arXiv research

Discusses the tight versus overtwisted dichotomy in 3D contact geometry.

Study on curvature decay in steady Ricci solitons, proving dichotomy.

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then …

The paper explores actions on metric spaces similar to 3D manifolds, proving rigidity results.

Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.

TQFT invariants are either easy or hard to compute, depending on the TQFT type.

This work refines Cover's theory for binary classification on low-dimensional data.

Given a sequence of complete Riemannian manifolds $(M_n)$ of the same dimension, we construct a complete Riemannian manifold $M$ such that for all $p \in (1,\infty)$ the $L^p$-norm of the Riesz transform on $M$ dominates the $L^p$-norm of the Riesz transform on $M_n$ for all $n$. Thus we establish the following dichoto…

Study of Matsumoto maps on foliated bundles over hyperbolic manifolds.

New geometric structures defined in contact metric geometry.

A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively splits the set of classes into two subsets, and trains a binary classifier to distinguish between each subset. Even though ensembles of nested dichotomies with random structure…

Nested dichotomies are used as a method of transforming a multiclass classification problem into a series of binary problems. A tree structure is induced that recursively splits the set of classes into subsets, and a binary classification model learns to discriminate between the two subsets of classes at each node. In …

A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively applies binary splits to divide the set of classes into two subsets, and trains a binary classifier for each split. Many methods have been proposed to perform this split, each …

Classifies actions of groups on hyperbolic spaces, proving dichotomy.

Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that $(M,g)$ has bounded geometry, or more generally is thick at infinity. Then th…

Anosov groups' measures on limit sets are uniquely determined by their dimension.

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the …

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humilière-Leclercq…

In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one…

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

Study free energy in spherical spin glasses, proving universality dichotomy.

Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.

New theory shows perishable goods markets are more stable and efficient.

The study quantifies how many objects can be linearly classified under all views.

New concept of regular separation for ODEs leads to improved Hardy field results.

Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.

The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.

We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we u…

Study homotopy groups of open books and their pages, pages, and bindings.

The paper studies proper discontinuity of actions on Weyl chamber flow spaces.

In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a 2-d analogue of this fact: there is a close relationship between the topology of the contact structure on a convex surfac…

We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy $b_1$ even or odd.

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.

Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generaliz…

Study of Milnor invariants and ropelength of spherical links.

A popular approach for predicting the future of dynamical systems involves mapping them into a lower-dimensional "latent space" where prediction is easier. We show that the information-theoretically optimal approach uses different mappings for present and future, in contrast to state-of-the-art machine-learning approac…

The paper develops a theory of conformal density at infinity for groups with contracting elements.

This work studies scaling laws for low-precision training in high-dimensional linear regression.

Breiman's data analysis dichotomy is outdated, offering a third approach: mechanistic models.

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove fini…

Optimal control problem for firm cash flow with dividend and capital injection strategies.

DET unifies geometric and functional alignment for high-dimensional scientific data.

Study traveling waves in hyperbolic space for Fisher-KPP equations.

The study classifies warped products with harmonic curvature on surfaces, showing two possibilities for the metric.

Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G in a binate (hence acyclic) group. In the other direction, there are no nontrivi…

The Higgs field growth is studied on special geometric spaces, confirming a conjecture.

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $Σ$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $Σ$ does not separate $M$ then $Σ$ is totally geodesic and $M\setminusΣ$ is isometric to the Riemannian product $Σ\times(a,b)$, and if $Σ$ se…

The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows fa…