Warwick EPSRC Symposium on Derived Categories and Applications

沃里克 EPSRC 派生类别及应用研讨会

基本信息

批准号：
EP/L018314/1
负责人：
Miles Reid
金额：
$ 20.37万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FL018314%2F1
关键词：
Warwick EPSRC Symposium Derived Categories

项目摘要

The planned 2014-2015 Warwick EPSRC symposium is a year long concentrated activity on the theory and applications of derived categories. The subject of derived categories emerged in the second half of the 20th century as a distillation of the ideas of homological algebra, which calculates invariants of a topological space such as its "number of n-dimensional holes". While more high brow and abstract than the more primary methods of attaching invariants to a mathematical or physical object, the derived category has a number of important advantages, allowing us to see so-called "quantum symmetries" of manifolds that are inaccessible to more conventional theories. They are thus an essential ingredient of attempts to understand the mathematics of physically important theories such as string theory, mirror symmetry and supersymmetry.Starting from the top, the more theoretical aspects covered during the year involve abstract notions such as higher category theory, DG enhancements and derived geometry. These are substantial generalisations of conventional geometry and category theory, and the theory is currently at the level of understanding and standardising the foundations of the new subject. This is an exciting stage in the development of a mathematical theory, but not one that can be convincingly explained in simple terms. Our symposium will run several schools and workshops at different levels expanding on these matters.At the other extreme, derived categories feed back into explicit calculations that can be applied to give useful results describing the properties of usual objects of algebra, geometry and theoretical physics. For example, derived categories have provided by far the best treatment of the McKay correspondence, that relates the representation theory of a finite subgroup G in SL(2,CC) or SL(3,CC) with the topology of a resolution of the orbifold quotient CC^n/G. In a similar vein, our symposium will include workshops studying derived category approaches to the study of different moduli spaces and their invariants (such as the classical moduli spaces of vector bundles, or of algebraic curves).In between these two extremes is a rich body of theories and problems in algebra, geometry and physics to which it is known or suspected that derived category methods can be applied. This includes issues arising from string theory, such as homological mirror symmetry, that works around the conjecture that the derived category mediates between the complex geometry of a Calabi-Yau 3-fold and the symplectic geometry of its mirror partner.

计划中的 2014-2015 年沃里克 EPSRC 研讨会为期一年，集中讨论派生类别的理论和应用。派生范畴这一主题出现于 20 世纪下半叶，是同调代数思想的升华，它计算拓扑空间的不变量，例如“n 维孔的数量”。虽然比将不变量附加到数学或物理对象的更主要方法更高级和抽象，但派生类别具有许多重要的优点，使我们能够看到流形的所谓“量子对称性”，这是更传统的方法无法实现的理论。因此，它们是尝试理解弦理论、镜像对称和超对称等物理重要理论的数学的重要组成部分。从顶部开始，这一年涵盖的更多理论方面涉及抽象概念，例如更高范畴论、DG 增强和导出的几何图形。这些是传统几何和范畴论的实质性概括，该理论目前处于理解和标准化新学科基础的水平。这是数学理论发展中令人兴奋的阶段，但无法用简单的术语令人信服地解释。我们的研讨会将在不同级别上举办几所学校和研讨会，以扩展这些问题。在另一个极端，派生类别反馈到显式计算中，这些计算可用于给出描述代数、几何和理论物理的常见对象的属性的有用结果。例如，派生类别提供了迄今为止对 McKay 对应关系的最佳处理，它将 SL(2,CC) 或 SL(3,CC) 中的有限子群 G 的表示理论与轨道折叠的分辨率拓扑联系起来商 CC^n/G。同样，我们的研讨会将包括研究派生类别方法的研讨会，以研究不同模空间及其不变量（例如向量丛或代数曲线的经典模空间）。在这两个极端之间是一个丰富的主体已知或怀疑可以应用派生范畴方法的代数、几何和物理学的理论和问题。这包括弦理论引起的问题，例如同调镜像对称性，它围绕着这样的猜想：派生范畴在卡拉比-丘三重的复杂几何形状和其镜像伙伴的辛几何形状之间进行调解。