题目：Prudent affine group schemes over a complete discrete valuation ring and parameterized differential Galois groups

报告人：Dao Van Thinh, Institute of Mathematics, Vietnam Academy of Science and Technology

时间地点：2025年12月25日（周四）16：00-17：00， 数学新楼312

                 2025年12月30日（周二）09：00-10：00， 二教2302

摘要：An affine group scheme is completely determined by its coordinate ring, a commutative Hopf algebra. Among flat affine group schemes over a Dedekind ring, those with projective coordinate rings over the base ring enjoy many favorable properties. In this series of talks, we focus primarily on the case where the base ring is a complete discrete valuation ring (cDVR), and we investigate conditions under which an affine group scheme has a projective coordinate ring. One such equivalent condition, known as prudence (as introduced in [1]), can be verified on finite quotients of the base ring. In the first talk, I will introduce the notion of prudence for affine group schemes over a cDVR and discuss several equivalent characterizations. I will also explain how to verify this property in the context of differential Galois groups. In the second talk, I will present some of our most recent work in this area, concerning parameterized differential Galois groups. Specifically, we study an irregular formal connection over R((x)), where R is a cDVR, and ask whether the associated differential Galois (Tannakian) group scheme is prudent. The rank-one case is now completely understood, while the higher-rank case remains largely open. I will conclude by discussing several open problems.

References:

[1] P. H. Hai and J. P 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.