Final Report Summary - WALLXBIRGEOM (Wall-crossing and Birational Geometry)
Algebraic Geometry studies the geometry of solutions sets of polynomial equations, called algebraic varieties. The ultimate goal of algebraic geometry is to classify algebraic varieties, and to understand relations between them, often in terms of understanding possible polynomials maps between them.
Algebraic Geometry has a long history of interplay between algebraic methods (exploiting properties of polynomials) and geometric properties. The proposal has helped build a new such bridge, between questions very much of classical flavour in algebraic geometry, and abstract categorical methods. In particular, the notion of stability conditions, originally introduced by Tom Bridgeland in order to give mathematical foundations to notions in theoretical physics, now has a number of deep but concrete and explicit in algebraic geometry. For example, a classical problem of algebraic geometry asks to understand explicitly the possible degrees and embedding dimensions of a given smooth algebraic curve (Riemann surface) into projective spaces; this is the topic of Brill-Noether theory. Stability conditions and wall-crossing, via work carried out in this proposal, now give a new approach to this topic, leading to more precise and powerful results for even the most basic questions in the field.
Algebraic Geometry has a long history of interplay between algebraic methods (exploiting properties of polynomials) and geometric properties. The proposal has helped build a new such bridge, between questions very much of classical flavour in algebraic geometry, and abstract categorical methods. In particular, the notion of stability conditions, originally introduced by Tom Bridgeland in order to give mathematical foundations to notions in theoretical physics, now has a number of deep but concrete and explicit in algebraic geometry. For example, a classical problem of algebraic geometry asks to understand explicitly the possible degrees and embedding dimensions of a given smooth algebraic curve (Riemann surface) into projective spaces; this is the topic of Brill-Noether theory. Stability conditions and wall-crossing, via work carried out in this proposal, now give a new approach to this topic, leading to more precise and powerful results for even the most basic questions in the field.