# Geometric Measure Theory.

**Geometric Measure Theory.**

**Lecturer: **John Andersson [johnan [at] kth.se]

**Course book: **Krantz-Parks "Geometric integration Theory" Birkhäuser (The book is free to download at the KTH library webpage).

**Course Description: **Geometric measure theory is an advanced subject that draws from many parts of mathematics. The main idea is to use integration and measure theory techniques to investigate geometric problems. In the course we will explain how good convergence properties of integrals can be used to prove existence of solutions to geometrical variation problems (mostly the minimal surface equation). We will also cover much other mathematics.

In the first part of the course we will, in particular, repeat the basics of measure theory (definition of measures, Hausdorff measures, outer measures, differentialtion of measures, Riesz representation Theorem), we will also discuss rectifiable sets and the area and coarea formula (think change of variables and polar coordinates), Stokes theorem and integration on mannifolds.

In the second part of the course we will consider more specialized topics. We will discuss currents and compactness properties of currents, this also involves some distribution theory. We will use currents to prove existence of minimizers of geometric problems. We will end the course with several lectures where we discuss regularity theory for minimizing currents.

**Suggested prerequisites: **Given the diversity of backgrounds of the students in our PhD program it is more or less impossible to demand certain courses. But I will assume that you have a masters degree in mathematics and that you understand basic measure theory (measures and integration). This is not intended to be a broad course and we will be covering many very technical theorems in analysis. It is therefore important that you have an interest in analysis and that you are willing to work.

**Examination: **Active participation in exercise classes, maybe homework assignments, oral exam in May.

**Schedule and Suggested Reading:** (All readings is in Geometric Int. Theory, and we always meet in Room 3418)

Lecture 1: 15th Jan (Wednesday) 13:15-15:00 Repetition of measure theory pp. 1-22

Lecture 2: 20th Jan (Monday) 15:15-17:00 Repetition measure theory pp. 22-33, 53-57, and 61-69 .

Exercise Class: 24th Jan (Friday) 15:15-17:00 Submitted questions for seminar 1

Lecture 3: 27th Janury (Monday) 13:15-15:00 Covering Theorems pp. 91-109

Lecture 4: 3d February (Monday) 13:15-15:00 Differentiation of Measures & Riesz Representation Thm

Exercise Class: 7th February (Friday): 15:15-17:00 Submitted questions for seminar 2

Lecture 5: 10th February (Monday) 13:15-15:00 The Area Formula pp. 125-137. We only had time to cover the pages 125-133 **so there will be some changes from the original schedule for the next several lectures.**

Lecture 6: 17th February (Monday) 13:15-15:00 ~~Coarea formula and rectifiable sets pp. 137-159 ~~We will cover Rademacher's Theorem and the area formula for Lipschitz functions pp 134-136. Then we will cover the co-area formula pp. 137-143.

Exercise Class: 21st February15:15-17:00 Submitted questions for Seminar 3

Lecture 7: 24th February 13:15-15:00 ~~Stokes Theorem and differential forms pp. 159-173~~ We will cover pp. 143-159. The focus of the lecture will be on rectifiable sets and the Poincare inequality.

Lecture 8: 2nd March 13:15-15:00 ~~Currents I pp. 173-189~~ Stokes Theorem and differential forms pp. 159-173

Exercise Class 6th March 15:15-17:00 Submitted questions for seminar 4

Lecture 9: 16th March 13:15-15:00 ~~Currents II pp. 189-204 ~~Currents I pp. 173-189

Lecture 10: 23d March 13:15-15:00 ~~Currents III pp. 204-224 ~~Currents II pp. 189-204

Exercise Class: 27th March 13:15-15:00

Lecture 11: 30th March 13:15-15:00 Currents III pp. 204-224

CANCELLED:~~ Lecture 12: 6th April 13:15-15:00 The Compactness Theorem pp. 225-241~~

CANCELLED: ~~Exercise Class: 8th April 15:15-17:00~~

CANCELLED: ~~Lecture 13: 17th April 15:15-17:00 The Flat Metric and Existence of Minimizers pp. 241-255~~

CANCELLED:~~ Lecture 14: 20th April 13:15-15:00 Regularity of Minimizers I pp. 255-269~~

CANCELLED: ~~Exercise Class: 24th April 13:15-15:00~~

Lecture 15: Rectifialbillity of currents pp225-241 Notes form the ZOOM lecture. Part 1 Part 2

Lecture 16: 4th May 13:15-15:00 ZOOM ROOM The Compactness Theorem pp. 225-241 Lecture Notes

Lecture 17: 11th May ZOOM ROOM The Flat Metric and Existence of Minimizers pp. 241-255

Lecture 18: 18th May ZOOM ROOM 13:15-15:00 Regularity of Minimizers I pp. 255-270 NOTES

Lecture 19: 22nd May ZOOM ROOM 13:15-15:00 Harmonic functions 270-286

Lecture 20: 27th May ZOOM ROOM 13:15-15:00 Regularity of Minimizers II pp. 286-308 SLIDES