My research is at the intersection of model theory, arithmetic, and algebraic geometry.
For an ordinary surface, the topological fundamental group takes a fixed base-point of the surface and computes all continuous loops on that surface around the base-point up to deformation of loops. This group-gadget gives one a computational way to detect holes on the surface without having to rely on visualizing the topology of the surface.
For an algebraic curve, the topological fundamental group as a construct makes less sense because the topology of a curve is coarse enough so as to make the idea of continuous loops around base-points fruitless. On the other hand, by using limits of covering spaces one can define an analogous algebraic fundamental group for a curve. This giant infinite group is highly non-commutative yet someone how has enough structure to not only detect holes, but completely reconstruct the geometry of the curve! Whereas the topological fundamental group of a surface can only ever see the number of holes on that surface, the algebraic fundamental group of a curve can often see the complete geometry of the curve!
Along with Boris Zilber at the University of Oxford, we have codified the algebraic fundamental group as a multi-sorted logical structure so as to use model-theoretic ideas, analogies, and language to go further with the study of hyperbolic curves over number fields. One such use-case is the application of geometric stability theory to use elimination of imaginaries to construct k-rational points on hyperbolic k-curves.
A model theory section conjecture (pdf) December, 2020