SCGA · Derived categories, stability conditions and geometric applications

Derived categories, stability conditions and geometric applications

Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory in physics, Bridgeland introduced the notion of stability conditions on derived categories. The main goal of this intradisciplinary research was to employ ideas and tools in algebra, geometry and mathematical physics to develop new techniques for solving long-standing problems in algebraic geometry. The central theme is the deformation of stability conditions on derived categories and wall-crossing. The main impact of this research programme will be to enhance the existing expertise in pure mathematics in Europe. My projects employed ideas and tools in different fields of mathematics to develop modern techniques for answering many fundamental questions in algebraic geometry. In particular, it strengthens existing links between algebraic geometry, homological algebra, birational geometry, symplectic geometry, enumerative geometry and mathematical physics, which are all very active areas of research in Europe. Hence my projects will increase the possibility of collaborations between researchers in these fields. A great occasion to transfer ideas and techniques between different research areas is intradisciplinary workshops which I have attended and delivered talks several times during my fellowship. These workshops will have a significant impact on Europe and international mathematical communities. Impact on the younger generation of scientists: I delivered several talks in Junior Geometry seminars and schools around the world. I have also introduced new projects to several Master/Phd students and I plan to supervise these projects. Although my research programme is focussing on topics in mathematics, it has the potential of influencing other sciences. Lots of objectives in the proposal are closely related to string theory, hence, the results and techniques would be helpful for physicists. Moreover, the second project involves a detailed study of stable vector bundles holding a specific number of global sections on algebraic curves, which will be useful for algebraic-geometric codes in coding theory.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз