Fourier–Mukai Transforms and Derived Equivalence: Derived reconstruction of curves, Fourier–Mukai partners of K3 surfaces, and K3 categories of cubic fourfolds.
Étale Cohomology: Étale cohomology of curves, base change theorems, the six-functor formalism, Poincaré duality, and arc-topology.
Spherical Functors and P_n-Functors: A modern proof of the 2-out-of-4 criterion for spherical functors and a construction of P_n-twists using coherent convolution.