In this talk, we will first introduce the Jarnik curve, which is a convex C^2-smooth curve with as many lattice points as possible. This is important (and rediscovered) in harmonic analysis since it serves as counterexamples to Mizohata-Takeuchi-type conjectures. Next, we would like to add extra smoothness conditions to the curve and see what will happen. I will talk about some literature (especially the work of Bombieri-Pila) and focus on the case when the curve is C^3-smooth and convex. The result is due to Swinnerton-Dyer. The proof uses an ”elementary” argument, but it is surprising that it has not been improved for the past 50 years.
Talk by Irving (Xuerui) Yang.