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Derived categories stability conditions and geometric applications

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... ver más

31/10/2023

UNIVERSITE PARISSA...

197K€

Presupuesto del proyecto: 197K€

Líder del proyecto

UNIVERSITE PARISSACLAY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2023-10-31

MSCA-IF-2019: Individual Fellowships

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UNIVERSITE PARISSACLAY

196.71K€ | Lider

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Duración del proyecto: 42 meses Fecha Inicio: 2020-04-22
Fecha Fin: 2023-10-31

Líder del proyecto

UNIVERSITE PARISSACLAY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 197K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto 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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