Wall-crossing: from classical algebraic geometry to differential geometry, mirror symmetry and derived algebraic Geometry
Lead Research Organisation:
University of Cambridge
Department Name: Pure Maths and Mathematical Statistics
Abstract
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.
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.
Organisations
Publications
F. Rezaee
(2024)
Conjectural criteria for the most singular points of the Hilbert schemes of points
in Experimental Mathematics
F. Rezaee; M. Swaminathan
(2023)
Constructing smoothings of stable maps
in Preprint on arXiv ("author's accepted manuscript")
Rezaee F
(2025)
Constructing smoothings of stable maps
in Advances in Mathematics
Rezaee F
(2024)
An interesting wall-crossing: failure of the wall-crossing/MMP correspondence
in Selecta Mathematica
Rezaee F
(2024)
Geometry of canonical genus 4 curves
in Proceedings of the London Mathematical Society
Rezaee F
(2024)
Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points
in Experimental Mathematics
Rezaee F, Swaminathan M
(2024)
An obstruction to smoothing stable maps
| Description | -Finding and proving a stunning and surprisingly simple closed formula for computing the evaluation of the Behrend function at the singular points of any affine scheme in any ambient space, which has been very subtle to date, even in zero dimensions. (The paper on this is being finalized.) -Resolving the problem of listing the full wall-crossing for the Hilbert scheme of n points in projective plane for any n, which was previously known only for n at most 9. The method here used the structure of the scattering diagram developed by the fellow and the supervisor since the beginning of this fellowship (the papers on these being finalized). -Making excellent progress towards resolving an almost 50-year open problem suggested by Briancon and Iarrobino in 1978 regarding the singularities of the Hilbert schemes of a tetrahedral number of points. The fellow (conjecturally) generalized the old conjecture to any number of points (the paper on this has already been published) and started working on the proof with summer research students. In joint work, the fellow and the students further (conjecturally) generalized the fellow's conjectures. The paper on this is being finalized. -Making significant progress regarding the smoothability of stable maps problem: the fellow and a collaborator gave effective sufficient and also necessary conditions for a generalized notion of eventual smoothability. The paper on the sufficient condition has already been published, and the paper on the necessary condition has been submitted for publication. |
| Exploitation Route | The findings can be taken forward, respectively, as follows: -Behrend function formula: One future direction would be to look at the moduli space of schemes and try to compute the Behrend function of the moduli space from the Behrend function of the individual elements. Anyone working on calculating sheaf theoretic invariants in enumerative geometry could be interested in such an investigation. -Scattering diagram: The development of the geometry of the scattering diagram for P^2 can be extended to higher dimensions, which is the next project the fellow and the supervisor will investigate. -Singularities of the Hilbert scheme: The fellow and her brilliant summer research students plan to work on proving the conjectures made by the fellow. -Smoothability: The fellow and her collaborator on this project plan to use the results and move it forward to ultimately give a more optimal compactification than the stable maps compactification. |
| Sectors | Other |