My research lies in the intersection between algebraic geometry, differential geometry and string theory. More specifically, I have been studying Donaldson-Thomas type theory on Calabi-Yau 4-folds, which is a way to count 'instantons' or coherent sheaves on CY4.
DT4 theory could be viewed as a complexification of Donaldson's theory on oriented 4-manifolds. Formally, DT4 theory should fit into a topological quantum field theory relating instanton countings on Spin(7), G2 and CY3 manifolds. Nevertheless, I prefer restricting to the algebraic subcase for CY4 and CY3. Basically, for a simple degeneration Xt of CY4 into two 4-folds glued along their anti-canonical divisor Y, we expect a gluing formula relating DT4 invariant of the generic fiber of Xt and relative DT4 invariants of those two 4-folds (which are elements in the DT3 cohomology of Y)