arXiv research

Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\text{polylog}(1/ε))$ which is $ε$-close, and similarly for any rational function there exists a ReLU network of size $O(\text{polylog}(1/ε…

CATR rationalizes text data to stabilize causal effect estimation.

Study stabilizers of isotropic classes in rational 4-manifolds, finding diffeomorphisms that almost preserve Lefschetz fibrations.

New presentation of Goussarov-Habiro Lie algebra using primitive Feynman diagrams.

Sparse text alignments learned via optimal transport improve model explainability.

In this paper, we develop a method for constructing left-orders on the fundamental groups of rational homology 3-spheres. We begin by constructing the holonomy extension locus of a rational homology solid torus $M$, which encodes the information about peripherally hyperbolic $\widetilde{\text{PSL}_2\mathbb{R}}$ represe…

The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.

The paper proves a cohomological injection for a specific group.

Study rational homology of moduli space via Morse functions, proving stability phenomena.

A new approach to rationalization identifies true rationales by considering causal relationships.

Many kinds of data are naturally amenable to being treated as sequences. An example is text data, where a text may be seen as a sequence of words. Another example is clickstream data, where a data instance is a sequence of clicks made by a visitor to a website. This is also common for data originating in the domains of…

Outer space for RAAGs is a contractible finite-dimensional space for automorphisms.

We examine certain symmetries in the deficiencies of a rational surgery on a knot in $S^3$ by comparing the $\text{Spin}^c$-structures on the rational surgery with those on a related integral surgery. We then provide an application of these symmetries in the form of a theorem that obstructs Dehn surgeries in $S^3$. Thi…

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a Kähler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal m…

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimension…

We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techni…

Modeling alignment as resource-limited cognitive processes, researchers derive performance bounds.

Article explores non-freeness of groups generated by two specific matrices, providing counterexamples and sequences.

Selection of input features such as relevant pieces of text has become a common technique of highlighting how complex neural predictors operate. The selection can be optimized post-hoc for trained models or incorporated directly into the method itself (self-explaining). However, an overall selection does not properly c…

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $π_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in th…

A torus manifold $M$ is a $2n$-dimensional orientable manifold with an effective action of an $n$-dimensional torus such that $M^T\neq \emptyset$. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If $M$ is a simply connected torus manifold which a…

We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $δ$--graded knot Floer homology is mutation--invariant. Using the Clifford module structure on $\widetilde…

New invariant detects infinite order cabled knots.

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / Γ$, where $G$ is a complex connected Lie group and $Γ$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_ρ$ associated to any irreducible representa…

Convolutional neural networks (CNNs) have recently emerged as a popular building block for natural language processing (NLP). Despite their success, most existing CNN models employed in NLP share the same learned (and static) set of filters for all input sentences. In this paper, we consider an approach of using a smal…

Nearby pinwheels are isotopic, solving Arnold's conjecture.

Study mapping class group actions on cohomology of configuration spaces.

Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…

Classifies real rational knots and curves in a specific quadric space.

New method for simplifying knots with specific properties.

The study calculates the average genus of rational knots and links.

We note that a rational $3$-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational $3$-tangle diagrams up to isotopy. However, there is no perfect classification about rational $3$-tangle diagrams such as the classification of rational $2$-tangle diagrams cor…

New rational band moves simplify knot classification.

Classifies surgeries on torus knots and cables that bound rational homology balls.

Jones polynomial coincidences explored for rational knots.

Lower bounds on rational slice genus using Heegaard Floer invariants.

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…

Study connects taffy pulling, fractions, and rational tangles.

New knots show linear independence in slice concordance.

The paper proves that rational concordance of double twist knots is reciprocal.

In this paper, we introduce a rational $τ$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives o…

This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangl…

New findings on knots that are both topologically and rationally slice.

The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.

Classifies torus bundles bounding 4-manifolds with rational homology.

Study shows conditions for rational ellipticity of manifolds with symmetries.