The development of classical Mathematical Analysis into several branches of contemporary Analysis has led to many interesting results, both from a theoretical and applied perspective. Established techniques and tools in Analysis as well as new techniques have been combined in a harmonious way to solve many previously unsolved problems. The fundaments of modern mathematics, in general, and Analysis in particular are classical tools and approaches. With this course, we plan to systematically study several classical topics in Analysis. Hence the course is intended for all PhD students, and will encompass several areas in mathematical analysis. The core idea is to see how Analysis can be applied in different branches of mathematics. The chosen topics may vary from one year to another, depending on the examiner. The topics will include several main areas in Analysis, with contemporary areas of applications. The areas covered range over the following topics: Functional Analysis, Geometric Measure Theory, Ergodic Theory, Probabilistic Techniques, Harmonic Analysis on Groups, Sobolev Spaces. In each of these topic, besides the basics, we shall have a few directions to study further at deeper level.
Information for research students about course offerings
Content and learning outcomes
The course will include (some or all of) the following areas, but may well be expanded, depending on participants and examiner.
Functional Analysis, Geometric Measure Theory, Ergodic Theory, Probabilistic Techniques, Harmonic Analysis on Groups, Sobolev Spaces.
Intended learning outcomes
After completing the course:
Students should have general knowledge of several classical topics in Analysis and their applications in other areas of mathematics.
Students are supposed to have in-depth knowledge of at least one area outside their own research, and its connection to other areas.
Students should be familiar with technical tools from the areas represented at the course.
Students should have a heuristic overview of the topics given at the course.
Lectures and presentation, self-study, homework.
Literature and preparations
Attendance are required to have god knowledge in Analysis and Algebra at master level, and some basic probability theory.
Robert Zimmer: Functional analysis
Pertti Mattila: Geometry of sets and measures in Euclidean spaces.
Peter Walters: An introduction to Ergodic theory
Richard F. Bass: Probabilistic Techniques in Analysis
Katznelson: An introduction to Harmonic Analysis.
Folland: A course in abstract harmonic analysis
Bressan: Lecture Notes on Sobolev Spaces
R. Adams, J.F. Fournier: Sobolev Spaces
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
- PRO1 - Project work, 7.5 credits, grading scale: P, F
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.
Examination will consist of one topic presentation by students, as well as homework.
Other requirements for final grade
Approved homework assignments, and oral presentation of a project with written report.
Opportunity to complete the requirements via supplementary examination
Opportunity to raise an approved grade via renewed examination
- 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.
Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.Course web FSF3631