From the inverse Galois problem to Brauer–Manin obstruction

Thời gian: 16:30-1800, thứ năm, 03/08/2023

Hình thức: Offline tại phòng 612 A6 và online qua google meet, cụ thể https://meet.google.com/yep-kbzk-eao?pli=1&authuser=1

Tóm tắt: Galois theory associates with each finite separable normal field extension a finite group. A basic yet widely open question is the "inverse Galois problem", which asks if a given finite group is realizable as a Galois group of a given ground field. We know, after the various works of Hilbert, Noether and Ekedahl, that over number fields, this problem can be reduced to arithmetico-geometric questions on homogeneous spaces of connected linear algebraic groups. This talk aims to present a brief introduction to the inverse Galois problem and its variants, namely the regular inverse Galois problem, Noether's problem, Grunwald's problem, and retract rationality. Known methods and results shall be discussed. We also mention the connection between these topics and the study of Brauer-Manin obstruction on rationally connected varieties, Colliot-Thélène's conjecture, as well as the descent conjecture.

Reference: https://arxiv.org/abs/2302.13719.