A sub-Riemannian endpoint map with no regular values
We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.
math.DG↗
arXiv subjects
Antonio Lerario
Publications and source records attributed to Antonio Lerario.
Sard property for rank 2 polarizations in metabelian Lie groups
We provide sharp bounds on the dimension of the abnormal set for rank $2$ polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. The proof is based on a novel approach that makes essential use of tools from tame geometry. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most $2$, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.
math.DG↗
Morse theory of Euclidean distance functions from algebraic hypersurfaces
Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications.
math.AG↗
Sard properties for polynomial maps in infinite dimension
Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.
math.DG↗
The Denario project: Deep knowledge AI agents for scientific discovery
We present Denario, an AI multi-agent system designed to serve as a scientific research assistant. Denario can perform many different tasks, such as generating ideas, checking the literature, developing research plans, writing and executing code, making plots, and drafting and reviewing a scientific paper. The system has a modular architecture, allowing it to handle specific tasks, such as generating an idea, or carrying out end-to-end scientific analysis using Cmbagent as a deep-research backend. In this work, we describe in detail Denario and its modules, and illustrate its capabilities by presenting multiple AI-generated papers generated by it in many different scientific disciplines such as astrophysics, biology, biophysics, biomedical informatics, chemistry, material science, mathematical physics, medicine, neuroscience and planetary science. Denario also excels at combining ideas from different disciplines, and we illustrate this by showing a paper that applies methods from quantum physics and machine learning to astrophysical data. We report the evaluations performed on these papers by domain experts, who provided both numerical scores and review-like feedback. We then highlight the strengths, weaknesses, and limitations of the current system. Finally, we discuss the ethical implications of AI-driven research and reflect on how such technology relates to the philosophy of science. We publicly release the code at https://github.com/AstroPilot-AI/Denario. A Denario demo can also be run directly on the web at https://huggingface.co/spaces/astropilot-ai/Denario, and the full app will be deployed on the cloud.
cs.AI↗
Thom-Milnor bounds for smooth manifolds
We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends previous results in the Euclidean setting by Lerario and Stecconi [J. Singul., 2021]. As a key step, we generalize the Thom-Milnor bound to polynomial maps on a nonsingular real algebraic variety, improving the dependence on the degree. Finally, inspired by the work of Bürgisser, Cucker and Tonelli-Cueto [Found. Comput. Math., 2020], we introduce a condition number for families of functions. Using this we extend existing bounds due to Basu, Pollack and Roy [Proc. Amer. Math. Soc., 2004], for the Betti numbers of semialgebraic sets described by closed conditions to what we call closed semialgebraic type sets, namely sets defined by closed inequalities involving smooth functions.
math.AG↗
$\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials
We prove that if $p>d$ there is a unique gaussian distribution (in the sense of Evans) on the space $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ which is invariant under the action of $\mathrm{GL}(n, \mathbb{Z}_p)$ by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space $\mathbb{R}[x_1, \ldots, x_n]_{(d)}$ (respectively $\mathbb{C}[x_1, \ldots, x_n]_{(d)}$). More generally, if $V$ is an $n$--dimensional vector space over a nonarchimedean local field $K$ with ring of integers $R$, and if $λ$ is a partition of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_λ(V)$ under the action of the group $\mathrm{GL}(n,R)$.
math.NT↗
Quantitative approximate definable choices
In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest.
math.AG↗
Probabilistic intersection theory in Riemannian homogeneous spaces
Let $M=G/H$ be a Riemannian homogeneous space, where $G$ is a compact Lie group with closed subgroup $H$. Classical intersection theory states that the de Rham cohomology ring of $M$ describes the signed count of intersection points of submanifolds $Y_1, \ldots, Y_s$ of $M$ in general position, when the codimensions add up to $\dim M$. We introduce the probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$, whose multiplication describes the unsigned count of intersection points, when the $Y_i$ are randomly moved by independent uniformly random elements of $G$. The probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$ has the structure of a graded commutative and associative real Banach algebra. It is defined as a quotient of the ring of Grassmann zonoids of a fixed cotangent space $V$ of $M$. The latter was introduced by the authors in [Adv. Math. 402, 2022]. There is a close connection to valuations of convex bodies: $\mathrm{H}_{\mathbb E}(M)$ can be interpreted as a subspace of the space of translation invariant, even, continuous valuations on $V$, whose multiplication coincides with Alesker's multiplication for smooth valuations. We describe the ring structure of the probabilistic intersection ring for spheres, real projective space and complex projective space, relying on Fu [J. Diff. Geo. 72(3), 2006] for the latter case. From this, we derive an interesting probabilistic intersection formula in complex projective space. Finally, we initiate the investigation of the probabilistic intersection ring for real Grassmannians, outlining the construction of a probabilistic version of Schubert Calculus.
math.DG↗
What is the probability that a random symmetric tensor is close to rank-one?
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 coefficients with respect to the Bombieri-Weyl metric. These results are obtained using techniques from Random Matrix theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our findings give a complete solution to the original problem. In the case of rational normal curves it leads to a simple formula describing explicitly exponential decay with respect to the degree of the tensor.
math.AG↗
The Grassmann distance complexity
Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept quantifies the complexity of solving the nearest point problem for subanalytic sets in the Grassmannian, using the intrinsic Riemannian distance. Unlike the Euclidean case, the Grassmannian distance is neither smooth nor semialgebraic, and its study requires using Lipschitz critical point theory and o-minimal geometry. We establish fundamental properties of GDC, including computable bounds for real algebraic varieties and conditions ensuring the finiteness of critical points. Our results also include a nonlinear version of the classical Eckart-Young theorem, which characterizes critical points of the distance function from a generic $k$-plane to simple Schubert varieties.
math.DG↗
What is... Random Algebraic Geometry?
We survey some ideas from the subject of Random Algebraic Geometry, a field that introduces a probabilistic perspective on classical topics in real algebraic geometry. This offers a modern approach to classical problems, such as Hilbert's Sixteenth Problem.
math.AG↗
Nonarchimedean integral geometry
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 prove a general integral geometric formula for $K$-analytic groups and homogeneous $K$-analytic spaces, analogous to the corresponding result over the reals. This generalizes the $p$-adic integral geometric formula for projective spaces recently discovered by Kulkarni and Lerario, e.g., to the setting of Grassmannians. Based on this, we outline the construction of a nonarchimedean probabilistic Schubert Calculus. For this purpose, we characterize the relative position of two subspaces of $K^n$ by a position vector, a nonarchimedean analogue of the notion of principal angles, and we study the probability distribution of the position vector for random uniform subspaces. We then use this to compute the volume of special Schubert varieties over $K$. As a second application of the general integral geometry formula, we initiate the study of random fewnomial systems over nonarchimedean fields, bounding, and in some cases exactly determining, the expected number of zeros of such random systems.
math.AG↗
The probabilistic method in real singularity theory
We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e. with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.
math.AG↗
How regularization affects the geometry of loss functions
What neural networks learn depends fundamentally on the geometry of the underlying loss function. We study how different regularizers affect the geometry of this function. One of the most basic geometric properties of a smooth function is whether it is Morse or not. For nonlinear deep neural networks, the unregularized loss function $L$ is typically not Morse. We consider several different regularizers, including weight decay, and study for which regularizers the regularized function $L_ε$ becomes Morse.
cs.LG↗
Optimal transport between algebraic hypersurfaces
What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini-Study one. The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is Kähler and the corresponding metric is of Weil-Petersson type. To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own. We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.
math.DG↗
A proof of the Hamiltonian Thom Isotopy Lemma
In this note we present a complete proof of the fact that all the submanifolds of a one parameter family of compact symplectic submanifolds inside a compact symplectic manifold are Hamiltonian isotopic.
math.SG↗
Symplectic Instability of Bézout's Theorem
We investigate the failure of Bézout'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 $C_ε$ with the property that the cardinality $\#C_ε\cap Z_d$ of the transversal intersection of $C_ε$ with an algebraic plane curve $Z_d$ of degree $d$, as a function of $d$ can grow arbitrarily fast. As a consequence we obtain that, although Bézout's Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is "arbitrarily false" for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon "instability of Bézout's Theorem").
math.SG↗