Matrix product states play an important role in quantum information theory to represent states of many-body systems. They can be seen as low-dimensional subvarieties of a high-dimensional tensor space. In this talk, I will introduce two variants: homogeneous matrix product states and uniform matrix product states. Studying the linear spans of these varieties leads to a natural connection with invariant theory of matrices. For homogeneous matrix product states, a classical result on polynomial identities of matrices leads to a formula for the dimension of the linear span, in the case of 2x2 matrices. I will explain this connection, and the difficulties that arise when generalizing it to the case of uniform matrix product states. This talk is partially based on joint work with Claudia De Lazzari and Harshit Motwani.

In this talk, we introduce a new approach for solving Nash equilibrium problems of polynomials. It is based on a hierarchy of semidefinite relaxations, which are constructed with Lagrange multiplier expressions and feasible extensions. Under some general assumptions, we show this approach either returns a Nash equilibrium or detects its nonexistence.

Gröbner's problem is a longstanding open question, posed by Gröbner in 1967, about determining when a monomial projection of a Veronese variety is an arithmetically Cohen-Macaulay variety. In this talk, we review the state of the art of Gröbner's problem and we present new contributions. Our approach takes advantage of the invariants of finite abelian groups and their combinatorics.

In the first part of the talk, it will be shown how the partition function in statistical mechanics can be interpreted as an algebraic variety. In accordance to earlier literature, the zero-temperature limit is equivalent to taking the tropical limit of the algebraic variety. Previous literature have also generalised the temperature parameter to an n-vector. Here, we show that in the case of n=2, the two components of this generalised quantity are the inverse temperature and inverse temperature times chemical potential, respectively. Other values of n can also be similarly interpreted as various intensive thermodynamic parameters.

The second part of the talk concerns null geodesics in four dimensional spacetimes. In particular, we observe that the condition for null circular orbits defines an A-discriminantal variety. A theorem by Rojas and Rusek for A-discriminants leads to the interpretation that there are two branches of null circular orbits for certain classes of spacetimes. A physical consequence of this theorem is that light rings around generic black holes with non-degenerate horizons are unstable. [Joint work with Mounir Nisse]