arXiv research

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the calculation of cohomology of certain configuration spaces.

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

The study finds billiard trajectories with infinitely many reflections in certain cones.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

We smooth the singularities of a strictly hyperbolized smooth cube manifold.

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not pro…

Billiard trajectories in curved spaces have predictable travel times.

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant Morse and Lusternik - Schnirelman theories. The paper continues results publishe…

Proves Goh conditions for singular curves with specific properties.

Study on overlaps of singular vectors in Gaussian matrix submatrices.

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize …

We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the dependence locus.

Rigidity of travel times for convex obstacles in Riemannian manifolds is proven.

Improving Bayesian filtering with strictly proper scoring rules

Extends Smale's principle to produce minimal graphs with singularities.

Study strict stability of cones with isolated singularities.

Study learns dynamics of linear systems from multiple short trajectories.

Estimates roughness of volatility from discrete variance data.

We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. For plane billiards (when m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schir…

Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.

Study of supervised learning from multiple non-independent sequences.

We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature $Θ^h$ as a matrix of currents. We …

Constructs flow lines connecting unstable to stable self-expanders.

A new neural network captures and explains trajectory patterns.

We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. suppor…

The paper constructs stable minimal hypersurfaces with specific singularities.

The paper proves weaker conditions for global smoothings of special Lagrangian submanifolds with conical singularities.

The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set o…

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a prescribed final point $B\in X$ and make a prescribed number $n$ of reflections at the bo…

We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Ampère equations to give a classification of the surfaces in 3-dimensional space forms which are em…

Algorithm learns weight matrix from single trajectory of nonlinear dynamical system.

We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a Killing submersion over a strictly Hadamard surface whose fibers are the trajectories of a unit Killing field. We prove that every complete surface whose principal curvatures are greater than a certain function (depending on the ambient manifold)…

This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…

The paper classifies vertices in planar polygons formed by convex domains.

In a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $ω\in Ω$, might be strictly greater than the …

Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

New Calabi-Yau metrics with conical singularities are created near complex lines.

We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.

We show that the recent work of Lee [23] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static soluti…

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…

Bounds on saddle connections on flat spheres with conical singularities.

The paper proves the exact number of singular points in the intersection of convex shapes.

Learning optimal feedback control laws capable of executing optimal trajectories is essential for many robotic applications. Such policies can be learned using reinforcement learning or planned using optimal control. While reinforcement learning is sample inefficient, optimal control only plans an optimal trajectory fr…

Study shows instability of naked singularities in scalar field models.

The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.