Welcome! This is the webpage for the student learning seminar on prismatic F-gauges at Northwestern University. The main goal is to learn about the recent development of the stacky approach (a.k.a. geometrization) in p-adic geometry and their applications.
Bhargav Bhatt’s Princeton notes on Prismatic $F$-gauges (2023)
Other References:
Bhatt-Lurie: Absolute prismatic cohomology (2022)
Bhatt-Lurie: The prismatization of $p$-adic formal schemes (2022)
Drinfeld: Prismatization (2020)
Drinfeld: A $1$-dimensional formal group over the prismatization of $\text{Spf}(\mathbb{Z}_p)$ (2021)
Li-Mondal: On endomorphisms of the de Rham cohomology functor (2021)
(Winter 2024) Mondays 3pm-4pm in Lunt 103
(Spring 2024) Wednesdays 12pm-1pm in Lunt 102
| Date | Topics | Reference | Speaker | |
|---|---|---|---|---|
| Week 1 | Jan. 22 | Introduction to the stacky approach, construction of $X^{dR}$ in char. 0, quasi-ideals | Yuanning Zhang | |
| Week 2 | Jan. 29 | Geometry of filtration, $X^{dR,+}$ and $X^{Hodge}$, Cartier duality between $\mathbb{G}_a^\sharp$ and $\widehat{\mathbb{G}_a}$ | (§2.2-2.4) | Yuanning Zhang |
| Week 3 | Feb. 5 | de Rham cohomology in mixed char., crystalline miracle, Witt vector models for $\mathbb{G}_a^\sharp$ and $\mathbb{G}_a^{dR}$ | (§2.5-2.6) | Junho Won |
| Week 4 | Feb. 12 | $X^{dR,c}$ and the Deligne-Illusie theorem | (§2.7) | Yuanning Zhang |
| Week 5 | Feb. 19 | Crystalline cohomology and Nygaard filtration | (§3.1-3.3) | Kirill Magidson |
| Week 6 | Feb. 26 | Mazur’s theorem of Newton-above-Hodge | (§3.4-3.5) | Junho Won |
| Spring break | ||||
| Week 7 | Mar. 27 | Syntomification and F-gauges in char. p | (§4.1-4.3) | Kirill Magidson |
| Week 8 | Apr. 3 | Serre duality in syntomic cohomology | (§4.4-4.5) | Junho Won |
| Week 9 | Apr. 10 | Serre duality in syntomic cohomology (cont.) | Junho Won | |
| Week 10 | Apr. 17 | Prismatization in mixed char. and WCart | (§5.1 and APC) | Yuanning Zhang |
| Week 11 | Apr. 24 | Admissible W-modules and filtered prismatization in mixed char. | (§5.2-5.5) | Yuanning Zhang |
| Week 12 | May 1 | Syntomification in mixed char. | (§6.1-6.2) | Kirill Magidson |
| Week 13+ | • Galois representations and F-gauges (§6.3-6.6) | |||
| • B. Bhatt’s Crystals and Chern classes | ||||
| • A. Petrov’s work on the Deligne-Illusie theorem and the Sen operator | ||||
| • Haoyang Guo and Shizhang Li’s Frobenius height of prismatic cohomology with coefficients | ||||
| • V. Vologodsky’s work on Fontaine-Laffaille modules | ||||
| • V. Drinfeld’s work on Barsotti-Tate groups | ||||
| • Toby Gee’s work on mod p crystalline Breuil-Kisin modules |