SCGA | Derived categories, stability conditions and geometric applications

Summary
Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. Although these objects have been studied for a long time, there are still lots of crucial open problems: If we are given a variety, can we embed it in other well-known varieties? For instance, can we find a ''nice'' surface which contains a given curve? If yes, how many such surfaces exist, and can we characterise them via some of the geometrical properties of the curve?

The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory, Bridgeland introduced the notion of stability conditions on derived categories. This topic has been highly studied due to its connections to various fields in mathematics and physics, and lots of ideas and techniques have been developed in the area.

Now is the time to employ the whole spectrum of modern tools in derived categories and stability conditions to solve so far intractable geometrical problems. My recent work proves that deformation of stability conditions and varying stability status of an object (wall-crossing phenomenon) are powerful new techniques for solving long-standing geometrical problems, that do not appear to involve derived categories. Surprisingly, stability conditions and wall-crossing truly provide the right context for studying those problems.


The main goal of this research programme is to draw upon ideas and tools in algebra, geometry and mathematical physics to describe some outstanding geometrical problems in terms of derived categories and stability conditions, and then apply wall-crossing techniques to solve those problems.
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More information & hyperlinks
Web resources: https://cordis.europa.eu/project/id/887438
Start date: 01-09-2021
End date: 31-10-2023
Total budget - Public funding: 196 707,84 Euro - 196 707,00 Euro
Cordis data

Original description

Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. Although these objects have been studied for a long time, there are still lots of crucial open problems: If we are given a variety, can we embed it in other well-known varieties? For instance, can we find a ''nice'' surface which contains a given curve? If yes, how many such surfaces exist, and can we characterise them via some of the geometrical properties of the curve?

The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory, Bridgeland introduced the notion of stability conditions on derived categories. This topic has been highly studied due to its connections to various fields in mathematics and physics, and lots of ideas and techniques have been developed in the area.

Now is the time to employ the whole spectrum of modern tools in derived categories and stability conditions to solve so far intractable geometrical problems. My recent work proves that deformation of stability conditions and varying stability status of an object (wall-crossing phenomenon) are powerful new techniques for solving long-standing geometrical problems, that do not appear to involve derived categories. Surprisingly, stability conditions and wall-crossing truly provide the right context for studying those problems.


The main goal of this research programme is to draw upon ideas and tools in algebra, geometry and mathematical physics to describe some outstanding geometrical problems in terms of derived categories and stability conditions, and then apply wall-crossing techniques to solve those problems.

Status

TERMINATED

Call topic

MSCA-IF-2019

Update Date

28-04-2024
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Horizon 2020
H2020-EU.1. EXCELLENT SCIENCE
H2020-EU.1.3. EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (MSCA)
H2020-EU.1.3.2. Nurturing excellence by means of cross-border and cross-sector mobility
H2020-MSCA-IF-2019
MSCA-IF-2019