On analytical applications of stable homotopy (the Arnold conjecture, critical points)

We prove the Arnold conjecture for closed symplectic manifolds with $\pi_2(M)=0$ and \cat M=\dim M. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.Comment: AMSTEX, 12 pages, submitted to Math. Zeitschrift, improvement (correction) of the line of the proof of the Arnold conjectur

On higher analogs of topological complexity

Farber introduced a notion of topological complexity \TC(X) that is related to robotics. Here we introduce a series of numerical invariants \TC_n(X), n=1,2, ... such that \TC_2(X)=\TC(X) and \TC_n(X)\le \TC_{n+1}(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.Comment: LATEX, 8 page