Antonio Lerario

Scuola Internazionale Superiore di Studi Avanzati di Trieste | SISSA

We address the general problem of estimating the probability that a real symmetric tensor is close to rank-one tensors. Using Weyl's tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety. More precisely, we give an explicit formula for its reach and curvature coef...

We study the optimal transport problem between complex algebraic projective hypersurfaces, by constructing a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space. The optimal transport problem between hypersurfaces is then defined through a constrained version of the Benamo...

Let a $K$ be a non-archimedean discretely valued field and $R$ its valuation ring. Given a vector space $V$ of dimension $n$ over $K$ and a partition $\lambda$ of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_\lambda(V)$ under the action of the group $\mathrm{GL}(n,R)$. When $K$ is a non-archimede...

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz (Proc Am Math Soc 143(5):1875–1889, 2015) on smooth com...

Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be viewed as random sections of the s-th complex tensor power of the tangent bundle of the 2-sphere. In this pape...

We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a Reeb function. We prove that for a Reeb function we can prescribe the set of minima (or maxima), as soon as this set is a PL subcomplex of the manifold. In analogy with Reeb's Sphere Theorem, we use such f...

The aim of this paper is to give a thorough insight into the relationship between the Rumin complex on Carnot groups and the spectral sequence obtained from the filtration on forms by homogeneous weights that computes the de Rham cohomology of the underlying group.

Let $K$ be a nonarchimedean local field of characteristic zero with valuation ring $R$, for instance, $K=\mathbb{Q}_p$ and $R=\mathbb{Z}_p$. We introduce the concept of an $R$-structure on a $K$-analytic manifold, which is akin to the notion of a Riemannian metric on a smooth manifold. An $R$-structure determines a norm and volume form on the tange...

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordere...

We prove that in a globally subanalytic family of convex bodies the subfamily of zonoids is log-analytic, and in particular it is definable in the o–minimal structure generated by globally subanalytic sets and the graph of the exponential function. This result establishes a connection between Model theory and Functional analysis.

We investigate the failure of B\'ezout's Theorem for two symplectic surfaces in $\mathbb{C}\mathrm{P}^2$ (and more generally on an algebraic surface), by proving that every plane algebraic curve $C$ can be perturbed in the $\mathscr{C}^1$-topology to an arbitrarily close smooth symplectic surface $C_\epsilon$ with the property that the cardinality...

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic m...

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordere...

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz on smooth complete intersections in the euclidean space...

We prove that with “high probability” a random Kostlan polynomial in $$n+1$$ n + 1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere $$\mathbb {S}^n$$ S n . The dependence between the “low degree” of the approximation and the “high probability” is quantitat...

We prove that in a tame family of convex bodies the set of zonoids is tame. Here "tame" means "definable in the o-minimal structure generated by globally subanalytic sets and the graph of the exponential function".

We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spa...

Let $D$ be a disk in $\mathbb{R}^n$ and $f\in C^{r+2}(D, \mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of $f$ meets a given semialgebraic set $W\subset J^{r}(D, \mathbb{R}^k).$ Examples of sets arising in this way are...

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and in...

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by δk,n the average number of projective k-planes in RPn that intersect (k+1)(n-k) many random, independent and uniformly distributed linear...

We define a new family of graph invariants, studying the topology of the moduli space of their geometric realizations in Euclidean spaces, using a limiting procedure reminiscent of Floer homology. Given a labeled graph $G$ on $n$ vertices and $d \geq 1$, $W_{G, d} \subseteq \mathbb{R}^{d \times n}$ denotes the space of nondegenerate realizations of...

We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called “thermodynamic” regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to conve...

Motivated by questions in real enumerative geometry we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, where $d_{k,n}=(k+...

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

In the recent paper [arXiv:1612.06893] P. B\"urgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $\delta_{k,n}$ the average number of projective $k$-planes in $\mathbb{R}\textrm{P}^n$ that intersect $(k+1)(n-k)$ many random, independent and uniformly distributed linear projec...

We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $\mathbb{R}\mathrm{P}^n$. Using a random spectral sequence, we prove an asymptotically exact estimate on the expected number of connected components in the complement of $s$ such hypersurfaces...

Given a polynomial map ψ:Sm→Rk with components of degree d, we investigate the structure of the semialgebraic set Z⊆Sm consisting of those points where ψ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning the upper estimate on the topological complexity of a polynomial...

Motivated by Hilbert's 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and in...

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterl\'e) and to the study of random $p$-adic polynomial systems of equations.

We introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces. This can be considered as a further step in the original Shub–Smale program of studying the real zeros of random polynomial systems. Our technique is general, and it also applies, for example, to the case of the enumerative geome...

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic m...

Motivated by numerous questions in random geometry, given a smooth manifold $M$, we approach a systematic study of the differential topology of Gaussian Random Fields (GRF) $X:M\to \mathbb{R}^k$, i.e. random variables with values in $C^\infty(M, \mathbb{R}^k)$ inducing on it a Gaussian measure. We endow the set of GRFs with the narrow topology and...

We investigate some geometric properties of the real algebraic variety \(\Delta \) of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in \(\Delta \). We exhibit connections of our...

We prove that with "high probability" a random Kostlan polynomial in $n+1$ many variables and of degree $d$ can be approximated by a polynomial of "low degree" without changing the topology of its zero set on the sphere $S^n$. The dependence between the "low degree" of the approximation and the "high probability" is quantitative: for example, with...

We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called "thermodynamic" regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to conve...

We investigate some geometric properties of the real algebraic variety $\Delta$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $\Delta$. We exhibit connections of our study...

Given a sequence |$\{Z_d\}_{d\in \mathbb{N}}$| of smooth and compact hypersurfaces in |${\mathbb{R}}^{n-1}$|⁠, we prove that (up to extracting subsequences) there exists a regular definable hypersurface |$\Gamma \subset {\mathbb{R}}\textrm{P}^n$| such that each manifold |$Z_d$| is diffeomorphic to a component of the zero set on |$\Gamma$| of some p...

Spectrahedral cones are linear sections of the cone of positive semidefinite symmetric matrices. We study statistical properties of random spectrahedral cones (intersected with the sphere) $$ \mathscr{S}_{\ell, n}=\{(x_0,\ldots,x_\ell)\in S^\ell \mid x_0 I + x_1 R_1 + \cdots+ x_\ell R_\ell\succ 0\}$$ where $I$ is the identity matrix and $R_1, \ldot...

Given a smooth manifold $M$ and a totally nonholonomic distribution $\Delta\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are critical points of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space $\Omega$ of horizontal paths st...

We discuss homotopy properties of endpoint maps for horizontal path spaces, i.e. spaces of curves on a manifold M whose velocities are constrained to a subbundle ? ? TM in a nonholonomic way. We prove that for every 1 = p < 8 these maps are Hurewicz fibrations with respect to the W1,p topology on the space of trajectories. We prove that the space o...

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree{\operatorname{edeg}G(k,n)} of the real Grassmannian {G(k,n)} as the average number of real k -planes meeting nontrivially {k(n-k)} random subspaces of {\mathbb{R}^{n}} , all of dimension {n-k} , where these subspaces are sampled uniformly and i...

We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$ in terms of the expected modulus of the determinant of a special random matrix. In the case $n=3$ we prove that the average number of real lines on a random cubic surface in $\mathbb{R}\textrm{P}^3$ is $6\sqrt{...

We consider the Betti numbers of an intersection of k random quadrics in (Formula presented.). Sampling the quadrics independently from the Kostlan ensemble, as (Formula presented.) we show that for each (Formula presented.) the expected ith Betti number satisfies (Formula presented.)In other words, each fixed Betti number of X is asymptotically ex...

We investigate the geometry of a random rational lemniscate $\Gamma$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational functions $r=p/q$ of degree $n$ by sampling $p$ and $q$ independently from the complex Kostlan ensemble of random pol...

Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study homotopy properties of the horizontal base-point free loop space $\Lambda$, i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities are constrained to $\Delta$ (for example: legendrian knots in a contact manifold). A key technical ingredient for our st...

A probabilistic approach to the study of the number of zeros of complex harmonic polynomials was initiated by W. Li and A. Wei (2009), who derived a Kac-Rice type formula for the expected number of zeros of random harmonic polynomials with independent Gaussian coefficients. They also provided asymptotics for a complex version of the Kostlan ensembl...

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critica...

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the nu...

Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale's alpha-theory to certify the results. Using...

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z) \in \mathbb{R}^{2n}...

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in |$\mathbb {R}\hbox {P}^n$| defined by a homogeneous polynomial f of degree d in the real Fubini–Study ensemble. We prove that for the expectation of the number of connected components, \begin{equation}\label{eq1} \mathbb{E} b_0(Z...

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal chang...

We explicitly compute the intrinsic volume of the set of real (and real symmetric) matrices of Frobenius norm one and given corank (the case of matrices with zero determinant as a special case). We give asymptotic formulas for our computations and we discuss several examples and applications.

We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics typically appear in families (c...

We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a first application, we consider the set of n x n (Real, Complex or Quaternionic) Hermitian matrices with Froben...

We demonstrate counterexamples to Wilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic fi...

We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated problems, from Real Algebraic Geometry to sub-Riemannian Geometry.

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic...

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RP^n defined by a Real Bombieri-Weyl distributed homogeneous polynomial of degree d. We prove that the expectation of the number of connected components of such hypersurface has order d^n, the asymptotic being in d for n fixed. We...

Let W be a linear system of quadrics on the real projective space ℝPn and X be the base locus of that system (i.e. the common zero set of the quadrics in W). We prove a formula relating the topology of X to that of the discriminant locus ∑W (i.e. the set of singular quadrics in W). The set ∑W equals the intersection of W with the discriminant hyper...

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit f...

Let Xℝ be the zero locus in ℝPn of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by b(Xℝ) the sum of the Betti numbers of Xℝ, we prove that(Formula Presented) The methods we use combine random matrix theory...

We prove that the total Betti number of the intersection X of three quadrics in RP^n is bounded by n(n+1). This bound improves the classical Barvinok's one which is at least of order three in n.

Given a smooth function f on R^n and a submanifold M, we prove that the set of diagonal quadratic forms q such that the restriction of f+q to M is Morse is a dense set (in the n-dimensional space of diagonal quadratic forms). The standard transversality argument seems not to work and we need a more refined approach.

We study the relation between a complex projective set C in CP^n and the set R in RP^(2n+1) defined by viewing each equation of C as a pair of real equations. Once C is presented by quadratic equations, we can apply a spectral sequence to efficiently compute the homology of R; using the fact that the (Z_2)-cohomology of R is a free H*(C)-module wit...

We study the topology of the set of the (spherical, projective and affine) solutions of a system of two quadratic inequalities: we give explicit formulas for its Betti numbers and give some sharp estimates for them; we also give a bound linear in n for the topological complexity of the intersection of two quadrics in RP^n. We study the geometry of...