The goal of this paper is to establish the precise connection between the centro-affine invariants of plane curves and the feedback invariants of nonlinear scalar control systems in the plane. We will also show how the centro-affine structure provides a shortcut to the structure equations for feedback equivalence one obtains by applying Cartan’s equivalence method.

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F (t1, . . . , tn) of WDVV equations of associativity polynomial in t1, . . . , tn−1, exp tn.

We present an new system of ordinary differential equations with affine Weyl group symmetry of type E (1) 6 . This system is expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian. 2000 Mathematics Subject Classification: 34M55, 17B80, 37K10. Introduction The Painlevé equations PJ (J = I, . . . ,VI) are ordinary differential equations of second order. It is know...

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit e...

We consider the Hilbert scheme Hilb(Cd) of (d + 1) points in affine d-space C (d ≥ 3), which includes the square of any maximal ideal. We describe equations for the most symmetric affine open subscheme of Hilb(Cd), in terms of Schur modules. In addition we prove that Hilb(Cd) is reducible for n > d ≥ 12.