2-torials / Deformation theory
What happens if we take an algebraic structure — say, multiplication in an algebra — and wiggle it a little bit? Formalising what we could actually mean by this and then understanding the consequences will lead us to start exploring the theory of first-order deformations. Following this path will eventually lead us to naturally stumble upon certain chain complexes whose cohomology describe solutions to our stated problems. Eventually we will have all we need to understand why people are so interested in so-called Maurer–Cartan elements and DGLAs, and what any of this has to do with algebra or geometry.
This series was based essentially entirely on the first two sections of Balázs Szendrői’s 1999 tutorial notes entitled “The unbearable lightness of deformation theory”.