Application of Chern class I am taking a course about differentiable manifold and I need to prepare a representation about Chern class.
Although now I am familiar with the properties of Chern class, but I can not find good examples of the applications of Chern class. As a topological invariant of vector bundle, I really want to find some example to use Chern class to distinguish some different vector bundle. I am also looking other applications which can illustrate why Chern class is useful. Thanks!
 A: The first time I really used Chern classes was when writing this blog post. The goal was to compute the cohomology ring of a hypersurface of degree $d$ in $\mathbb{CP}^3$, and I ended up computing its Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes to do this. It was fun. 
A: I think there is a really nice example from physics, though I am not sure this is exactly what you're looking for I will try to explain a bit in any case. It might be a bit too involved for your presentation.
It is in the context of the integer quantum Hall effect. The original reference for what I'm about to explain is a paper by Kohmoto.
We are interested in computing the current through some rectangular conductor. I will spare you the details, they can be found in the paper linked above. But by some generalization of Bloch's theorem we obtain a map
\begin{equation}
 u: \mathbb{R}^{2} \rightarrow \mathbb{C},(k_x,k_y)\mapsto u(k_x,k_y),
\end{equation}
such that for some fixed $a,b \in \mathbb{R}$ we have two real-valued maps $\theta_x,\theta_y$ such that
\begin{align}
 u(k_x+a, k_y) = e^{i \theta_x(k_x,k_y)}u(k_x,k_y), \\
 u(k_x, k_y+b) = e^{i \theta_y(k_x,k_y)}u(k_x,k_y).
\end{align}
We might say that $u$ is almost periodic, so it doesn't quite descend to the torus $\mathbb{T}^2 = \mathbb{R}^2/(a \mathbb{Z} \times b \mathbb{Z})$. We can construct a $U(1)$-bundle over the torus $\mathbb{T}^2$ such that the map $u: \mathbb{R}^2 \rightarrow \mathbb{C}$ is a section of this bundle. We then obtain a connection one-form $A$ by the formula
\begin{equation}
A(k_x,k_y) = \left(u^{\dagger}(k_x,k_y) \vec{\nabla} u(k_x,k_y) \right) \cdot \vec{dk}.
\end{equation}
This connection one-form induces a connection, the curvature of which we denote by $F$.
If one now write out the formula for the first Chern class
\begin{equation}
 \int_{\mathbb{T}^2} F,
\end{equation}
it turns out that the transverse current in the conductor is proportional to it.
More details are in the paper I linked above, but I am not sure its very accesible to mathematicians.
If you like this example I could add more details.
A: The Chern numbers of a smooth almost-complex manifold are a complete cobordism invariant.
