Derived invariance by syzygy complexes

We study derived invariance through syzygy complexes. In particular, we prove that syzygy-finite algebras and Igusa--Todorov algebras are invariant under derived equivalences. Consequently, we obtain that both classes of algebras are invariant under tilting equivalences. We also prove that derived equivalences preserve AC-algebras and the validity of the finitistic Auslander conjecture.

Two papers which changed my life: Mil nor's seminal work on flat manifolds and bundles

This chapter surveys developments arising from John Milnor's 1958 paper, “On the existence of a connection with curvature zero” and his 1977 paper, “On fundamental groups of complete affinely flat manifolds.” The former deeply influenced the theory of characteristic classes of flat bundles, and the latter clarified the theory of affine manifolds, setting the stage for its future flourishing. This chapter begins with some reminiscences on Milnor. It then describes the history of the Milnor–Wood inequality and the Auslander Conjecture and then proceeds to more recent developments, including a description of Margulis space-times, a startling example of an affine three-manifold with free fundamental group.