NWO-DFG Hilbert schemes: Modern Views and Synergies

• Applicant: Dr. Andreas Krug
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567118383

Moduli spaces are ubiquitous constructions in algebraic geometry. They describe classifications of objects when continuous parameters are involved. They are our most natural source of new algebraic varieties, constructed out of existing ones. Their geometric properties are reflected, and reflect upon, the properties of the objects used to define them. There are often connections to mathematical physics and representation theory, and they are a rich source of inspiration for new mathematical tools and questions. In this project, we focus on Hilbert schemes of points on curves and surfaces, parameterizing how n points on a variety can move around and come together. These are some of the most tractable examples of moduli spaces of sheaves, and they form the starting point for understanding more complicated moduli spaces. We will study them using modern tools from category theory and representation theory, with the main novelty being the extension of the usual input in the constructions to objects of a “quantum” (i.e., noncommutative) nature or objects having additional symmetries. In these extended settings we aim to establish suitable analogues of the foundational results in the classical case, relating to the structure and properties of these new moduli spaces. This leads to exciting synergies between different fields of mathematics, where these quantum or symmetric objects have been studied from other perspectives. We will moreover develop computational tools to facilitate the study of these moduli spaces, to build a firm foundation for future research in this area.

• DFG Programme: Research Grants
• International Connection: Netherlands
• Cooperation Partner: Professor Pieter Belmans, Ph.D.