FSF3710 Advanced Topics in Differentiability and Integrebility 7.5 credits

Mathematical analysis is a rich and varied subject with many different techniques and methods. There is hardly any branch of mathematics that has not been influenced by the powerful methods of analysis. This course will cover a large number of these techniques. We will, in particular, cover real function theory (Sobolev, Lipschitz and BY-spaces), singular integrals (Calderon-Zygmund theory), convexity (Alexandrov's Theorem), some advanced techniques in partial differential equations (such as de Giorgi- Nash-Moser theory) and some convergence properties of Fourier series. After the course the student will have a good understanding of a variety of advanced topics in analysis.

After completing this course the student should:

• Have a good understanding of a variety of different subjects in mathematical analysis. The subject areas studied can, to a certain extent be influenced by the students own researchareas. But the understanding should include real function theory, singular integrals, convexity, de Giorgi- Nash-moset theory, convergence  properties of Fourier series.
• Be able to independently  read, understand and present advanced mathematics.
• Be able to discuss and synthesize mathematics.
• Be able to put the above mentioned subjects in perspective and have some insight of their possible applications.

This course is open for PhD students. It is desirable to have a solid background in mathematical  analysis (such as SF2713 Foundations of Analysis) and measure theory

(such as SF2743 Advanced Real Analysis). Some basic understanding of POE and Sobolev spaces is also desirable.

The course literature will be depend on the topics presented and will be agreed upon between the course participants and the course leaders. Individual chapters form the following books might be used:

L Caffarelli, X Cabre -Fully nonlinear elliptic equations

R Courant, D Hilbert- Methods of Mathematical Physics, Volume 1

M G. Crandall, H Ishii, P-L- Lions user's guide to viscosity solutions of second order partial differential  equations

M G. Crandall, H Ishii, P-L - Lions user's guide to viscosity solutions of second order partial differential  equations

L.C. Evans -Partial  Differential Equations

L C Evans, R F Gariepy- Measure theory and fme properties of functions

D Gilbarg, N Trudinger- Second order partial differential equations

E Giusti- Direct Methods in the Calculus of Variations

E Di Nezza, G Palatucci, E Valdinoci- Hitchhiker's guide to the fractional Sabalev spaces

J Maly, W Ziemer Fine regularity properties for solutions of elliptic PDEs

E Stein - Harmonic analysis

E Stein - Singular Integrals and Differentiability Properties of Functions

L Tartar- The General Theory of Homogenization

W Ziemer- Weakly differentiable functions Sobolev spaces and functions of bounded variation

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

If the course is discontinued, students may request to be examined during the following two academic years.

• Presentation of one topic during a seminar.
• Suggest home assignments  for the fellow students.
• Active participation  in seminars.
• Oral exam at the end of the course.

Oral exam.

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.