Perfectoid Spaces (Summer 2019)

Instructor: Daniel Gulotta
Location and time: Monday and Thursday 12pm-1pm in room C3, with some make-up sessions Tuesday 12pm-1pm; see below.
Prerequisites: I will assume some familiarity with p-adic numbers and with algebraic geometry.

References

Main references:

[SW1] P. Scholze, J. Weinstein, Berkeley lectures on p-adic geometry
[SW2] P. Scholze, J. Weinstein, Moduli of p-divisible groups
[W] J. Weinstein, Arizona Winter School 2017: Adic spaces

Additional references for the local Langlands part of the last lecture:

P. Colmez, G. Dospinescu, W. Niziol, Cohomologie p-adique de la tour de Drinfeld: le cas de la dimension 1
T. Gee, Modularity lifting theorems (section 2.6)
M. Harris, Supercuspidal representations in the cohomology of Drinfelʹd upper half spaces; elaboration of Carayol's program
P. Scholze, On the p-adic cohomology of the Lubin-Tate tower
T. Wedhorn, The local Langands correspondence for GL(n) over p-adic fields

Schedule of lectures

Date	Topic	References
29 Apr	Motivation, Huber rings, continuous valuations	[W] 1.1-1.5, 1.7, [SW1] 2
2 May	Adic spaces	[W] 1.6, 1.8, [SW1] 2-3
6 May	Adic spaces	[W] 1.8, [SW1] 3-4, 5.2
9 May	Adic spaces, perfectoid rings, tilting	[SW1] 4.4, 6
20 May	Tilting and untilting, sousperfectoid spaces	[SW1] 6
21 May^*	Perfectoid spaces, almost mathematics, untilts	[SW1] 7, [W] 2.4
23 May	Pro-étale site, diamonds	[W] 3.2-3.4, [SW1] 8
27 May	Analytic adic spaces and diamonds	[W] 3.4-3.5 [SW1] 9-10.1
28 May^*	Examples of diamonds	[W] 4.5, [SW1] 10.2
30 May	Fargues-Fontaine curve	[SW1] 11.2-11.3, 13.5 [W] 2.5, 4.2
10 Jun	Banach-Colmez spaces, p-divisible groups	[SW1] 15.2, [W] 4.4, [SW2] 4.1
11 Jun^*	p-divisible groups and vector bundles on the curve	[W] 2.6-2.7, [SW1] 15.2, [SW2] 5.1
14 Jun	Classification of p-divisible groups, Rapoport-Zink spaces	[SW2] 5.2, 6.1
17 Jun	Rapoport-Zink spaces	[SW2] 6.1-6.4, [SW1] 24.2
18 Jun^*	Period maps	[SW2] 6.1, 7.1 [SW1] 24.1-24.2
20 Jun	Duality of Rapoport-Zink spaces, local Langlands	[SW2] 6.5, 7.2, [SW1] 24.3

^*Location changes (time is still 12pm-1pm):

21 May: C5
28 May: L6
11 Jun: C5
18 Jun: C5