Geometry of singularities and homotopy theory

I will outline a recent proof of the Donovan–Wemyss Conjecture, which strikingly links the birational geometry of threefold singularities with derived equivalences of finite-dimensional algebras. Central to this approach is a homological reinterpretation: contraction algebras arising from crepant resolutions of cDV singularities can be realized as derived endomorphism algebras of 2Z-cluster tilting objects in the singularity category. Leveraging the derived Auslander–Iyama correspondence, we deduce that derived equivalences of such algebras reflect isomorphisms of the underlying singularities. A key player in this story is the restricted universal Massey product, which acts as a fine invariant distinguishing derived structures. This is joint work with Gustavo Jasso and Bernhard Keller. This talk aims to showcase how geometry, representation theory, and homotopy theory intertwine in proving a deep and elegant conjecture.