Neeman has recently initiated the study of (triangulated) categories equipped with a "metric". This allows for notions such as Cauchy sequences and completions from the classical theory of metric spaces. It turns out that this simple idea has many interesting applications. In particular, it has been used to settle several conjectures, and prove many interesting results with applications mainly in (non-)commutative algebra and algebraic geometry, although there are connections with homotopy theory and representation theory also.
In this talk, we will give a broad overview of these techniques, and illustrate various applications. In particular if time permits, we will see a representability theorem and its application to a new proof of GAGA theorems, a result on transferring semiorthogonal decompositions, and a result on the non-existence of bounded t-structures for perfect complexes on singular finite-dimensional Noetherian schemes.
