Some applications of complex Floer theory to topology

I will describe an approach to defining diffeomorphism invariants of low-dimensional manifolds and knots. Morally, these invariants are based on counting solutions to certain equations coming from quantum field theory. However, their rigorous construction relies on ideas from (complex)
symplectic topology and algebraic geometry. Parts of this talk are based on joint work with Ciprian Manolescu and with Ikshu Neithalath, as well as work in progress.