Perverse Schobers and the McKay Correspondence

范畴化问题是代数几何和表示论的混合。他们研究代表函子衍生类别的代数簇。导出范畴是簇的终极同调不变量。作用于它就把作用于更传统的不变量（如上同调或K-理论）归为范畴。不幸的是，派生范畴是围绕有严重缺陷的三角范畴公理建立的。DG增强理论是在90年代初由Bondal和Kapranov提出的，以解决这些缺陷。在过去的十年中，Keller，Toen等人的一系列革命性结果使它迅速发展起来。许多以前无法解决的问题现在都在我们的范围之内。从事这个项目的学生将研究这些新技术，并将其应用于证明一个长期存在的猜想：广义辫子的范畴作用于完全和部分旗簇的派生范畴（余切束）。一般化的辫子是允许其股以某种方式接触的辫子。它们有多个端点配置，并且可以是不可逆的，因此形成一个类别而不是一个群。这仍然是开放的十年，因为它的大部分构建块还没有被发现：球形，P-和格拉斯曼函子以及它们诱导的等价性。其中，只有球面函子现在完全解决了：通过Anno和Logvinenko的DG技术。P-函子我们只有有限的例子，而Grassmanian函子是一个未知的领域。该项目的学生将加入一个跨越英国，美国，日本和丹麦的国际合作团队。他将致力于进一步发展目前的需求理论的球形和P-函子，或计算我们的第一个例子格拉斯曼函子，或计算广义辫子关系的degretured分类行动。如果成功，他的工作将有助于实现这一雄心勃勃的目标。

Categorification problems are a mix of algebraic geometry and representation theory. They study representations by functors on derived categories of algebraic varieties. The derived category is the ultimate homological invariant of a variety. Acting on it categorifies acting on more conventional invariants such as cohomology or K-theory.Unfortunately derived categories were built around the seriously flawed axiomatics of triangulated categories. The theory of DG-enhancements was conceived in early 90s by Bondal and Kapranov to fix these flaws. It was rapidly developed over the last decade in a series of revolutionary results by Keller, Toen, et al. Many previously inaccessible problems are now within our reach.The student working on this project would study these new techniques and apply them towards proving a long-standing conjecture: the category of generalised braids acts on the derived categories of (the cotangent bundles of) full and partial flag varieties. Generalised braids are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group.This remained open for a decade because most of its building blocks weren't discovered yet: spherical, P- and Grassmanian functors and the equivalences they induce. Of these, only spherical functors are now completely worked out: via DG-techniques by Anno and Logvinenko. P-functors we only have limited examples of, while Grassmanian functors are an uncharted territory.The student working on the project would join an international team of collaborators spanning UK, US, Japan and Denmark. He will work either on further developing the currently in-demand theories of spherical and P-functors, or on computing our first examples of Grassmanian functors, or on computing the generalised braid relations in the conjectured categorical action. If successful, his work would contribute to the ambitious goal of turning this conjectured action into reality.