Description of the Calogero–Moser spaces (PhD dissertation discussion)

Let $M_n$ be the space of all $n \times n$ matrices. Then the product space $M_n \times M_n$ has the canonical symplectic structure. Moreover, the general linear group $GL_n$ acts by simultaneous conjugation on this product space. It is easy to show that the symplectic structure descends to a Poisson structure on the corresponding orbit space. The latter induces the Poisson bracket on the algebra $A$ of $GL_n$ invariant polynomial functions on $M_n \times M_n$. In this presentation we talk about the Calogero–Moser spaces, their Poisson algebra structure, and the affine Cremona group action on these spaces. Also, we will give some results related to the description of the ring of $GL$-invariant matrices and the coordinate ring of the Calogero–Moser spaces. Moreover, we will discuss how Poisson bracket helps uncover defining relations of the coordinate ring of the Calogero–Moser space in general.