Wenhan Dai

p-adic Geometry Learning Seminar: On Banach-Colmez Spaces

The goal of this series of talks is to learn the basics of finite-dimensional Banach Spaces (aka. Banach-Colmez spaces). More details can be found on Shizhang’s organizing page. There is a small chance we might have to deviate from this schedule if some topics end up taking longer/shorter than we expect.

• Organizers: Koji Shimizu (Tsinghua) and Shizhang Li (MCM).
• Time: Every Monday, 3:00-5:00pm.
• Venue: MCM 110 & Online (Zoom ID: 4663562952; Password: mcm1234).

Please email Shizhang (lishizhang@amss.ac.cn) if you would like to join the mailing list for this seminar.

Schedule

• (9/5) Koji Shimizu: Introduction and organization; [Col02, Introduction]. (tentative plan)
• (9/19) Shizhang Li: Sympathetic algebras and $\widehat{\overline{C{X}}}$; [Col02, §2-3]. (notes)
• (9/26) Yongquan Hu: Sympathetic closure; [Col02, §5]. (notes)
• Meeting canceled on the National Day’s holiday.
• (10/10) Qijun Yan: A funny field $\mathfrak{C}$; [Col02, §4, 6.1-6.5]. (notes)
• (10/17) Yupeng Wang: Vector Spaces; [Col02, §6.6-6.9, 7.1-7.2]. (notes)
• (10/24) Zhefan Zheng: Finite-dimensional Banach Spaces; [Col02, §7.3-7.8]. (notes)
• (10/31) Daxin Xu: p-adic period rings; [Col02, §8]. (notes)
• (11/7) Yong-Suk Moon: Fundamental exact sequence; [Col02, §9]. (notes)
• (11/14) Zekun Chen: Weakly admissible implies admissible; [Col02, §11].
• (11/21) Jiedong Jiang: Banach-Colmez spaces; [LB18, §2]. (notes)
• (11/28) Ruochuan Liu: Pro-étale cohomology of affine spaces; [LB18, §3]. (notes)
• (12/5) Heng Du: The category $\mathscr{BC}$ in terms of the Fargues-Fontaine curve; [LB18, §5-7]. (notes)
• (12/12) Koji Shimizu: Overview and local syntomic computations; [CN17, §1-3]. (notes)
• (12/19) Shanwen Wang: Syntomic cohomology and $(\phi, \Gamma)$-modules; [CN17, §4]. (notes)
• (12/26) Yong-Suk Moon: Semistable comparison theorem; [CN17, §5]. (notes)

References

1. [Col02] Pierre Colmez, Espaces de Banach de dimension finie, J. Inst. Math. Jussieu 1 (2002), no. 3, 331-439. MR 1956055. (Note that this has different numbering than the one on his website.)
2. [LB18] Arthur-César Le Bras, Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine, Duke Mathematical Journal 167 (2018), no. 18, 3455-3532.
3. [CN17] Pierre Colmez and Wieslawa Nizioł, Syntomic complexes and p-adic nearby cycles, Inventiones mathematicae 208 (2017), no. 1, 1-108.
4. [WN19] Wieslawa Nizioł, Geometric syntomic cohomology and vector bundles on the Fargues-Fontaine curve, J. Algebr. Geom. 28 (2019), no. 4, 605-648.