Interaction Networks via Grassmannians

Abstract: When can a network of mutually reinforcing N components remain stable? To approach such questions, we describe the interactions through generalized Lotka–Volterra equations—a broad class of dynamical systems modeling how components influence one another over time. This formulation leads to a family of semi-algebraic sets determined by the sign pattern of the parameters. These sets encode positivity conditions defining regions of potential coexistence, with polynomial degrees growing exponentially in N. Embedding the parameter space into the real Grassmannian Gr(N,2N) transforms these conditions into sign relations governed by the Grassmann–Plücker equations and oriented matroids. This geometric reformulation yields a realization problem through which we detect impossible interaction networks and study the algebraic structure underlying stability. If time permits, we will also touch on how these structures connect to algebraic curves. This talk is based on our recent work arXiv:2509.00165.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars