Wall-crossing: from classical algebraic geometry to differential geometry, mirror symmetry and derived algebraic Geometry

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This project involves the field of algebraic geometry and connections to other areas including algebraic topology, symplectic/diffen geometry, and programming. I use my expertise in Bridgeland stability conditions and wall-crossing, and also use some modern tools to tackle this project which is related to very active research subjects in today's pure mathematics. Half of this project studies so-called moduli (or parameter) spaces which are fundamental in algebraic geometry. One effective way to study moduli spaces is via wall-crossing. As the wall-crossing method can get easily un-controllable in higher dimensions, I propose to incorporate the technology from the modern language of "Derived Algebraic Geometry" and also "Quadratic integer Programming" to make the process more tractable. Then I will use this in studying some moduli spaces which are called Hilbert schemes. Another half of the project is connecting algebraic geometry to differential geometry and mirror symmetry via Bridgeland stability conditions to produce some new Bogomolov-Gieseker type inequalities (which will be a step towards extending the celebrated Hitchin-Kobayashi type correspondence to Bridgeland stability conditions, which is very interesting to differential geometers). On the other hand, we take a further step and relate these to mirror symmetry via finding some corresponding "mirror inequalities", which in turn would shed light on the solvability of special Lagrangian type equations (which are very important to symplectic geometers these days), and also could help to find mirror equations to some Partial Differential Equations which cannot be tackled directly. The theory of Bridgeland stability is well established in Europe and the applications as above are considered as cutting edge in the area. During this fellowship, I will also learn other skills e.g. teaching, supervision, interviewing, etc, which will be important for my next career which will be hopefully a good academic job.