I am currently working on obtaining my PhD in Algebraic Geometry, specifically working with the derived categories of schemes.
I am co-organiser for both the Geometry in Cardiff (GiC) and CALF seminars.
My current research is focussed on providing explicit formulas for the derived tensor product in various situations, namely when a collection of local complete intersections (lcis) inside of a smooth ambient variety intersect in a non-lci. Essential to this study is the fact that the resulting object can be non-equidimensional, that is, different components of the geometric object may have different dimensions. I hope to obtain some results dependent on what kind of non-equidimensionality our resulting scheme has.
Skein-Triangulated Representations Of Generalised Braids
The ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). There are many examples where Br_n acts categorically on the derived category of an algebraic variety: the minimal resolutions of Kleinian singularities, the cotangent bundles of flag varieties, etc.
To understand the derived category for the cotangent bundles of partial flag varieties, one needs a generalisation of the braid group to the generalised braid category. One may hope to understand categorical actions (so called skein-triangulated representations) of this new object on the derived category of (cotangent bundles of) partial flag varieties.