Spring 2023, Thursdays, 13:15--15:00

Lecture room 3721, Lindstedtsvägen 25.

Instructors: :

Danijela Damjanovic (KTH)

Henrik Shahgholian (KTH)

The course is intended for all PhD students, and will encompass many areas in mathematics at a reasonable level.

The core idea is to see how Analysis can be applied in different branches of mathematics.

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.

Examination: (Dates/deadlines will be announced later)

The examination is done in a slightly unconventional way, as explained during the first lecture.

1) Each participant chooses a topic and will be called "topic-responsible" (TR) for that topic.

2) Each TR will have a presentation, approximately 70-80 min. (We try to have non-overlapping topics, and it should be not too close to your own research-work.)

3) Each TR writes a report/note so all other participants will have a copy of it (pdf format, either well-handwritten or typeset).

4) Include exercise/homeworks in the report for other members.

5) Every participant will choose homework from at least 5 other TR to solve and hand in to the TR.

6) Each TR will grade the HW, and in case there is a need for make up, it can go one more round.

7) Each TR after grading hands in the list of students who did HW in his topic, along with the graded HW, to instructors, for a final check.

8) Your grade Pass/Fail will be judged based on all points above, and your active participation during the lectures of instructors, but more importantly the lectures of other TRs.

9) Time frames: Part 1) Instructors' lectures, Part 2) Consultation (mid March-mid April). Part 3) Presentations (late April-late May.)

Eligibility

Prerequisite is to have good knowledge in Analysis and Algebra at master level, and to have some basic probability theory.

Literature: (Notes to be added)

1) Geometric measure theory: [Notes] Pertti Mattila: Geometry of sets and measures in Euclidean spaces.

2) Functional Analysis:lecture 2, Functional Analysis I, Simon-Reed.

3) Probabilistic Techniques: [Notes], Books: An Introduction to Stochastic Differential Equations (Lawrence C. Evans), Probabilistic Techniques in Analysis (Richard F. Bass)

4) Ergodic Theory: Peter Walters: An introduction to Ergodic theory. SF3631_ Ergodic theory.pdf

5) Sobolev Spaces: A.Bressan:[Notes], Lecture Notes on Sobolev Spaces , Links to an external site.R. Adams, J.F. Fournier: Sobolev SpacesLinks to an external site. See also Fractional Sobolev spaces.

6) Harmonic Analysis on Groups: Folland: A course in abstract harmonic analysis. SF3631 Rep. theory.pdf

All students should upload their presentation as ONE pdf file here: Presentations (after all is uploaded, I can link files in the below list, for a more convenient access)

60 min. each, with 5 min. break. Observe the time and room below. Thursday and Friday we start at 12:15.

	Name	Chosen Topic	Presentation date		Upload solution to exercise (1 file only)
1)	Bernardo	Fractional Sobolev Spaces	April 19: 13:15-14:15 room 3424	presentation	upload solutions 1
2)	Jon-Magnus	Marstrands projection teorem	April 19: 14:20-15:20 room 3424	presentation	upload solutions 2
3)	Danai	Fourier analysis on polytopes	April 19: 15:25-16:25 room 3424	presentation	upload solutions 3
4)	Sven	Applications of ergodic theory to additive combinatorics	April 20: 12:15-13:15 room 3721	presentation	upload solutions 4
5)	Federica	Central limit theorem in dynamics	April 20: 13:20-14:20 room 3721	presentation	upload solutions 5
6)	Nedialko	Ergodic theorems for group actions	April 20: 14:25-15:25 room 3721	presentation	upload solutions 6
7)	Andres Franco	Generalized implicit function theorems on Frechet spaces	April 20: 15:30-16:30 room 3721	presentation	upload solutions 7
8)	Elmo	Haar measure on abelian locally compact topological groups	April 21: 12:15-13:15 room 3418	presentation	upload solutions 8
9)	Kirthana	Frostman's Lemma	April 21: 13:20-14:20 room 3418	presentation	upload solutions 9
10)	Rasmus	Schrödinger operators	April 21: 14:25-15:25 room 3418	presentation	upload solutions 10
11)	Jan McGarry	Applications of ergodic theory to the study of lattices	April 21: 15:30-16:30 room 3418	presentation	upload solutions 11