Higher Order Fourier Analysis reading course
What
A reading course run by Yaël Dillies, Zhichen Zhou and Othmane Rih where we learn together about the various aspects of higher order Fourier analysis, building up to a proof of Szemerédi’s theorem for $k = 4$.
Everyone is welcome to join!
The reading course consists of weekly online lectures/presentations as well as in-person exercise sessions. The online sessions are meant to form the core of the course. The in-person sessions, while making a nice addition, remain fully optional. After 9 student-run lectures based on Ben Green’s nilsequences notes leading up to Szemerédi’s theorem, there will be student-run presentations by groups of two.
Examination
PhD students at SU or KTH can take this course for 7.5 ETCS credits.
The examiner is Olof Sisask and to pass the course one needs to give a 2x45-min presentation in pairs on an important paper in higher order Fourier or an application to additive combinatorics, ergodic theory or model theory. Here are a few suggestions:
- A new proof of Szemerédi’s theorem, W. T. Gowers
- The primes contain arbitrarily long arithmetic progressions, B. Green, T. Tao
- Improved Bounds for Szemerédi’s Theorem, J. Leng, A. Sah, M. Sawhney
- A point of view on Gowers uniformity norms, B. Host, B. Kra
- An inverse theorem for the Gowers $U^{s+1}[N]$-norm, B. Green, T. Tao, T. Ziegler
Where
Online lectures/presentations: Zoom. Details to be confirmed.
In-person meetings: Mathematics Department, Albano House 1, Floor 3, Stockholm University. See the schedule for the room.
See the location on Open Street Map.
When
The schedule beyond the first meeting is provisional, and will be decided upon during the first meeting.
| Week | Time UTC+2 | Room | Title |
|---|---|---|---|
| v 38 | 17/09 11-12 | Cramér | First meeting |
| v 39 | 23/09 11-12 | TBD | Lec 1 - Sec 1.1-1.2: Nilsequences, Heisenberg group |
| v 39 | 25/09 14-16 | TBD | Exercise session 1 |
| v 40 | 30/09 11-12 | TBD | Lec 2 - Sec 1.3+2.1: Gowers norms, filtration |
| v 40 | 02/10 14-16 | TBD | Exercise session 2 |
| v 41 | 07/10 11-12 | TBD | Lec 3 - Sec 2.2+2.4-2.5: Lattice automorphic functions, inverse theorem, linear nilsequences |
| v 41 | 09/10 14-16 | TBD | Exercise session 3 |
| v 42 | 14/10 11-12 | TBD | Lec 4 - Sec 3.1: Host-Kra cube groups |
| v 42 | 16/10 14-16 | TBD | Exercise session 4 |
| v 43 | 21/10 11-12 | TBD | Lec 5 - Sec 3.2-3.3: Polynomial sequences, Taylor expansions |
| v 43 | 23/10 14-16 | TBD | Exercise session 5 |
| v 44 | 28/10 11-12 | TBD | Lec 6 - Sec 4.1-4.2: Differentiating nilsequences |
| v 44 | 30/10 14-16 | TBD | Exercise session 6 |
| v 45 | 04/11 11-12 | TBD | Lec 7 - Sec 5.1: Gowers norm on cyclic groups |
| v 45 | 06/11 14-16 | TBD | Exercise session 7 |
| v 46 | 11/11 11-12 | TBD | Lec 8 - Sec 5.2-5.3 Gowers norm on $[N]$, nilsequence obstruction |
| v 46 | 13/11 14-16 | TBD | Exercise session 8 |
| v 47 | 18/11 11-12 | TBD | Lec 9 - No ref yet: Szemerédi’s theorem |
| v 47 | 20/11 14-16 | TBD | Exercise session 9 |
| v 48 | 25/11 10-12 | TBD | Presentation 1 - TBD |
| v 49 | 02/12 10-12 | TBD | Presentation 2 - TBD |
| v 50 | 09/12 10-12 | TBD | Presentation 3 - TBD |
| v 51 | 16/12 10-12 | TBD | Presentation 4 - TBD |