Skein-triangulated representations of generalised braids

从事这个项目的学生将在简单约化群的齐次空间上构造广义辫的绞三角表示。让我们来解开这个密集的陈述，勾勒出它的背景，它的跨学科相关性以及它为学生提供的机会。本项目是拓扑学、几何学、表示理论和理论物理的结合。低维拓扑的主要研究对象是辫状结构和链路。辫子是具有固定端点的开放式字符串的配置，而链接是类似的环状字符串配置。在他们的研究中，一个关键的进步是菲尔兹奖得主沃恩·琼斯在20世纪80年代提出的量子链接不变量。他通过一个简单的过程将一个多项式与每个环节联系起来，这个过程反复使用一个单绞关系将复杂的环节分解成简单的闭环。在20世纪90年代末，这被推广到著名的Khovanov链同调，类似地使用串串关系的同调版本构造。在2000年代中期，通过在群SLn的齐次空间的某些片的派生范畴上表示群SL2，利用几何表示理论计算了Khovanov同调。关键因素来自于理论物理学，通过同调镜像对称。衍生方案的交集作为结构复合体携带了我们所相交的方案的衍生张量积。对于基础格式的交点，交点结构复合体的上同调承载着交点的重要几何信息

the student working on this project would construct skein-triangulated representations of generalised braids on homogeneous spaces of simple reductive groups. Let us unpack this dense statement, sketch out its context, its interdisciplinary relevance and the opportunities it offers to the student. This project lies on the interface of topology, geometry, representation theory, and theoretical physics. Key objects of study in low-dimensional topology are braids and links. Braids are configurations of open-ended strings with fixed endpoints, while links are similar configurations of looped strings. A crucial advance in their study was the development in 1980s of a quantum link invariant by Fields medallist Vaughn Jones. He associated a polynomial to each link via a simple procedure which repeatedly used a single skein relation to break complicated links down to simple closed loops. In late 1990s this was generalised to the celebrated Khovanov homology of links, constructed similarly using a homological version of the skein relation. In mid 2000s Khovanov homology was computed via geometrical representation theory by representing the group SL2 on the derived categories of certain slices of the homogeneous spaces of the group SLn. The key ingredient came from theoretical physics via the homological mirror symmetry. he intersection of derived schemes carries as structure complex the derived tensor product of structure sheaves of the schemes we are intersecting. For intersections of underived schemes, the cohomologies of the intersection structure complex carries important geometric information about the intersection