Enumerative geometry is a branch of mathematics that deals with the following question: How many geometric objects are there that satisfy certain constraints? The simplest example of such a question is How many lines pass through two points?. A more interesting question is How many lines are there in three dimensional space that intersect four generic lines?. An extremely important class of enumerative question is to ask How many rational (genus 0) degree d curves are there in CP^2 that pass through 3d-1 generic points? Although this question was investigated in the nineteenth century, a complete solution to this problem was unknown until the early 90’s, when Kontsevich-Manin and Ruan-Tian announced a formula. In this talk we will discuss some natural generalizations of the above question; in particular we will be looking at rational curves on del-Pezzo surfaces that have a cuspidal singularity. We will describe a topological method to approach such questions. If time permits, we will also explain the idea of how to enumerate genus one curves with a fixed complex structure by comparing it with the Symplectic Invariant of a manifold (which are essentially the number of curves that are solutions to the perturbed d-bar equation).

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