Enumerative geometry is a branch of mathematics that deals with the following question: "How many geometric objects satisfy certain constraints". A well known class of enumerative question is to count curves in a linear system H^0(X,L) that have some prescribed singularities. In this talk we will describe a topological method to approach this problem. We will express the enumerative numbers as the Euler class of an appropriate Bundle. We will then go on to explain how we compute the degenerate contribution of the Euler class using a topological method.

The polynomial method in combinatorics has recently emerged as a simple but strikingly powerful method to answer many fundamental questions about combinatorial parameters associated with geometric objects. I will survey some of the old and new applications of this method (focussing on the simpler proofs!) including: 1) "List-decoding bounds for Reed-Solomon codes": Given $n$ points in the plance, how many polynomials of degree $d$ can pass through $t$ of them? (Guruswami and S. '98) 2) "Bounds on Kakeya and Nikodym sets": How small can a subset of $F_q^n$ (the $n$-dimensional vector space over the finite field of cardinality $q$) be so that it contains a line in every direction. (Dvir 2008) 3) "Bounds on Joints in R^3": What is the largest number of "joints" (non-coplanar intersection of three lines) that can be formed by a set of L lines? (Guth and Katz, 2008) 4) "Bound on capsets in F_3^n": How large can a subset in F_3^n so that it contains no complete line? (Ellenberg, Gijswijt; based on Croot-Lev-Pach 2016).