Warwick EPSRC Symposium on Derived Categories and Applications

计划于2014-2015年间举行的华威EPSRC研讨会是一项为期一年的关于派生范畴理论和应用的集中活动。派生范畴作为同调代数思想的升华出现在20世纪下半叶，它计算一个拓扑空间的不变量，如它的n维洞的数量。虽然比将不变量附加到数学或物理对象的更主要的方法更高雅和抽象，但派生范畴有许多重要的优势，使我们能够看到更传统理论无法获得的所谓的流形的“量子对称”。因此，它们是试图理解弦理论、镜像对称和超对称等物理上重要理论的数学的基本要素。从顶层开始，这一年涵盖的更多理论方面涉及抽象概念，如更高范畴理论、DG增强和派生几何。这些都是传统几何和范畴理论的实质性概括，该理论目前处于理解和标准化新学科基础的水平。这是数学理论发展的一个激动人心的阶段，但不是一个可以用简单的术语令人信服地解释的阶段。我们的研讨会将举办几个不同层次的学校和研讨会，对这些问题进行扩展。在另一个极端，派生范畴反馈到显式计算中，这些计算可以应用于给出有用的结果，描述代数、几何和理论物理等常见对象的性质。例如，派生范畴提供了迄今为止对McKay对应的最好处理，该对应将SL(2，CC)或SL(3，CC)中的有限子群G的表示理论与有序商CC^n/G的分解的拓扑联系起来。在类似的脉络中，我们的研讨会将包括研究派生范畴方法的研讨会，这些工作坊研究不同的模空间及其不变量(如向量丛的经典模空间，或代数曲线的经典模空间)。在这两个极端之间，有大量的理论和代数、几何和物理问题，已知或怀疑派生范畴方法可以应用于这些理论和问题。这包括弦理论产生的问题，例如同调镜像对称性，它围绕着这样一个猜想工作：派生范畴在Calabi-Yau三重折叠的复杂几何和它的镜像伙伴的辛几何之间起中介作用。

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.