WEEKLY ACTIVITIES

Thời gian: 16h30, thứ năm, 27/06/2024

Tóm tắt: Let (R,m) be a complete local ring with the residue field of characteristic 0. Our object of study is flat group schemes over Spec(R). And the question is: can we characterize all flat group schemes over R? This is a hard question andthe difficulty is that group schemes can be of infinite type. In the series of papers [1], [2], Duong, Hai, and dos Santos introduced the concept of infinite formal blow-ups and showed that those formal blow-ups are typical among flat group schemes of infinite type. That construction appeared before in [3].

The above question has an Inverse Galois Differential Problem version: which flat group schemes over R are differential Galois groups of some connections over R? In other words, (in case that R=C[[t]] a discrete valuation ring) given a flat group scheme G/R, could we find an abstract group Gamma which is a subgroup of G(R) such that the images of Gamma on both generic and special fibers of G are dense?

In this talk, I will recall the concept of prudence of a group scheme and give a partial answer to the above question. Someprecise results on the structure of formal prudent connection will be provided also.

References:

1. N. D. Duong, P. H. Hai and J. P. dos Santos. On the structure of affine flat group schemes over discrete valuation rings, I. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVIII (2018), 977-1032.
2. Phùng Hô Hai, João Pedro dos Santos, On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II, International Mathematics Research Notices, Volume 2021, Issue 12, June 2021, Pages 9375–9424, https://doi.org/10.1093/imrn/rnaa247
3. Prasad, Gopal & Yu, Jiu-Kang. (2004). On quasi-reductive group schemes. Journal of Algebraic Geometry. 15. 10.1090/S1056-3911-06-00422-X.

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