I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

Abstract. A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The principal result is that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension. In fact orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane. Limit sets are nowhere dense, unknotted and unlinked. In dimension 2 every trajectory is eventually monotone.In dimension 3 a compact limit set which does not contain an equilibrium is a closed orbit or a cylinder of closed orbits.

A compact, connected, combinatorial 4-maifold is embeddable in R^7 if its twisted normal Stiefel-Whitney class in dimension 3 is trivial.

Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M.  When N = Rⁿ , the fibre of E is the Stiefel manifold of m-frames in n-space.

Let Y and X denote Ck vector fields on a possibly noncompact surface with empty boundary, 1 ≤ k< ∞. Say that YtracksX if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. Theorem Assume the Poincaré–Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of Ck vector fields that track X, then the elements of g have a common zero in K. Applications are made to the dynamics of attractors and transformation groups.

Let G be a connected nilpotent Lie group with a continuous local action on a real surface M, which might be non-compact or have non-empty boundary @M. The action need not be smooth. Let φ be the local flow on M induced by the action of some one-parameter subgroup. Assume K is a compact set of fixed points of φ and U is a neighborhood of K containing no other fixed points. THEOREM. If the Dold fixed-point index of φt |U is non-zero for sufficiently small t > 0, then Fix(G) ∩ K ≠ Ø.

The activation dynamics of nets are considered from a rigorous mathematical point of view. A net is identified with the dynamical system defined by a continuously differentiable vector field on the space of activation vectors, with fixed weights, biases, and inputs. Chaotic and oscillatory nets are briefly discussed, but the main goal is to find conditions guaranteeing that the trajectory of every (or almost every) initial activation state converges to an equilibrium. Several new results of this type areproved. These are illustrated with applications to additive nets. Cascades of nets are considered and a cascade decomposition theorem is proved. An extension of the Cohen-Grossberg convergence theorem is proved for certain nets with nonsymmetric weight matrices.