arXiv research

Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.

Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.

New neural network criterion connects RH to minimization problem.

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultima…

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ri…

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfac…

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and t…

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a polynomial-length certificate that can be verified in polynomial time to prove that K…

We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming $S^3$-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assumin…

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…

Virtual links are generalizations of classical links that can be represented by links embedded in a ``thickened'' surface $Σ\times I$, product of a Riemann surface of genus $h$ with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the $1/N^2$ expansion of an i…

The study limits intersections of curves on a torus.

Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases wh…

Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…

A method to describe Riemann surfaces using graph profiles is proposed.

Study on the spectrum of drift Laplacian on Ricci expanders.

We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…

The study examines Riemann solitons and almost solitons on specific Kenmotsu manifolds.

Study Liouville action for harmonic maps between Riemann surfaces.

Every open Riemann surface can be triangulated with equilateral triangles.

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…

The article investigates almost Riemann solitons and gradient almost Riemann solitons in a specific type of manifold.

Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.

Study classifies Riemann solitons on specific 3D Lorentzian groups.

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…

Paper examines conditions making Riemann solitons trivial and estimates their scalar curvature.

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

Riemann's mathematical papers contain many ideas that arise from physics, and some of them are motivated by problems from physics. In fact, it is not easy to separate Riemann's ideas in mathematics from those in physics. Furthermore, Riemann's philosophical ideas are often in the background of his work on science. The …

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…

New symmetries found in Riemann-Cartan geometries.

Groupoids help define Riemann sums on manifolds.

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origin…

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by S…

Criterion found for Teichmüller extremal maps on infinite Riemann surfaces.

A framework for hypothesis testing on attributed graphs using sampling.

The study examines Fisher-Riemann geodesics for nonparametric probability densities.

New invariant for Riemann surfaces connects to moduli space classes.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…

Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold $M$ is equi…

Study curve shortening flow on Riemann surfaces with conic singularities.

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

Derives the derivative of the Riemann-Hilbert map for surface connections.

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.