Degenerate CR singularities and polynomial convexity of real submanifolds in $\mathbb C^n$

We will discuss some notions of degeneracy for CR singularities of an $m$-dimensional real submanifold $M$ in $\mathbb C^n$, when $m\geq \frac{2}{3}(n+1)$. Our geometric interpretations of these degeneracies allows us to compute the dimensions of the loci of such degeneracies when $M$ is in general position. This yields an application to the problem of finding the minimum complex dimension $n$ such that all closed $m$-dimensional real manifolds admit polynomially convex embeddings into $\mathbb C^n.$ This is joint work with Rasul Shafikov.