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An isocrystal is a type of linear algebraic datum that describesÌý various objects arising in arithmetic algebraic geometry. On the one hand, they have a relatively concrete description as vector spacesÌýÌý
over a field together with some special endomorphisms. This makes themÌý very amenable to computation. On the other, they are linked to veryÌý important structures in Arithmetic Geometry such as p-divisible groupsÌý and the crystalline cohomology of algebraic varieties. This dualÌý nature makes them indispensable tools for tackling many problems. InÌý this talk, we will define the category of isocrystals over a field ofÌý characeristic p, explain its basic properties, and give someÌýÌý
applications.