EPSRC Reference: 
EP/T018658/1 
Title: 
Derived categories, stability conditions and geometric applications. 
Principal Investigator: 
Feyzbakhsh, Dr S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Imperial College London 
Scheme: 
EPSRC Fellowship 
Starts: 
01 October 2020 
Ends: 
30 September 2023 
Value (£): 
405,383

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Geometry studies higherdimensional 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 wellknown 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 (wallcrossing phenomenon) are powerful new techniques for solving longstanding geometrical problems, that do not appear to involve derived categories. Surprisingly, stability conditions and wallcrossing truly provide the right context for studying those problems.
The main goal of this research programme is to draw on 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 wallcrossing techniques to solve them.

Key Findings 
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Summary 

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Organisation Website: 
http://www.imperial.ac.uk 