# Mathematical Analysis for PhD - Students

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## Analysis for all!

This course is intended as a PhD level course in analysis for new PhD students and for PhD students not specializing in mathematical analysis or in any closely related field. We will cover the basics of advanced analysis: measure theory and integration, some functional analysis and the theory for the weak derivative (the fundamental theorem of calculus done right!).

An important part of the course will be to motivate the results that we cover. We will not just follow a book and memorize the theorems, proofs and definitions. Instead we will pick a very interesting problem in mathematical analysis (minimization of the Dirichlet energy) whose solution requires much abstract analysis. Throughout the course we will use the minimization problem to motivate the theory we develop. The idea is that you should not only learn the theory; but also be able to motivate why we develop the theory and why abstract analysis is needed and natural when solving advanced problems. Hopefully this approach will allow everyone to get a feel for analysis and for some basic research methods.

## Course plan.

Pre-course assignment: A homework assignment will be distributed before the course starts. This assignment is pedagogical and you are not supposed to solve it. The purpose of the assignment is to introduce the minimization problem of the Dirichlet energy and to give you the chance to discover its difficulties on your own. You should spend, maybe, 1.5-2 hours on it and discover that is is a difficult problem. During the first lecture we will discuss the research program we will use to solve it. Here is a link to the pre-course assignment.

Lectures: Fridays 10-12 in room 3418 (1st Lecture 2nd September, last lecture December).

Plan of the lectures (with suggested exercises, et.c. VERSION 1st December) The notes from 25th of November contains notes on the fundamental Theorem of Calculus.

Lectures 2 and 3 - what I think everyone should know about measures. (Changed Due Date, Hand in at the day of the final lecture!!!)

The lecture plan contains suggested reading and exercises as well as your assigned reading. The current version is semifinal: no new material will be added (except maybe exercises) but typos (which are legion) will be fixed and I might clarify some passages in the future.

## Examination

During the course we will have two homework assignments and we will end the course with an oral exam. Participation during the lectures is expected (exceptions can be made).

The oral exam will take place on the 9th of January 14-17 in room 3418 and on the 10th of January 14-17 in room 3806. You need to sign up for a time so if you couldn't attend the final lecture and haven't signed up you need to contact me. The oral exam will be based on the following questions:

Questions to study before the oral exam.

First Set of Mandatory Homework Assignments  (Corrected 12:09 Friday the 11th Nov) Due on the 18th of November!!!!

Second Set of Mandatory Homework Assignments

## Course literature:

Most results from the course will be found in many good analysis textbooks. However, I have been unable to find one book that contains all that we need. In order to deal with this I have chosen to assign reading from Walter Rudin's Real and Complex analysis (which also will be used in a reading course at KTH so some of you will have to buy it in any case). Most stuff in the course can also be found in Royden and Fitzpartick's Real analysis (a book I personally like better, but it is somewhat more advanced than Rudin). The course material that is not covered in these books (such as Sobolev spaces) will be provided as lecture notes.

## My Lecture notes

Lecture 1 (Background and introduction to the problem).

Lecture 2 and 3 (This contains the measure theory we will cover, it will take two lectures, the original plan was one)

Lecture 4

Lecture 5

Lecture 6