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Stability and wall-crossing in algebraic and differential geometry

Final Report Summary - STABAGDG (Stability and wall-crossing in algebraic and differential geometry)

The ERC StG project StabAGDG focused on new research directions in geometry, originating from the study of close analogues of the Einstein equations for gravity and of the Yang-Mills equations in particle physics.
Powerful mathematical and physical methods allow to approach the circle of questions around these equations via the two key concepts of stability (selecting a natural class of candidates for solutions) and wall-crossing (how stability changes with the natural parameters of the theory, a rigid analogue of phase transition).
We obtained a better mathematical understanding of both stability and wall-crossing, and we found new ways of using the relevant calculations in theoretical physics to prove new unexpected results in algebraic and differential geometry.
Notable outcomes include the emergence of new refined enumerative invariants counting algebraic curves from wall-crossing and an innovative use of infinite-dimensional analogues of classical ordinary differential equations in the study of enumerative aspects of Calabi-Yau geometries.