Bhargav Bhatt: Algebraic Geometry in Mixed Characteristic

Explore recent developments in algebraic geometry over p-adically complete commutative rings in this one-hour lecture by Bhargav Bhatt. Delve into topics such as Hensel's p-adic world, prismatic cohomology, semistable reduction, and connections to algebraic K-theory. Examine the shape of prismatic cohomology and its applications, including new calculations in algebraic K-theory and Kodaira vanishing in mixed characteristic. Gain insights into Serre's global-to-local dictionary and various variants and applications of Kodaira vanishing. Access accompanying slides for visual support and deeper understanding of the presented concepts.

Intro
Motivating goal
The geometric backdrop: Hensel's p-adic world (1997)
Progress from the 20th century
A 21st century upgrade: prismatic cohomology
The shape of prismatic cohomology
Semistable reduction
Connections to algebraic K-theory
New calculations in algebraic K-theory
Kodaira vanishing in mixed characteristic
Interlude: Serre's global-to-local dictionary
Kodaira vanishing: variants and applications