arXiv research

Example of non-smooth isotopy using K3 surfaces.

New example of non-smooth isotopy after stabilization in 4-manifolds.

New gauge theory invariant detects non-smooth isotopy of $P^2$-knots.

MARINA-P improves non-smooth federated optimization with adaptive stepsizes.

Extends curve theory to non-smooth data with finite curvature and torsion.

AsylADMM improves gossip-based learning for non-smooth objectives.

Survey on preserving curvature bounds for non-smooth Ricci flow.

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly sin…

We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.

A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.

New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.

The paper explores various stationarity concepts in non-smooth optimization.

New bounds explain deterministic non-smooth deep nets without large Lipschitz constants.

We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…

Smoothness analysis of adversarial training reveals $L_\infty$ constraints cause more non-smoothness.

Approaches 4D Schoenflies via pseudo-isotopy.

In the framework of Lorentzian warped products, we study the Friedmann-Robertson-Walker cosmological model to investigate non-smooth curvatures associated with multiple discontinuities involved in the evolution of the universe. In particular we analyze non-smooth features of the spatially flat Friedmann-Robertson-Walke…

This work speeds up hyperparameter selection for non-smooth convex models using implicit differentiation.

Classifies curves up to symplectic isotopy.

New algorithms optimize non-smooth, non-convex objectives with improved complexity.

In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar …

In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration compl…

Smooth isotopy on cube saves energy with extra dimensions.

A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in gen…

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Paper tackles private optimization for non-smooth objectives efficiently.

Refined 1-cocycle for knots helps quantify isotopies.

Classifies isotopy classes of links from Thompson's group F and its subgroup.

Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.

We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…

Advances smooth over-parameterization for solving non-smooth optimization problems.

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.

New methods improve convergence in non-convex non-smooth learning problems.

Minimal surfaces and curves can have singularities removed by isotopy.

Adaptive data fusion boosts efficiency in multi-task optimization.

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.

Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.

The study constructs examples of pseudo-isotopic but not isotopic 4-manifold diffeomorphisms.

In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.

Veering triangulations link Thurston norm and isotopy of surfaces.

Proof confirms Hamiltonian isotopy of submanifolds.

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies.

This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop fram…

The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], but fur…

Abstracts a theorem for non-smooth maps in infinite dimensions.

Proof of isotopy invariance in Khovanov link homology.

New SPS variant improves non-smooth optimization without small gradients.

Classifies smooth isotopy of a specific Lagrangian embedding.