Mathematics Colloquia and Seminars

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Classification problems are ubiquitous and related to all areas of mathematics in one way or another. I will explain why viewing moduli problems as functors allows one to recover the structure of a variety on the set of isomorphism classes of objects, and then I will talk about modern methods of studying moduli problems. The modern theory "Beyond GIT" provides a "coordinate-free" way of thinking about classification problems. Among giving a uniform philosophy, this allows to treat problems that can't necessarily be described as global quotients.

For example, when studying objects in an abstract abelian category, one doesn't know a priori if the set of isomorphism classes can be expressed as a set of orbits of a group acting on a variety, e.g., an affine space. Yet, one often encounters moduli problems in an abstract setting, e.g., when studying Bridgeland stable objects. While we cannot tackle this general case, we show how the methods of BGIT can be applied to prove existence and projectivity of moduli spaces of objects in a class of abelian categories. This is based on a joint work in progress with Andres Fernandez Herrero, Emma Lennen.

Further, applying these methods to moduli of quiver representations allows us to focus on geometry and obtain new results. I will define a determinantal line bundle which descends to a semiample line bundle on the moduli space and provide effective bounds for its global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus the adequate moduli space is projective over an arbitrary noetherian base. This part is based on a preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka (https://arxiv.org/abs/2210.00033).