In this talk, we explore the geometry underlying totally mixed Nash equilibria for
n-player games. We model these equilibria using vector bundles and introduce the Nash equi-
librium scheme, which parametrizes all totally mixed equilibria of a given game. Depending
on the format of the game, we define two key geometric objects: the Nash discriminant and
the Nash resultant varieties. These capture games whose Nash equilibrium schemes exhibit
unexpected behavior - such as being nonreduced or having positive-dimensional components.
No prior background in game theory will be assumed, and we will illustrate the main ideas with
concrete examples. This is joint work with Hirotachi Abo and Luca Sodomaco.

In this talk we study a class of polynomials whose positive support is the set of vertices of a simplex and which may have several negative support points in the simplex. Various groups of authors have provided an exact characterization for the global nonnegativity of a polynomial in this class in terms of circuit polynomials and that characterization provides a tractable nonnegativity test. We generalize this characterization to the constrained nonnegativity over a set $X$ under an additional convexity precondition in the moment space using methods from duality theory and convex analysis. This provides a tractable nonnegativity test over $X$ for the class in terms of a power cone program.

In this talk we classify linear operators on polynomials which preserve non-negativity of polynomials on any given closed subset of Rⁿ. Desprite the purely algebraic formulation of this question, we show that moments as a functional analytic tool solve this questions.

Additionally, we introduce the concept of generators of linear operators on polynomials, i.e., we treat evolution equations on polynomials. We classify all generators of linear operators on polynomials and show that this is larger as a regular Fréchet Lie algebra. We classify a large group of operator algebras in the set of generators. We give a resolvent characterization of generators of K-positity preserving semi-groups for any closed set K in Rⁿ.

This is jount work with Konrad Schmüdgen and Lars-Luca Langer.

This talk investigates a subspace of the space of symmetric tensors (forms) and its properties in relation to (positive) Waring decompositions. Given a polynomial substitution map $q$, we show that $q$-Symmetric forms are precisely those forms that admit a Waring decomposition over the variety $V_q$ parametrized by $q$. After characterizing the structure of this subspace, we focus on the cone of positive $q$-Symmetric forms and its dual. Finally we apply these results to finite dimensional truncated moment problems and present an explicit algorithm for computing Waring decompositions over $V_q$. Based on joint work with S. Kuhlmann and B. Mourrain.