MAT1501: Geometric Measure Theorey

Basic information

Lecture notes

• week 1 lecture notes (including exercises). See also errata.
• An exercise, for people who like set theory and general topology, is: read, understand, and be prepared to explain to me in detail the proof of Theorem 2.2.4 from Federer's book . (If you look at this page, please ignore 2.2.5, which uses a different definition of Radon measures from ours.)
• week 2 lecture notes (including exercises). (current version posted 8:30pm, October 28, with some corrections)
• week 3 lecture notes (including exercises). (current version posted 8:30pm, October 28, with some corrections)
• week 4 lecture notes (including exercises). (current version posted 8:30pm, October 28, with some corrections)
• lecture notes, week 5 + half of week 6 (including exercises). (current version posted 8:25pm, October 15)
• lecture notes, second half of week 6 + week 7 (including exercises). (current version posted 8:00pm, November 12)
• week 8 lecture notes (including exercises). (current version posted 8:00pm, November 12)
• week 9 lecture notes (including exercises). (current version posted 8:00pm, November 12)
• more lecture notes (including exercises). (current version posted 3:00pm, December 3)

The typed notes in particular will be full of mistakes, most of which will just be typos (I hope). (In this way they will accurately reflect the lectures, which will also be full of mostly small mistakes, although in general not the same ones as in the notes.) A standing assignment in the class is to let me know about any mistakes that you spot, no matter how small. If you do not tell me about any mistakes, I will suspect that you are not reading the notes, since the mistakes are certainly there.

Marking

Option 1 is the default. I recommend option 2 only for students who either already know something about GMT, or advanced students who are already working on some research topic with some connection, possibly tenuous, to the subject.

Option 1: Do enough exercises. The exercises will appear in the lecture notes. Many of them will be easy, more or less only requiring some simple manipulations using definitions. So at a minimum, this will force you to keep up, to some extent, with the lecture notes, and to be familiar with some basic definitions. Of course, some problems are harder, for people who like a challenge. But to obtain a good mark it is only necessary to do any exercises, say about 2 per week if you do easy ones. If you prefer harder exercises, a smaller number is fine.

If you choose this option, I will ask you to come to my office about every 4-5 weeks, tell me some exercises you have done, and be prepared to explain to me in detail the solution to one or two of them (consulting any notes you have, if you wish.)

Option 2: presentation + consulting. This will involve making some sort of presentation (details to be determined) about some research-related topic. For example, if you are an advanced graduate student working on some research topic that has some connection to GMT, you might explain to me something about that; or you might read a paper and tell me about it. Consulting means that you should also be available to answer questions, or at least to discuss questions, with students who are pursuing option 1.

If you prefer option 2, then please talk to me about possible presentation topics within the first 4-5 weeks of the term.

Texts

• Lecture notes on Geometric Measure Theory by L. Simon;
  concise lecture notes that provide a good introduction to fundamental results about currents and varifolds, and some aspects of regularity theory for area-minimizing currents.
• Measure Theory and Fine Properties of Functions by L. C. Evans and R. Gariepy;
  contains a very clear discussion of general measure theory in Euclidean spaces and some topics related to geometric measure theory (eg, area and coarea formulas, BV functions) with more details than many of the other references listed here.
• Cartesian Currents in the Calculus of Variations, vol. I by M. Giaquinta, G. Modica, and J. Soucek;
  The first two chapters contain a discussion of general measure theory and an introduction to the theory of rectifiable currents, with useful examples, and some modern improvements on classical proofs as found in the books of Federer and Simon. (However, it also has a lot of mistakes.) Later chapters also interesting but less relevant for this course.
• Geometric Measure Theory by H. Federer
  the classic, encyclopedic reference on the subject, giving a complete overview of the field as it was in 1969. Not always easy to read and in some ways dated, but also in many ways still unsurpassed.
• some papers, in particular "Currents in metric spaces", by L. Ambrosio and B. Kirchheim, Acta. Math., 2000.