Periodic Reporting for period 1 - ModSingLDT (Moduli Spaces associated with Singularities)
Reporting period: 2021-01-01 to 2022-12-31
Moduli spaces are central objects of modern algebraic geometry. They have a strong relationship to natural enumerative questions. In many cases they are associated to a base variety and they contain a large amount of information about the geometry and topology of this base. In this project we investigated moduli spaces that were associated with singular spaces. These spaces have special points, or 'singularities', where the behavour of the space is different from what is expected. As our problems were in pure mathematics, our work is in general important for the wider scientific community including researchers in other areas such as theoretical physics or pure algebra. The main aim of the project was to find enumerative invariants of certain moduli spaces, for example Hilbert schemes and to obtain mathematical results on them.
The project proposed to contribute to enumerative invariants of the Hilbert schemes parametrizing zero-dimensional subschemes of some basic classes of surface singularities as well as of its analogues, and find connections between these enumerative invariants and vertex algebras. The project also made progress to the categorification of Heisenberg algebras and the quantum spectrum arising from Gromov-Witten invariants of surfaces.
We obtained several new and interesting results on Hilbert schemes and other moduli spaces. The research topic was at the meeting point of many different mathematical disciplines (such as enumerative algebraic geometry, mathematical physics, representation theory and so on). This interdisciplinary nature of the project makes it likely that the results will find their applications in various mathematical areas.