We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.

In this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

The paper is an addendum to D. Andrica and L. Funar, ‘On smooth maps with finitely many critical points’, J. London Math. Soc. (2) 69 (2004) 783–800.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli–Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.

In this paper, we present a smooth framework for some aspects of the “geometry of CW complexes”, in the sense of Buoncristiano, Rourke and Sanderson. We then apply these ideas to Morse theory, in order to generalize results of Franks and Iriye-Kono.

More precisely, consider a Morse function $f$ on a closed manifold $M$. We investigate the relations between the attaching maps in a $\text{CW}$ complex determined by $f$, and the moduli spaces of gradient flow lines of $f$, with respect to some Riemannian metric on $M$.