Triangulated structures in algebra, geometry, and topology

    PI: Fernando Muro

Abstract: Triangulatec categories play a central role in several areas of mathematics and theoretical physics. However, this algebraic structure is not yet known well enough. There are upsetting results related to invariants such as K-theory: we know there cannot be any reasonable K-theory for triangulated categories, yet we can recover the K-theory of ring from its derived category. Triangulated categories arise in nature as localizations of more complex structures, called models, whose essence they try to capture. For a long time, it was believed that all triangulated categories had models, but we recently found counterexamples. The goal of this project is measuring the distance between triangulated categories and their models, finding where in the midway invariants such as K-theory can be computed. In order to reach this goal we will study the structures interpolating between models and triangulated categories. We will treat this problem from a geometric perspective by means of moduli spaces in homotopical algebraic geometry.