From moduli spaces to algebras - A survey of modern Donaldson-Thomas theory
Classifying objects has a long history in mathematics. Moduli spaces turned out to be a useful tool if the classification is wild, i.e. if the answer is not a discrete list as for Lie algebras. On the other hand, moduli spaces are very hard to understand from a geometrical point of view. Hence, replacing them with related vector spaces might simplify the subject, in particular as these vector spaces should carry an algebra structure inherited from a similar structure on moduli spaces. In the first part of my talk I'll sketch this strategy for easier classification problems for which we can use the cohomology of moduli spaces. This includes the classification of vector bundles on algebraic curves or quiver representations. In the second part we'll try to extend this machinery to include classification problems motivated by string theory involving sheaves on Calabi-Yau 3-folds, representations of quivers with potential or flat connections on oriented real 3-folds. This is joint work with Ben Davison and based on work of Kontsevich/Soibelman and Joyce.