

Lectures

Weeks 3-9 and 11-20, Wednesday 14-16 in room B322 and Thursday 14-16 in room C122.

There will be no lecture nor exercise session on Wednesday, April 22 and no lecture on Thursday, April 23. The session for exercise 9 will be on Wednesday, April 29. There will be no lecture on Thursday, April 30. The last events of the course will be

Wednesday, April 29, Anne Isabel Gaudreau: Nikodym sets, and the exercise session,

Wednesday, May 6, PM: Removable sets, and the exercise session,

Thursday, May 7, Jesse Jääsaari: Number theoretic sets

Wednesday, May 13: Francesca Corni: Cookie-cutters

Easter Holiday 2.-8.4.

Course description

Fractal sets in this course mean both very general (closed or Borel) sets in Euclidean spaces and special fractal sets such as various Cantor-type sets and graphs of continuous nowhere differentiable functions. They play a significant role in mathematical analysis and also in other parts of mathematics and its applications. Constructions and properties, mainly geometric measure theoretic ones, of such sets will be studied.  Possible  topics for the course include  

Hausdorff dimension, Frostman's lemma, Riesz potentials and capacities, and Fourier transform

Minkowski and packing dimensions

Hausdorff dimension, orthogonal projections, and distance sets

Self-similar and self-affine sets

Graphs of continuous functions (e.g., Weierstrass nowhere differentiable function)

Brownian motion

Cantor sets of uniqueness and multiplicity for Fourier series

Besicovitch (or Kakeya) sets and Fourier transform

Peter Jones's version of the travelling salesman theorem

Removable sets for bounded analytic functions

Lecture notes

Bibliography

C. J. Bishop and Y. Peres, Fractal Sets in Probability and Analysis, Cambridge University Press, 2015.

K. J. Falconer, Geometry of Fractal Sets, Cambridge University Press, 1985.

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.

P. Mattila, Fourier Analysis and Hausdorff dimension, Cambridge University Press, 2015.

Registration

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Exercises

Laura Venieri will conduct exercise sessions on Wednesdays 16 - 18 in B322.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9 

Exercise 10