Cones of locally non-negative polynomials

The study of non-negative polynomials is motivated by the obvious fact that the value at a global minimum of a real polynomial $f$ is the maximal value $c$ such that $f-c$ is globally non-negative. This shows its connection to optimization. Similarly, a local minimum $x_0$ of $f$ induces the polynomial $f-f(x_0)$ which takes value $0$ and is locally non-negative at $x_0$.

I will present results from my PhD thesis on the convex cone of locally non-negative polynomials. We will see geometric interpretations and examples of faces of this cone, some general theory of cones in infinite-dimensional vector spaces and classifications of faces using tools from singularity theory. I will also give a short outlook on an application to sums of squares in real formal power series rings.