Dynamical systems solvable by algebraic operations

In this talk I shall focus on systems of nonlinear Ordinary Di§erential Equations, and introduce the notion of their solvability by algebraic operations: implying that their general solution, considered as a function of complex time, feature at most a Önite number of rational branch points, or equivalently deÖne a Riemann surface with a Önite number of sheets. Some properties of these systems shall be reviewed, including the subclasses of them featuring such remarkable properties as isochrony or asymptotic isochrony (as functions of real time). Techniques to identify such systems shall be reviewed, and several examples reported, including new classes of such systems. References: F. Calogero, Isochronous Systems, Oxford University Press, 2008 (264 pages, paperback 2012); Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge University Press, 2018 (168 pages). F. Calogero and F. Payandeh, ìPolynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactionsî, J. Math. Phys. 60, 082701 (2019). F. Calogero, R. Conte and F. Leyvraz, "New solvable systems of two autonomous Örst-order ordinary di§erential equations with purely quadratic right-hand sides" (in preparation).