n this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this.

Chapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5.

How can you figure out which points lie on a certain curve? And how many possible curves do you count by a given number of points? These are the kinds of questions Pim Spelier of the Mathematical Institute studied during his PhD research. Spelier received his doctorate with distinction on June 12.