Wedge product = set intersection? In a research article [1] I found the following formulation:

The wedge product may be considered as set intersection. For
  example, surfaces of constant $f(x,y,z)$ and surface of constant $g(x,y,z)$
  intersects along the lines given by $df \wedge dg$. The notion
  of interpreting the wedge product as set intersection is appealing
  from a topological standpoint.

The article almost does not make use of exterior differential systems or other sophisticated math. It also gives no precise reference for the above statement.
Questions:


*

*Does this viewpoint (wedge product = set intersection) make sense at all?

*What is meant with "the lines given by $df \wedge dg$"? Edit: E.g. When I set $f(x,y,z) = \frac{1}{2}(x^2 + y^2 + z^2) $ and $g(x,y,z) = ax + by + cz$ with real constants $a,b,c$. Then we have $df\wedge dg = (x dx + y dy + z dz)\wedge(a dx + b dy + c dz)$. But what line does correspond to this 2-form. (I would suspect that we obtain circles.)


[1] http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=990890

The statement in the article has to be taken with a huge grain of salt. Even for $1$-forms, there's no consistent way to interpret all wedge products of $1$-forms as intersections.
If you restrict attention to nonvanishing decomposable forms (ones that can be written locally as wedge products of $1$-forms), and add a few extra restrictions, then there is something that can be said. 
If $\omega$ is any differential form on an $n$-manifold $M$, let us define the kernel (sometimes called the radical) of $\omega$ at a point $p\in M$ to be the set of vectors $v\in T_pM$ such that $\omega_p(v,\cdot,\dots,\cdot) = 0$. If $\omega$ is a nonvanishing decomposable $k$-form, its kernel at $p$ is an $(n-k)$-dimensional subspace of $T_pM$ (the common kernel of $\eta_1,\dots,\eta_k$ when we write $\omega_p = \eta_1\wedge\dots\wedge\eta_k$), and these subspaces fit together to form a smooth $(n-k)$-dimensional distribution on $M$. 
Now if both $\omega$ and $\theta$ are nonvanishing decomposable forms, and their distributions are transverse to each other (i.e., the two distributions span the tangent space at each point), then $\omega\wedge\theta$ is a nonvanishing decomposable form whose distribution is the intersection of the distributions determined by $\omega$ and $\theta$.
If the distributions in question happen to be integrable, then this can be rephrased in terms of the leaves of the corresponding foliations: Each leaf determined by $\omega\wedge\theta$ is a (connected component of) an intersection of a leaf determined by $\omega$ and one determined by $\theta$.  
If the forms in question are allowed to vanish, or the distributions happen not to be transverse somewhere, then all of this falls apart because $\omega\wedge\theta$ will be zero at all such points.
