Kakeya sets and applications

(Reading seminar Winter 2021-2022)

Phan Thành Nam - Uni2work

General Information

Introduction: A Kakeya set is a subset in Euclidean space which contains a unit line segment in every direction. In 1919, Besicovitch showed that there are Kakeya sets which are very small, namely they are of measure zero. On the other hand, the Kakeya conjecture states that Kakeya sets are not too small, namely every Kakeya set in n-dimensional space must be n-Hausdorff dimensional. In the seminar, we will discuss basic properties of Kakeya sets as well as some interesting applications in real and harmonic analysis.

Audience : Bachelor and Master students of Mathematics and Physics.

Time and place: Thursday 10:00-12:00, room B 252. The next meeting takes places on November 11th.

Schedule:

28.10.2021. Introduction. Plan.

11.11.2021. Phan Thành Nam: Besicovitch-Perron’s construction of Besicovitch's sets.

18.11.2021. Martin Weirich: Baire category theorem and Korner's construction of Besicovitch's sets.

25.11.2021. Charlotte Dietze: Fourier transform, the disc conjecture and the Bochner-Riesz conjecture.

20.01.2022. Sean Disarò: Hausdorff dimension, Minkovski dimension, and Kakeya set conjecture.

27.01.2022. Jinyeop Lee: Kakeya maximal function conjecture

03.02.2022. Lukas Kienesberger: Schrödinger equation and the local smoothing conjecture (cancelled)

10.02.2022. Yunseok Lee: Kakeya sets in vector spaces over finite fields