The chromatic spectral sequence is an algebraic spectral sequence constructed from the Brown-Peterson spectrum, which converges to the second sheet of the Adams-Novikov Spectral sequence, the primary tool to understand stable homotopy. In this talk we build this spectral sequence and show how it helps us to understand the stable homotopy ring.

It is well known that the set of all $n \times n$ symmetric matrices of rank $k$ is a smooth manifold. This set can be described as those symmetric matrices whose ordered vector of eigenvalues has exactly $n-k$ zeros. The set of all vectors in $\mathbb{R}^n$ with exactly $n-k$ zero entries is itself an analytic manifold.

In this work, we characterize the manifolds $M$ in $\mathbb{R}^n$ with the property that the set of all $n \times n$ symmetric matrices whose ordered vector of eigenvalues belongs to $M$ is a manifold. In particular, we show that if $M$ is a $C^k$ manifold then so is the corresponding matrix set for all $k \in \{2,3,\ldots, \infty, \omega\}$. We give a formula for the dimension of the matrix manifold in terms of the dimension of $M$.

This is a joint work with A. Daniilidis and J. Malick.