##### Nathan Lowry, September 15, 2023

Through Grothendieck's functor of points perspective, every category carries with it an "internal language." In the presence of extra structure, one can even use this internal language to describe and generalize the basic notions of mathematical logic from a purely categorical perspective. In this talk, we will take a closer look at the "internal logic" definable on the category of sheaves over a scheme and use it to prove non-trivial result in algebraic geometry.