Community Research and Development Information Service - CORDIS



Project ID: 337039
Funded under: FP7-IDEAS-ERC
Country: United Kingdom

Mid-Term Report Summary - WALLXBIRGEOM (Wall-crossing and Birational Geometry)

Algebraic geometry is the study of geometric objects ("algebraic varieties") that have purely algebraic descriptions, as solutions sets of systems of polynomial equations: higher-dimensional generalisations and analogues of cylinders, spheres, ellipses, etc. It has many connections across all areas of pure mathematics; recently it has also become central to many applications, and important for computational tools. It also became widely used in string theory over the last 20 years, where objects of algebraic geometry were used to describe compact directions of space-time.

The gist of this proposal is to reverse this transfer of methodologies: it uses methods that were developed in the last decade with a view towards string theory and applies them in order make progress on the fundamental question of algebraic geometry: the classification of algebraic varieties, and maps between them.

The key such method is the notion of stability conditions on derived categories. The project has, on the one hand, dramatically improved the foundations stability conditions in various ways; on the other hand, it has shown a number of deep and useful connections to classical topics in algebraic geometry. It has led to a much deeper and more complete understanding of the geometry of moduli spaces; this is substantial progress towards the most basic question of algebraic variety: the classification of algebraic varieties, and maps between them.
The latest such application to classical algebraic geometry concerns the Brill-Noether theorem on the existence of rational functions on Riemann surfaces. The proof of the Brill-Noether theorem was one of the highlights of algebraic geometry in the 20th century. The results of the project give a completely new approach towards Brill-Noether statements, and have immediately added improvements and new perspectives to these classical results.

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United Kingdom
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