arXiv subjects

Jean-Yves Welschinger

Publications and source records attributed to Jean-Yves Welschinger.

At least 19 recordsLinked to original sources

arXiv2024

Amoeba Measures of Random Plane Curves

We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grow to $+\infty$.

math.AG↗

arXiv2022

Morse shellings out of discrete Morse functions

From the topological viewpoint, Morse shellings of finite simplicial complexes are {\it pinched} handle decompositions and extend the classical shellings. We prove that every discrete Morse function on a finite simplicial complex induces Morse shellings on its second barycentric subdivision whose critical tiles-or pinched handles-are in oneto-one correspondence with the critical faces of the function, preserving the index. The same holds true, given any smooth Morse function on a closed manifold, for any piecewise-linear triangulation on it after sufficiently many barycentric subdivisions.

math.CO↗

arXiv2021

Spectral sequences of a Morse shelling

We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a property shared by the classical shellable complexes. We now observe that every such tiling supports a quiver which is acyclic precisely when the tiling is shellable and then, that every shelling induces two spectral sequences which converge to the relative (co)homology of the complex. Their first pages are free modules over the critical tiles of the tiling.

math.AT↗

arXiv2021

Shellable tilings on relative simplicial complexes and their h-vectors

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

math.CO↗

arXiv2021

Morse shellings and compatible discrete Morse functions

We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated closed surface is Morse shellable while every closed three-manifold carries Morse shellable triangulations. Finally, any shelling encodes a class of discrete Morse functions whose critical points are in one-to-one correspondence, preserving the index, with the critical tiles of the shelling.

math.AT↗

arXiv2020

Morse shellings on products

We recently defined a property of Morse shellability (and tileability) of finite simplicial complexes which extends the classical one and its relations with discrete Morse theory. We now prove that the product of two Morse tileable or shellable simplicial complexes carries Morse tileable or shellable triangulations under some tameness condition, and that any tiling or shelling becomes tame after one barycentric subdivision. We deduce that any finite product of closed manifolds of dimensions less than four carries Morse shellable triangulations whose critical and h-vectors are palindromic. We also prove that the h-vector of a Morse tiling is always palindromic in dimension less than four or in the case of an h-tiling, provided its critical vector is palindromic.

math.AG↗

arXiv2018

Tilings, packings and expected Betti numbers in simplicial complexes

Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In codimension one, we use canonical filtrations of $\text{Sd}^d (K)$ to upper estimate these limits and get a monotony theorem which makes it possible to improve these estimates given any packing of disjoint simplices in $\text{Sd}^d (K)$. We then introduce a notion of tiling of simplicial complexes having the property that skeletons and barycentric subdivisions of tileable simplicial complexes are tileable. This enables us to tackle the problem: How many disjoint simplices can be packed in $\text{Sd}^d (K)$, $d \gg 0$?

math.PR↗

arXiv2017

Asymptotic topology of random subcomplexes in a finite simplicial complex

We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.

math.PR↗

arXiv2017

Asymptotic measures and links in simplicial complexes

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $Sd^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka.

math.GT↗

arXiv2015

Universal components of random nodal sets

We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ 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 $m\textgreater{}0$, bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed $n$-dimensional manifold $M$ equipped with some Lebesgue measure. In fact, for every closed hypersurface $Σ$ of $\mathbb R^n$, we prove that there exists a positive constant $p\_Σ$ depending only on $Σ$, such that for every large enough $L$ and every $x\in M$, a component diffeomorphic to $Σ$ appears with probability at least $p\_Σ$ in the vanishing locus of a random section and in the ball of radius $L^{-1/m}$ centered at $x$. These results apply in particular to Laplace-Beltrami and Dirichlet-to-Neumann operators.

math.SP↗

arXiv2014

Topology of random real hypersurfaces

These are notes of the mini-course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July $2014$. The subject was my joint work with Damien Gayet on the topology of random real hypersurfaces, restricting myself to the case of projective spaces and focusing on our lower estimates. Namely, we estimate from (above and) below the mathematical expectation of all Betti numbers of degree $d$ random real projective hypersurfaces. For any closed connected hypersurface $Σ$ of $\mathbb{R}^n$, we actually estimate from below the mathematical expectation of the number of connected components of these degree $d$ random real projective hypersurfaces which are diffeomorphic to $Σ$.

math.AG↗

arXiv2014

Betti numbers of random nodal sets of elliptic pseudo-differential operators

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.

math.SP↗

arXiv2014

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.

math.AG↗

arXiv2013

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 Kählerian 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}).

math.SG↗

arXiv2013

Open Gromov-Witten invariants in dimension four

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic discs with boundary on L, homologous to d and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure J. We furthermore get analogous open Gromov-Witten invariants by counting, for every non-negative integer k, unions of k discs instead of single discs.

math.SG↗

arXiv2012

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

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.

math.AG↗

arXiv2012

Open Gromov-Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X, ω)$. We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$. Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$, every relative homology class $d \in H_2 (X, L ; \Z)$ and adequate number of incidence conditions in $L$ or $X$, the weighted number of $J$-holomorphic discs with boundary on $L$, homologous to $d$, and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$, provided that at least one incidence condition lies in $L$. These numbers thus define open Gromov-Witten invariants in dimension six, taking values in the ring $A$.

math.SG↗

arXiv2011

What is the total Betti number of a random real hypersurface?

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve Hörmander's theory of peak sections as well as the formula of Poincaré-Martinelli.

math.AG↗