Topics in singularity theory and algebraic geometry

We will work on selected topics in singularity theory and algebraic geometry. We will focus on uncovering the geometric details of the contact loci of polynomials inside jet spaces. There are two possible directions for applications. One is in arithmetic, where contact loci play a prominent role in the monodromy conjecture. Another one is in symplectic geometry, where contact loci are conjectured to provide an algebraic formulation of the Floer cohomology of the iterations of the monodromy map. In this thesis both directions will be explored.