"Applications of Homological Algebra in Algebra, Geometry, and Physics"

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No two maple leaves are alike and neither are two snow flakes. However, nobody will mistake the former for the latter. This simple metaphor is at the basis of the deformation and classification theory of mathematical structures. Given an algebraic or geometric object that is specified by axioms, structural data, or equations, what remains of the structure and which aspects change if one (slightly) varies the defining parameters? The instruments and techniques mathematicians employ to answer these questions are contained in the ever expanding toolbox of homological algebra, which strives to classify that which is of equal shape or form, i.e. ``homological''. A coarse classification is usually obtained through discrete invariants, such as a dimension that may correspond to the number of spikes of a leaf; a finer one may necessitate continuously varying moduli that can be numbers again or manifest themselves through simpler structures. The metaphorical analogue here could be the shade of colour of a leaf. A deep fact of nature is that snow flakes have six spikes and that nothing with sevenfold symmetry can ever be a snow flake. Homological Algebra aims at similar fundamental results: the hunt for invariants and moduli employs highly abstract concepts such as triangulated categories, deformation groupoids, or controlling differential graded Lie algebras, leading to structures that have a simple general definition, but are difficult to exploit in any given concrete case. One thus looks for more manageable secondary invariants, among them cohomology groups, obstruction theories or concrete geometric objects such as discriminants or Frobenius structures. Applications range from classification theory in Algebra to fundamental questions in Geometry and Physics.