arXiv research

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

Geometric proof of contractibility of unitary group in strong topology.

Strong rigidity proven for non-compact surfaces.

A new knot invariant is fast, strong, topologically meaningful, and fun.

The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.

A new method for graph-structured data improves transformer performance by incorporating topology.

We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the t…

Study shows curvature constraints force submanifolds to have specific topology or geometry.

We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,ω)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,ω)$ of hamiltonian homeomorphisms introduced by Oh and Mull…

CNN accurately reconstructs lattice topology with strong thermal fluctuations.

Current Deep Reinforcement Learning algorithms still heavily rely on handcrafted neural network architectures. We propose a novel approach to automatically find strong topologies for continuous control tasks while only adding a minor overhead in terms of interactions in the environment. To achieve this, we combine Neur…

Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γgenerated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to …

We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the excepti…

Graph braid groups' complexity stabilizes for most graphs.

An algorithm preserves topological features in dimensionality reduction.

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which…

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies th…

This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …

New example solves topological dynamics problem.

Study submanifolds in spheres with Ricci curvature bounds.

In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is $(a)$-regular. Here…

This paper proposes a deep Convolutional Neural Network(CNN) with strong generalization ability for structural topology optimization. The architecture of the neural network is made up of encoding and decoding parts, which provide down- and up-sampling operations. In addition, a popular technique, namely U-Net, was adop…

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

We provide a strengthening of Jordan separation, to the setting of maps from a compact topological space X into a sphere, where the source space X is not necessarily a codimension one sphere, and the map is not necessarily injective.

Sharp pinching conditions restrict the geometry and topology of submanifolds.

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…

Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.

This study examines the topology of singularities in optimal semicouplings between unequal spaces.

Persistent homology enhances graph classification by capturing long-range graph properties.

New research shows sparse topologies can lead to faster convergence in distributed optimization.

In a previous paper, under the assumption that the Riemannian metric is special, the author proved some results about the moduli spaces and CW structures arising from Morse theory. By virtue of topological equivalence, this paper extends those results by dropping the assumption on the metric. In particular, we give a s…

Topological Flow Matching: A Generative Modeling Framework for Structured Spaces

Computational method approximates homology groups of compact metric spaces.

Python library for integrating TDA with machine learning.

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In …

Proves common stellar subdivisions for all PL homeomorphic polyhedra.

In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part …

This paper provides a topological method for filling contact structures on the connected sums of $S^2\times S^3$. Examples of nonsymplectomorphic strong fillings of homotopy equivalent contact structures with vanishing first Chern class on $\#_k S^2\times S^3$ $(k\geq2)$ are produced.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Formula estimates pseudo-Anosov maps' fixed points, linking to surface properties.

Flexible approach for normal approximations in geometric and topological statistics.

We present atomistic molecular dynamics simulations of two Polyethylene systems where all entanglements are trapped: a perfect network, and a melt with grafted chain ends. We examine microscopically at what level topological constraints can be considered as a collective entanglement effect, as in tube model theories, o…

Profinite rigidity proven for many hyperbolic manifolds.

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.