In this lecture series I will explain how one can use deformation theory to study derived categories in positive characteristic.
I will start by giving an overview on what does it mean to 'lift' something 'to characteristic 0' and when is this possible. Then I will present a baby example: the study of the Fourier-Mukai partners of products of elliptic curves over algebraically closed fields of characteristic at least 5. After that, I will present Lieblich-Olsson deformation technique which allows us to deform derived equivalence. This is a very versatile tools with many applications (not just in positive characteristic!). I will conclude the series by going over some of these applications in greater details.
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