New geometry arises in the study of algebraic varieties over characteristic p fields; this course will be a largely example-driven introduction to some aspects of the theory. We will start by getting a feel for curves and surfaces, with topics including: the Weil conjectures for curves, supersingular elliptic curves, the classification of surfaces, supersingular K3 surfaces, counterexamples to Kodaira vanishing, and counterexamples to the Bogomolov inequality. We will also treat lifting to characteristic 0 in detail, particularly for curves and some surfaces. Finally, time permitting we will also discuss proofs in characteristic 0 using characteristic p methods, such as Deligne--Illusie's paper of the degeneration of the Hodge-to-de Rham spectral sequence.
The room has been changed to 410.