We'll take a look at physical models coming from gauge theory. Starting from (parts of) the standard model as usual, followed by the NCG example of almost commutative manifolds and onto the goal of a gauge theory over a ``fuzzy'' manifold as in Perez-Sanchez.

We will go over some applications of the Atiyah-Bott localization formula. We will evaluate the integral of the volume form of $S^2$ as a sum over the fixed point set of the rotation action of $S^1$ on $S^2$. A more non-trivial application will be in the enumerative problem of counting number of lines in a cubic surface in $\mathbb{CP}^3$. We will explain why this problem is well-posed and then compute the number using Atiyah-Bott formula.