Moduli Spaces associated with Singularities

Project Information

ModSingLDT

Grant agreement ID: 891437

DOI

10.3030/891437

Project closed

EC signature date 31 August 2020

Start date 1 January 2021

End date 31 December 2022

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 151 850,88

EU contribution

€ 151 850,88

Coordinated by

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Hungary

Periodic Reporting for period 1 - ModSingLDT (Moduli Spaces associated with Singularities)

Reporting period: 2021-01-01 to 2022-12-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

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