The link of an extrovert divide
The linking number of singular maps.
The Lyusternik-Shnirel'man category, vector bundles, and immersions of manifolds.
The Łojasiewicz gradient inequality in a neighbourhood of the fibre
Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.
The Maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants.
The minimum number of zeros of Lipschitz-Killing curvature.
The Moduli of Super-Null-Correlation Bundles on IP3.
The Monodromy of a Curve with Ordinary Double Points.
The multiplicity of the Lyashko-Looijenga mapping on the discriminant strata of even and odd polynomials
The Mumford conjecture
The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and...
The Nash-Kuiper process for curves
A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then adress the question of Holder regularity of the limit map.
The non-criticality of certain families of affine varieties.
The nonuniqueness of Chekanov polynomials of Legendrian knots.
The Novikov conjecture and low-dimensional topology.
The number of critical points in Morse approximations
"The Optiverse" and Other Sphere Eversions
The parity of the Maslov index and the even cobordism category
We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over ℝ. We define an index two subcategory (the even subcategory) of a 3-dimensional cobordism category. The objects of the category are surfaces equipped with Lagrangian subspaces of their real first homology. This generalizes a result of the first author where surfaces are equipped with Lagrangian subspaces of their rational first homology.
The periodic Floer homology of a Dehn twist.
The Poincaré-Bendixson theorem and arational foliations on the sphere
Foliations on the 2-sphere with a finite number of non-orientable singularities are considered. For this class a Poincaré-Bendixson theorem is established. In particular, the work gives an answer to a problem of H. Rosenberg concerning labyrinths.
The Positvity of the Chern Classes of an Ample Vector Bundle.