Damien Gayet

Lower estimates for the expected Betti numbers of random real hypersurfaces

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 of a constant which only depends on the dimension n of the manifold with the Kahlerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).

Expected topology of random real algebraic submanifolds

Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k <=n, that X, E and L are defined over the reals and denote by RX the real locus of X. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in RX of holomorphic real sections of E tensored with L^d, where d is a large enough integer. Moreover, given any closed connected codimension k submanifold S of R^n with trivial normal bundle, we prove that a real section of E tensored with L^d has a positive probability, independent of d, to contain around the square root of d^n connected components diffeomorphic to S in its vanishing locus.

Deformations of associative submanifolds with boundary

Abstract Let M be a topological G 2 -manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. For a connected boundary ∂Y of genus g, the index is given by ∫ ∂ Y c 1 ( ν X ) + 1 − g , where ν X denotes the orthogonal complement of T ∂ Y in T X | ∂ Y and c 1 ( ν X ) the first Chern class of ν X with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order

Convexité rationnelle des sous-variétés immergées lagrangiennes

{L^{\frac{n}{m}}}

Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Lnm for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order

Smooth moduli spaces of associative submanifolds

{m > 0}