Mixed superposition rules: geometry and applications of Lie systems --- Javier de Lucas (UW)

Lie systems are time-dependent systems of differential equations whose general solutions can be expressed through finitely many particular solutions and constants. Mixed superposition rules extend this construction by allowing the particular solutions to belong to different auxiliary systems. They therefore provide a way to relate differential equations and reconstruct solutions from simpler systems. In this talk, we will introduce mixed superposition rules and explain how their construction is guided by the Lie-algebraic and geometric structures underlying the equations. We will discuss methods based on invariant foliations and semidirect sums of Lie algebras, together with a mixed coalgebra method for Lie systems admitting compatible Dirac structures. Applications will illustrate the role of symplectic and contact geometry, with examples involving Riccati equations, time-dependent oscillators, and thermodynamical systems. Finally, we will discuss extensions to Lie systems of partial differential equations and their use in the formal construction of solutions near a given particular solution, including an example related to the Tzitzéica equation. The talk is based on joint work with R. Campoamor-Stursberg, O. Carballal, and F. J. Herranz, published in Studies in Applied Mathematics 157, e70297 (2026).