Homological algebra and data

These lectures are a quick primer on the basics of applied algebraic topology with emphasis on applications to data. In particular, the perspectives of (elementary) homological algebra, in the form of complexes and co/homological invariants are sketched. Beginning with simplicial and cell complexes as a means of enriching graphs to higher-order structures, we define simple algebraic topological invariants, such as Euler characteristic. By lifting from complexes of simplices to algebraic complexes of vector spaces, we pass to homology as a topological compression scheme. Iterating this process of expanding to sequences and compressing via homological algebra, we define persistent homology and related theories, ending with a simple approach to cellular sheaves and their cohomology. Throughout, an emphasis is placed on expressing homological-algebraic tools as the natural evolution of linear algebra. Category-theoretic language (though more natural and expressive) is deemphasized, for the sake of access. Along the way, sample applications of these techniques are sketched, in domains ranging from neuroscience to sensing, image analysis, robotics, and computation.

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