Topology of Algebraic Varieties

Let $A$ be an affine variety inside a complex $N$ dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of $A$ with a very small sphere turns out to be a contact manifold called the link of $A$. If the first Chern class of our link is torsion (I.e. the singularity is numerically $\mathbb Q$ Gorenstein) then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that if the link of $A$ is contactomorphic to the link of $\mathbb C^3$ and $A$ is normal then $A$ is smooth at 0. This generalizes a Theorem by Mumford in dimension 2. I will give an introduction to the main result (including a very small introduction to contact geometry) in the first half of the lecture and a basic idea of the proof in the second half of the lecture.