The goal of enumerative geometry is to study a geometric space by counting certain subspaces within it. The first result in enumerative geometry is Euclid's observation that, given 2 distinct points in the plane, there is a single line through these points. A harder question is, given 2 points and 3 random lines in the plane how many conics (degree 2 curves) pass through both points and are tangent to each of the lines. These types of questions have interested geometers since the 1800's and earlier, but they are famously difficult. However, a breakthrough occurred in the 1990's when a surprising connection was made with physics. It was discovered that techniques and intuitions from string theory could be used to answer longstanding questions in enumerative geometry. The phenomenon behind this remarkable connection came to be known as mirror symmetry. In this talk I will give an introduction to mirror symmetry and its connection to enumerative geometry. At the end of the talk I will mention some current directions of inquiry and open questions.

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