Showing papers in "Contemporary mathematics in 2022"

On F-Contractions: A Survey

Abstract: D. Wardowski proved in 2012 a generalization of Banach Contraction Principle by introducing F-contractions in metric spaces. In the next ten years, a great number of researchers used Wardowski's approach, or some of its modifications, to obtain new fixed point results for single- and multivalued mappings in various kinds of spaces. In this review article, we present a survey of these investigations, including some improvements, in particular concerning conditions imposed on function F entering the contractive condition.

Equivalence of finite group actions on Riemann surfaces and algebraic curves

Abstract: We consider conformal actions of the finite group G G on a closed Riemann surface S S , as well as algebraic actions of G G on smooth, complete, algebraic curves over an arbitrary, algebraically closed field. There are several notions of equivalence of actions, the most studied of which is topological equivalence, because of its close relationship to the branch locus of moduli space. A second important equivalence relation is that induced by representation of G G on spaces of holomorphic q q -differentials. The notion of topological equivalence does not work well in positive characteristic. We shall discuss an alternative to topological equivalence, which we dub equisymmetry, that may be applied in all characteristics. The relation is induced by families of curves with G G -action, and it works well with rotation constants and q q -differentials, which are also defined in positive characteristic. After giving an overview of the various equivalence relations (conformal/algebraic, topological, q q -differentials, rotation constants, equisymmetry) we focus on the interconnections among rotation constants, q q -differentials, and equisymmetry.

An introduction to 𝑝-adic and motivic integration, zeta functions and invariants of singularities

Abstract: Motivic integration was introduced by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. To do so, he constructed a certain motivic measure on the arc space of a complex variety, taking values in a completion of the Grothendieck ring of algebraic varieties. Later, Denef and Loeser, together with the works of Looijenga and Batyrev, developed in a series of articles a more complete theory of the subject, with applications in the study of varieties and singularities. In particular, they developed a motivic zeta function, generalizing the usual (pp-adic) Igusa zeta function and Denef-Loeser topological zeta function. These notes are a basic introduction to geometric motivic integration, the precedentpp-adic ideas associated with it, and the theory of the above zeta functions related to them. We focus in practical computations and ideas, providing examples and a recent formula obtained by means of partial resolutions.

One dimensional equisymmetric strata in moduli space

Abstract: The moduli space M g \mathcal {M}_{g} of surfaces of genus g ≥ 2 g\geq 2 is the space of conformal equivalence classes of closed Riemann surfaces of genus g g . This space is a complex, quasi-projective variety of dimension 3 g − 3 3g-3 . The singularity set of the moduli space, which is roughly the same as the branch locus, becomes increasingly complicated as the genus grows. To better understand the branch locus, the moduli space may be stratified into a finite, disjoint union of smooth, irreducible, quasi-projective subvarieties called equisymmetric strata. Each stratum corresponds to a collection of surfaces of the same symmetry type. The topology of these strata is largely unknown. In this paper we explore the topology of the complex 1-dimensional strata, which are smooth, connected, complex curves with punctures. We are able to describe the topology of these strata explicitly, as punctured Riemann surfaces, in terms of the action of the automorphism group of the surfaces in the stratum.

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

Abstract: We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of ( 2 , 2 ) (2,2) -isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the ( 2 , 2 ) (2,2) -isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.

On the Riemann-Hurwitz formula for regular graph coverings

Abstract: The aim of this paper is to present a few versions of the Riemann-Hurwitz formula for a regular branched covering of graphs. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. We consider a finite group acting on a graph, possibly with fixed and invertible edges, and the respective factor graph. Then, the obtained Riemann-Hurwitz formula relates genus of the graph with genus of the factor graph and orders of the vertex and edge stabilizers.

Multiradical isogenies

Abstract: We argue that for all integers N ≥ 2 N \geq 2 and g ≥ 1 g \geq 1 there exist “multiradical” isogeny formulae, that can be iteratively applied to compute ( N k , … , N k ) (N^k, \ldots , N^k) -isogenies between principally polarized g g -dimensional abelian varieties, for any value of k ≥ 2 k \geq 2 . The formulae are complete: each iteration involves the extraction of g ( g + 1 ) / 2 g(g+1)/2 different N N th roots, whence the epithet multiradical, and by varying which roots are chosen one computes all N g ( g + 1 ) / 2 N^{g(g+1)/2} extensions to an ( N k , … , N k ) (N^k, \ldots , N^k) -isogeny of the incoming ( N k − 1 , … , N k − 1 ) (N^{k-1}, \ldots , N^{k-1}) -isogeny. Our group-theoretic argumentation is heuristic, but it is supported by concrete formulae for several prominent families. As our main application, we illustrate the use of multiradical isogenies by implementing a hash function from ( 3 , 3 ) (3,3) -isogenies between Jacobians of superspecial genus- 2 2 curves, showing that it outperforms its ( 2 , 2 ) (2,2) -counterpart by an asymptotic factor ≈ 9 \approx 9 in terms of speed.

Chen-Ricci inequalities for Riemannian maps and their applications

Abstract: Riemannian maps between Riemannian manifolds, originally introduced by A.E. Fischer in [Contemp. Math. 132 (1992), 331–366], provide an excellent tool for comparing the geometric structures of the source and target manifolds. Isometric immersions and Riemannian submersions are particular examples of such maps. In this work, we first prove a geometric inequality for Riemannian maps having a real space form as a target manifold. Applying it to the particular case of Riemannian submanifolds, we recover a classical result, obtained by B.-Y. Chen in [Glasgow Math. J. 41 (1999), 33–41], which nowadays is known as the Chen-Ricci inequality. Moreover, we extend this inequality in case of Riemannian maps with a complex space form as a target manifold. We also improve this inequality when the Riemannian map is Lagrangian. Applying it to Riemannian submanifolds, we recover the improved Chen-Ricci inequality for Lagrangian submanifolds in a complex space form, that is a basic inequality obtained by S. Deng in [Int. Electron. J. Geom. 2 (2009), 39-45] as an improvement of a geometric inequality stated by B.-Y. Chen in [Arch. Math. (Basel) 74 (2000), 154–160].

On the Classical Solutions for the Kuramoto-Sivashinsky Equation with Ehrilch-Schwoebel Effects

Abstract: The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface morphology during Molecular Beam Epitaxy growth, provoked by an interplay between deposition of atoms onto the surface and the relaxation of the surface profile through surface diffusion. It consists of a nonlinear fourth order partial differential equation. Using energy methods we prove the well-posedness (i.e., existence, uniqueness and stability with respect to the initial data) of the classical solutions for the Cauchy problem, associated with this equation.

Galerkin Method and Its Residual Correction with Modified Legendre Polynomials

Abstract: Accuracy and error analysis is one of the significant factors in computational science. This study employs the Galerkin method to solve second order linear or nonlinear Boundary Value Problems (BVPs) of Ordinary Differential Equations (ODEs) with modified Legendre polynomials to seek numerical solutions. The residual function of a differential operator is used as non-homogeneous term information of an error differential equation. The Galerkin approximation is then improved or corrected by solving the error differential equation by the Galerkin method using the same polynomials. Thus we apply the double layer Galerkin method to a variety of instances. We compare approximate solutions with exact ones and results available in the literature, and in every case, we find better accuracy.

Initial state reconstruction on graphs

Abstract: The presence of noise is an intrinsic problem in acquisition processes for digital images. One way to enhance images is to combine the forward and backward diffusion equations. However, the latter problem is well known to be exponentially unstable with respect to any small perturbations on the final data. In this scenario, the final data can be regarded as a blurred image obtained from the forward process, and that image can be pixelated as a network. Therefore, we study in this work a regularization framework for the backward diffusion equation on graphs. Our aim is to construct a spectral graph-based solution based upon a cut-off projection. Stability and convergence results are provided together with some numerical experiments.

Automorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics

Abstract: | Automorphisms. | Group theory. | AMS: Functions of a complex variable – Riemann surfaces. | Algebraic geometry – Curves in algebraic geometry. | Group theory and generalizations – Other groups of matrices – Fuchsian groups and their generalizations (group-theoretic aspects). | Group theory and generalizations – Permutation groups – Finite automorphism groups of algebraic, geometric, or combinatorial structures. | Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Arithmetic aspects of dessins d’enfants, Bely˘ı theory. | Manifolds and cell complexes – Low-dimensional topology in specific dimensions – 2-dimensional topology (including mapping class groups of surfaces, Teichm¨uller theory, curve complexes, etc.).

Universal 𝑞-gonal tessellations and their Petrie paths

Abstract:

In [Sin88] the second author showed that the Farey map F \mathbb {F} is a model of the universal triangular tessellation. This is a tessellation that covers every other triangular tessellation on an orientable surface. More precisely, the automorphism group of F \mathbb {F} is the classical modular group Γ = P S L ( 2 , Z ) \Gamma =PSL(2,\mathbb {Z}) and every triangular map is the quotient of F \mathbb {F} by a subgroup of Γ \Gamma . The aim of this paper is to describe universal q q -gonal tessellations. Here the modular group will be replaced by Hecke groups.

In [SS16] it was shown that the Petrie paths of the Farey map pass through vertices whose numerators and denominators are Fibonacci numbers. In section 8 we consider Hecke-Fibonacci sequences which arise out of universal q q -gonal tessellations.

A Jordan-Schur Algorithm for Solving Sylvester and Lyapunov Matrix Equations

Abstract: This paper presents a version of the Bartels-Stewart algorithm for solving the Sylvester and Lyapunov equations that utilizes the Jordan-Schur form of the equation matrices. The Jordan-Schur form is a type of Schur form which contains additional information about the Jordan structure of the corresponding matrix. This information can be used to solve more efficiently the Sylvester and Lyapunov equations in some cases. A two-level algorithm is implemented which allows us to find directly non-scalar blocks of the solution matrix. These blocks have sizes that are determined by the Weyr characteristics associated with the eigenvalues of the equation matrices. In the case of large elements of the Weyr characteristics associated with multiple eigenvalues, the determination of the solution blocks can be done more efficiency. Also, the blocks equations can be more appropriate in solving the Sylvester and Lyapunov equations in the case of parallel computations. Results obtained from numerical experiments confirm that the accuracy of the new algorithm is comparable with the accuracy of the Bartels-Stewart algorithm.

Reduction of superelliptic Riemann surfaces

Abstract:

For a superelliptic curve X \mathcal {X} , defined over Q \mathbb {Q} , let p \mathfrak {p} denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of X \mathcal {X} such that: i) the corresponding moduli point p \mathfrak {p} has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point p \mathfrak {p} with minimal weighted height, as defined in \cite{b-g-sh}, and part ii) by the classical reduction of the binary forms.

Introduction to 𝑝-adic Igusa zeta functions

Abstract: These notes constitute the content of the series of lectures Introduction to local zeta functions by the second author, in the Lluís Santaló Research Summer School 2019: p p -Adic Analysis, Arithmetic and Singularities, June 24-28, at the Palacio de la Magdalena in Santander. We want to thank the organizers of the school for their excellent work. The lectures were intended as an elementary introduction to p p -adic Igusa zeta functions and related topics. We hope that these notes reflect that goal. Our text is complementary to the one of León-Cardenal and Zúñiga-Galindo [Rev. Integr. Temas Mat., 37, 2019, 45–76]. The notes of Nicaise [MSJ Mem., 21, 2010, 141–166] are an introduction to p p -adic and motivic zeta functions. Substantial survey articles are the \lq old\rq Bourbaki report of Denef [Séminaire Bourbaki, 1990/91, 1992, 359–386] and the more recent paper of Meuser [Amer. J. Math., 138, 2016, 149–179].

Planar representations of group actions on surfaces

Abstract: The space of skeletal signatures was introduced as a simple but generally much coarser two-dimensional representation of the space of all signatures with which a group can act on a compact oriented surface. In this paper, we provide a complete characterization of the groups for which the skeletal signature space provides a complete and accurate picture of the structure of the full space of signatures. We show that such groups fall into three distinct families, and for two of these families, we show that for sufficiently large genus, the skeletal signature space depends essentially only on group order, and we explicitly describe these spaces. For the third family, we show that the skeletal signature space depends much more strongly upon group structure and provide some partial analysis of the structure of this space.

Dessins d’enfants with a given bipartite graph

Abstract:

Let G \mathcal {G} be a finite connected bipartite graph. We describe the set D e s s G { \mathcal Dess}_{\mathcal G} consisting of all the isomorphism classes of the dessins d’enfants whose underlying bipartite graph is isomorphic to G \mathcal {G} . Such a description is explicit and it permits, for each dessin, to provide some of its Galois invariants, such as its passport, its automorphism group, its monodromy group and its Z-orientability property.

Extending Harvey’s surface kernel maps

Abstract: Let S S be a compact Riemann surface and G G a group of conformal automorphisms of S S with S 0 = S / G S_0=S/G . S S is a finite regular branched cover of S 0 S_0 . If U U denotes the unit disc, let Γ \Gamma and Γ 0 \Gamma _0 be the Fuchsian groups with S = U / Γ S= U/\Gamma and S 0 = U / Γ 0 S_0 = U/\Gamma _0 . There is a group homomorphism of Γ 0 \Gamma _0 onto G G with kernel Γ \Gamma and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of Γ 0 \Gamma _0 . In his 1971 1971 paper Harvey showed that when G G is a cyclic group, there is a unique simplest representative for this equivalence class. His result has played an important role in establishing subsequent results about conformal automorphism groups of surfaces. We extend his result to some surface kernel maps onto arbitrary finite groups. These can be used along with the Schreier-Reidemeister Theory to find a set of generators for Γ \Gamma and the action of G G as an outer automorphism group on the fundamental group of S S putting the action on the fundamental group and the induced action on homology into a relatively simple format. As an example we compute generators for the fundamental group and an integral homology basis together with the action of G G when G G is S 3 \mathcal {S}_3 , the symmetric group on three letters. The action of G G shows that the homology basis found is not an adapted homology basis.

Recent progress of biharmonic hypersurfaces in space forms

Abstract: In the past two decades, biharmonic submanifolds have attracted much attention from mathematicians all over the world. In particular, concerning Chen’s conjecture and the generalized Chen’s conjecture, many meaningful results have been obtained. In this survey paper, we will restrict our attention to biharmonic hypersurfaces in the real space forms. In the first three parts, we give a short survey on some new developments on biharmonic hypersurfaces in Euclidean spaces, hyperbolic spaces and Euclidean spheres, respectively. In the last section, we outline the proof of Chen’s conjecture for hypersurfaces of R 5 \mathbb {R}^5 [Adv. Math, 383 (2021), Paper No. 107697, 28] and point out the potential importance of the method and the key techniques used in the proof for further studies on biharmonic hypersurfaces in space forms.

Charting the 𝑞-Askey scheme

Abstract: Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q$-Askey scheme together with their $q$-hypergeometric representations by three sequences $x_k, h_k, g_k$ of Laurent polynomials in $q^k$, two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme.

Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

Abstract: Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra’s radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G G and a one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first-order quantum Hamiltonians. We also explicitly describe the Schrödinger operator.

On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

Abstract: The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.

Freudenthal Suspension Theorem And James-Hopf Invariant of Spheres

Abstract: This paper relates more precisely the image of the suspension map Σ:π2n+k(Sn) → π2n+k+1(Sn+1) with the kernel of James-Hopf invariant h2:π2n+k+1(Sn+1) → π2n+k+1(S2n+1) for k ≤ 9.

Future directions in automorphisms of surfaces, graphs, and other related topics

Abstract: The study of Riemann surfaces and the groups which act on them is a classical area of research dating back to the latter half of the 19th century. Research in this field has wide-reaching implications in geometry and topology, algebra, combinatorics, analysis, and number theory through related topics such as the study of dessins d’enfants, mapping class groups, and graphs on surfaces. Today, this is still a rich area of research with many open questions. In this expository article we pose 78 open problems, contextualize them within the field, and discuss partial results or progress toward answering the questions, when relevant.

Geodesic Triangles in H² with Short Sides

Abstract: We prove that if M is an orientable hyperbolic surface without boundary (possibly compact, possibly with infinitely generated fundamental group) and γ is a closed geodesic in M then any side of any triangle formed by distinct lifts of γ in H2 is shorter than γ.

Cyclic and dihedral actions on Klein surfaces with 2 boundary components

Abstract:

In this paper, we obtain the cyclic and dihedral groups which occur as a group of automorphisms of Klein surfaces of topological genus g g with 2 boundary components, either orientable or non-orientable. For each of those groups G G we determine the values of g g such that G G acts on a surface of genus g g both in orientable or non-orientable cases. As a noteworthy result we obtain that whenever the cyclic group C N C_N acts on a surface of genus g g and given orientability character, the dihedral group D N D_N of order 2 N 2N also acts on a surface of the same topological type. Besides, we give a topological action of the group on the surface, by using the uniformization by means of NEC groups. The results are computable, and we exhibit explicitly, distinguishing according to the orientability, all groups C N C_N and D N D_N acting on surfaces of each genus g g for 1 ≤ g ≤ 5 1\leq g\leq 5 .

On infinite octavalent polyhedral surfaces

Abstract: In this paper, we find octavalent triply periodic polyhedral surfaces that have identifiable underlying Riemann structures. We focus our study to eightfold cyclic covers over the thrice punctured sphere. There are three such surfaces up to equivalency, where one of them was studied as the cover of Fermat’s quartic in \cite{Lee2017} by the second author. In this paper, we study the remaining two cases. Results include a polyhedral surrogate of Schwarz minimal CLP surface and a polyhedral surface that is not a cover of the Bolza surface despite its construction.

Solvability of Higher Order Iterative System with Non-Homogeneous Integral Boundary Conditions

Abstract: The aim of this paper is to establish the existence of positive solutions by determining the eigenvalue intervals of the parameters μ1, μ2, ..., μm for the iterative system of nonlinear differential equations of order p wi(p) (x) + μi ai (x) fi (wi+1 (x) ) = 0, 1 ≤ i ≤ m, x∈ [0,1], wm+1 (x) = w1 (x), x ∈ [0,1], satisfying non-homogeneous integral boundary conditions wi (0) = 0, wi' (0) = 0, ..., wi(p-2) (0) = 0, wi(r) - ηi ∫01gi(τ)wi(r)(τ)dτ = λi, 1 ≤ i ≤ m, where r ∈ {1, 2, ..., p−2} but fixed, p ≥ 3 and ηi, λi ∈ (0, ∞) are parameters. The fundamental tool in this paper is an application of the Guo-Krasnosel'skii fixed point theorem to establish the existence of positive solutions of the problem for operators on a cone in a Banach space. Here the kernels play a fundamental role in defining an appropriate operator on a suitable cone.