Lissajous varieties and their applications
Speaker: Francesco Mascarin (Max Planck Institute for Mathematics and Science)
Abstract:
We introduce Lissajous varieties as affine algebraic varieties parametrized by sine and cosine functions. They generalize algebraic Lissajous curves in the plane to higher dimensions by applying a componentwise trigonometric map to a rational affine linear space.
We show that their degree (namely, the number of intersection points with a generic affine linear space of complementary dimension) can be computed from the volume of a corresponding polytope, and that their defining equations arise from rank constraints on an associated matrix encoding polynomial multiplication.
We discuss their applications to dynamical systems in the study of equilibria, with a particular focus on the Kuramoto model. Here we explore their connections with convex optimization and Lissajous discriminants.
Finally, in the context of semidefinite programming, we highlight the crucial role these varieties play in describing the algebraic boundary of the elliptope for certain graph classes.
This is based on joint work with Monique Laurent and Simon Telen.
