In this article, we prove that there exists at least one chord orbit of Hamilton vector field connecting the Legendre submanifold on a given energy stable hypersurface in some symplectic manifolds. Démonstration de existence de chord orbital de hamiltonien en une stable hypersurface dans une variété symplectique. Résumé. Soit Σ une hypersurface close en variété symplectique, Il est de montrer q...

On the Complexity of Real Equation Solving by Bernd Bank, Humboldt–Universität zu Berlin The first part of the speech deals with the particular case of integer points in a semi-algebraic set described by quasi-convex polynomial inequalities in the n–dimensional real space. The simply exponential bounds are “optimal” with respect to the dense codification of the polynomials. Without changing the...

In this article, we prove that there exists at least one periodic orbit of Hamilton vector field on a given energy hypersurface in R2n which proves the longstanding Seifert’s conjecture. The more general results are also obtained. Démonstration de la Conjecture de Seifert sur existence de périodique orbital de hamiltonien en une hypersurface dans une variété symplectique. Résumé. Soit Σ une hyp...

We present a large class of homogeneous 2-nondegenerate CR-manifolds M , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CRequivalence between domains U,V inM extends to a global real-analytic CR-automorphism ofM . We show that this class containsG-orbits in Hermitian symmetric spaces Z of compact type, where G is a real form of the complex L...

We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the microlocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1. These algorithms use a normal form algorithm for the Brieskorn lattice, standard basis methods f...

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W (n,F)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W (n,F)-Willmore hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hy...

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov [16] for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product o...