Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similarly to the role of the braid group in braided categories, a group naturally acts on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked, real, genus-zero stable curves

In this article, we describe the asymptotic behavior of sequences of solutions to some semilinear elliptic equations with critical exponential growth in planar domains. We prove, in particular, a result analogous to that of Struwe [12] in higher dimensions and extend the two-dimensional result of Adimurthi and Struwe [3] to arbitrary energies. We thus answer a question explicitly asked in this last article

(Equivariant $K$-theory of Bott towers. Application to the multiplicative structure of the equivariant $K$-theory of flag varieties)

We construct a basis of the equivariant $K$-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant $K$-theory of flag varieties in the basis constructed by Kostant and Kumar in [18]

Seidel and Smith [33] have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety $Y.$ This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra ${\mathfrak{sl}}_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting $Y$ as an open subset of the Hilbert scheme of a Milnor fiber

We give a geometric criterion that implies a singular maximal spectral type for a dynamical system on a Riemannian manifold. The criterion, which is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we check it for an ad hoc class of smooth mixing flows on ${\mathbb{T}}^3$ obtained from linear flows by time change and thus providing natural examples of mixing smooth diffeomorphisms and flows with purely singular spectra