arXiv research

The study explores mapping degree sets and their properties for manifolds.

Paper solves whether zero sets are mapping degree sets.

Every closed oriented manifold $M$ is associated with a set of integers $D(M)$, the set of self-mapping degrees of $M$. In this paper we investigate whether a product $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor $D(N)$ contains $d$. We find sufficient conditions so that $D(M\times N)$ contains e…

Analytic sets with unique infinite tangent cone are algebraic.

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

New algorithms find half-optimal independent sets in sparse graphs.

The study explores which sets of integers can be realized as the degrees of maps between manifolds.

New study shows limits of low-degree algorithms in finding large independent sets in sparse hypergraphs.

Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.

By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any closed orientable 3-manifold $M$?

For each closed oriented 3-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.

We compute the sets of degrees of maps between principal $SU(2)$-bundles over $S^5$, i.e. between any of the manifolds $SU(2)\times S^5$ and $SU(3)$. We show that the Steenrod squares provide the only obstruction to the existence of a mapping degree between these manifolds, and construct explicit maps realizing each in…

New measure for nonrationality of toric quasifolds.

In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are es…

We propose a novel method for network inference from partially observed edges using a node-specific degree prior. The degree prior is derived from observed edges in the network to be inferred, and its hyper-parameters are determined by cross validation. Then we formulate network inference as a matrix completion problem…

We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two…

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …

The paper approximates smooth hypersurfaces with algebraic ones, controlling the degree.

For harmonic maps of degree 2, a similar quantitative stability estimate does not hold uniformly.

New covering moves for 3-manifolds up to degree 4.

Study on flat singular points of area-minimizing currents, defining a singularity degree.

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichmüller space of $S$. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

We prove in this paper that, under suitable coinditions on an initial data set, we can obtain Area and Curvature Estimates for simple marginally outer trapped surfaces (or MOTS). Using this estimates, we derive a Compactness Theorem for MOTS. Moreover, the Compactness Theorem will allow us to adapt the recent Degree Th…

New findings on computational limits for estimating hidden structures.

New bounds on inscribed triangles in arbitrary planar domains.

We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the …

Alexander polynomial degree correlates with knot defect, proving conjecture for defect zero.

In order to find a way of measuring the degree of incompleteness of an incomplete financial market, the rank of the vector price process of the traded assets and the dimension of the associated acceptance set are introduced. We show that they are equal and state a variety of consequences.

This paper investigates the model degrees of freedom in k-means clustering. An extension of Stein's lemma provides an expression for the effective degrees of freedom in the k-means model. Approximating the degrees of freedom in practice requires simplifications of this expression, however empirical studies evince the a…

Study axisymmetric $σ_k$-Nirenberg problem on spheres.

We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a q…

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations…

The paper explores how different network architectures learn logical functions under GOTU, finding that a min-degree-interpolator is learned.

In-degree quiver polynomials for surface-links computed.

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such emb…

Fewer degrees of freedom can train deep networks, showing a sharp phase transition.

The study identifies two minimal orbits of foliations on complex projective plane and explores their properties.

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…

New degree theory proves existence of solitons on 4D manifolds.

The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.

This paper tests the multivariate normality of node degrees in Erdős-Rényi graphs.

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the to…

Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and torus semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-ma…

There has been a large amount of interest, both in the past and particularly recently, into the power of different families of universal approximators, e.g. ReLU networks, polynomials, rational functions. However, current research has focused almost exclusively on understanding this problem in a worst-case setting, e.g…

In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …