arXiv research

This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.

We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for semi-algebraic sets.

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from describing these results, we discuss briefly the background as well as the importanc…

We prove a formula that relates the Euler-Poincaré characteristic of a closed semi-algebraic set to its Lipschitz-Killing curvatures

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the bool…

We characterize value functions in partially observable MDPs as semi-algebraic sets.

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that…

Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.

Paper discusses conditions for global injectivity of semi-algebraic local diffeomorphisms.

Let $\R$ be a real closed field, ${\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and ${\mathcal P} \subset \R[X_1,...,X_k]$ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$. Let $S \subset \R^{\ell+k}$ be a semi-alg…

Let $\R$ be a real closed field, ${\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and ${\mathcal P} \subset \R[X_1,...,X_k]$ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebr…

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real close…

In this paper we construct a compactification for the parameter space of convex projective structures on a fixed n-manifold M. This parameter space is a closed semi-algebraic subset of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary is the inverse limit of an inve…

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exh…

Characterizes G2-structures on Lie algebras with non-trivial center.

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects…

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Ho…

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary was …

Looking for an efficient algorithm for the computation of the homology groups of an algebraic set or even a semi-algebraic set is an important problem in the effective real algebraic geometry. Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez wrote a paper [1], which made a step forward by giving an algorithm …

Characterizes structures preserved by groups in high-dimensional spaces.

Graev's nerve implies invariant Einstein metrics on homogeneous spaces.

This paper connects real closed fields to Hitchin representations and their properties.

In this article we give an explicit algorithm which will determine, in a discrete and computable way, whether a finite piecewise Euclidean complex is non-positively curved. In particular, given such a complex we show how to define a boolean combination of polynomial equations and inequalities in real variables, i.e. a …

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the int…

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topolo…

In this paper we study geometric, algebraic, and computational aspects of flexibility and infinitesimal flexibility of Kokotsakis meshes. A Kokotsakis mesh is a mesh that consists of a face in the middle and a certain band of faces attached to the middle face by its perimeter. In particular any 3x3-mesh made of quadran…

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications…

Study of curvature flow on specific Lie groups, leading to soliton solutions.

Metric problem solved for real analytic Riemannian manifolds.

We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the…

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inferen…

In this paper we investigate how germs of real functions can change under deformation. In particular we look at deformations of germs of isolated singularities from R_n to R_k (n >= k) and the relation with there natural stratification in some tame categorie (algebraic, analytic, semi-algebraic, subanalytic, o-minimal …

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-ba…

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…

Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the complet…

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the cov…

New framework for data-driven hyperparameter tuning with structured loss.

For any positive integer $k$, there exist neural networks with $Θ(k^3)$ layers, $Θ(1)$ nodes per layer, and $Θ(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $Ω(2^k)$ nodes. This result is proved here for a class o…

In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to $\mathbb R^n$. In the g…

Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection ${\mathcal A} = \{A_1,...,A…

Variant of previous work on smooth algebraic functions with compact and non-compact preimages.

Study of differential spinors on three-manifolds with skew-torsion.

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theo…

This paper presents a new mathematical framework to analyze the loss functions of deep neural networks with ReLU functions. Furthermore, as as application of this theory, we prove that the loss functions can reconstruct the inputs of the training samples up to scalar multiplication (as vectors) and can provide the numb…

Integrates estimation and optimization for uncertain parameters.

We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $τP$, $τ> 0$, simple to compute. This problem i…