Kaisa_2012_3_photo by Veikko Somerpuro

Ilmoittaudu
11.12.2017 klo 09:00 - 2.5.2018 klo 23:59

Aikataulu

Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

PäivämääräAikaOpetuspaikka
Ma 15.1.2018
10:15 - 12:00
Ti 16.1.2018
12:15 - 14:00
Ma 22.1.2018
10:15 - 12:00
Ti 23.1.2018
12:15 - 14:00
Ma 29.1.2018
10:15 - 12:00
Ti 30.1.2018
12:15 - 14:00
Ma 5.2.2018
10:15 - 12:00
Ti 6.2.2018
12:15 - 14:00
Ma 19.2.2018
10:15 - 12:00
Ti 20.2.2018
12:15 - 14:00
Ma 26.2.2018
10:15 - 12:00
Ti 27.2.2018
12:15 - 14:00
Ma 12.3.2018
10:15 - 12:00
Ti 13.3.2018
12:15 - 14:00
Ma 19.3.2018
10:15 - 12:00
Ti 20.3.2018
12:15 - 14:00
Ma 26.3.2018
10:15 - 12:00
Ti 27.3.2018
12:15 - 14:00
Ma 9.4.2018
10:15 - 12:00
Ti 10.4.2018
12:15 - 14:00
Ma 16.4.2018
10:15 - 12:00
Ti 17.4.2018
12:15 - 14:00
Ma 30.4.2018
09:15 - 12:00

Muu opetus

18.01. - 08.02.2018 To 14.15-16.00
22.02. - 01.03.2018 To 14.15-16.00
15.03. - 22.03.2018 To 14.15-16.00
05.04. - 19.04.2018 To 14.15-16.00
Tuomas Hytönen, Timo Hänninen
Opetuskieli: englanti

Materiaalit

The course is mainly based on the Finnish lecture notes "Moderni reaalianalyysi" by Ilkka Holopianen, MoRA.pdf.
Two English versions are available: (a) a translation made at the University of Turku, MoRAeng.pdf, covers Chapters 1-4; (b) an adaptation of the lecturer, mora-eng.pdf, covers most but not all of Chapters 1-4, but also the whole Chapter 5.
There is also an appendix with some additional material not contained in the other two files.

Luentomateriaalit

Tehtävät

1st exercise set for Jan 18

Tuomas Hytönen

2nd exercise set for Jan 25

Tuomas Hytönen

3rd exercise set for Feb 1

Tuomas Hytönen

4th exercise set for Feb 8

Tuomas Hytönen

5th exercise set for Feb 22

Tuomas Hytönen

6th exercise set for Mar 1

Note: In Ex. 6.2, the upper bound for the Hausdorff measure of a product set is not true in the stated generality, but it is true for instance when K, L are the same Cantor sets. This is enough for the claim concerning existence of sets of a given Hausdorff dimension.

Tuomas Hytönen

7th exercise set for Mar 15

Tuomas Hytönen

8th exercise set for Mar 22

Tuomas Hytönen

9th exercise set for Apr 5

Tuomas Hytönen

10th exercise set for Apr 12

Corrections: Ex. 10.2: The task should be estimating the lower s-density, not lower 1-density. Ex. 10.6: Add the assumption that E has finite Hausdorff 1-meausure.

Tuomas Hytönen

11th exercise set for Apr 19

Ex 11.2 is not correct as stated. A correct statement to prove is obtained by assuming that the range of mu is [0,1] instead of [0,infinity]. This has been corrected in Lemma A.3 of the notes on Apr 18, and this version of the lemma is all that is needed for Theorem A.4.

Tuomas Hytönen

Kurssin suorittaminen

The course exam will be held on Monday 30 April at 9:15-12:00 in Exactum B121. (The usual Monday lecture room, but observe that we start one hour earlier!) The actual examination time is 2 h 30 min, the additional 15 min is reserved for settling down in the beginning and returning the papers in the end. The examination will start as soon as we settle down, so be there in time!

Kuvaus

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

Belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Measure and integral, Real analysis I

Barchelor studies

Advanced knowledge on real analysis, incl. fractals.

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Hausdorff measure and dimension, self similar fractals, differentiation of Radon measures, Radon-Nikodym derivative, signed measures.

Required: Moderni reaalianalyysi, luentomoniste.

Recommended: Evans-Gariepy: Measure theory and fine properties of functions. K. Falconer: The Geometry of Fractal Sets.

Lectures and exercise classes.

Exam and excercises, Course will be graded with grades 1-5.

Exam, other methods will be described later