Constructive Category Theory with Applications to (non)commutative Algebra and Algebraic Geometry

In this lecture course I will demonstrate how category theory can be understood as an extremely powerful and flexible high-level algorithmic/programming language which enables an elegant constructive (re)organization of many concrete but also many abstract parts of algebra and geometry, which were thought to be beyond algorithmization. I will exemplify this point of view showing several applications including Chevalley's Theorem on images of rational maps, categories of finitely presented modules over computable rings, categories of coherent sheaves, spectral sequences, normal forms of linear PDE systems, homotopy and derived categories (the latter being the most elaborate way to compute cohomological invariants)