Cohomologies of Derived Intersections

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