In a joint work with Marie-Charlotte Brandenburg and Jesús A. De Loera, we develop exact polynomial time algorithms for ham-sandwich and centerpoint problems for rational polytopal measures. Our approach encodes cuts of prescribed proportions via a piecewise rational cap-volume function, reducing the problem to semialgebraic feasibility. This yields algorithms to decide existence, describe, count, and sample cuts. We also show that spaces of deep affine flats are semialgebraic, recovering the centerpoint set of a convex body as a floating body.

A multivariate polynomial with real coefficients is called real zero if it has only real zeros along each line through the origin and does not vanish at the origin. The Euclidean closure of the connected component of the non-vanishing set of a real zero polynomial containing the origin is called a rigidly convex set. These convex sets generalize spectrahedra. It is not known whether all rigidly convex sets are spectrahedra; this is known as the Generalized Lax Conjecture. We introduce a nested hierarchy of spectrahedra that relaxes a given rigidly convex set. We do this by adapting another spectrahedral relaxation developed by Schweighofer and by using an approximation result due to Fang and Fawzi. If a real zero polynomial has only two variables, our hierarchy of relaxations converges to the rigidly convex set of that real zero polynomial.

We discuss quantifier elimination in a natural language extension for finite-dimensional algebras over real closed and algebraically closed fields. The first part concerns constructive quantifier elimination for matrix rings via reduction to the underlying field. The second part generalizes the invariant theoretic methods from the first part to obtain quantifier elimination results for more general finite-dimensional algebras, including quaternions, and octonions. The emphasis is on explicit elimination procedures and the encoding of definable sets via separation of orbits under the automorphism group.

Free spectrahedra are noncommutative counterparts of classical spectrahedra. From the operator-system viewpoint, a free spectrahedron is precisely an operator system with a finite-dimensional C*-envelope (equivalently, a finite-dimensional realisation). Building on this, we introduce d-subhomogeneous operator systems as those having a d-subhomogeneous C*-envelope. We also discuss how this relates to notions of maximality of operator systems and the existence of certain types of realisations. Finally, we present a recent result: The class of projected subhomogeneous operator systems is closed under taking duals, paralleling the fact that duals of free spectrahedra are projected free spectrahedra.