ARNOLD  MATHEMATICAL  JOURNAL
Editor-in-Chief:
     Sergei Tabachnikov
Managing Editor:
       Maxim Arnold


A  Journal  of  the IMS,
Stony Brook University
Published by

Recent Papers

Research Papers
  1. Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces
    Gabriel Katz
    Arnold Mathematical Journal (2024)
    Published: 16 May 2024
    Abstract
    Let $A$ denote the cylinder ${\mathbb {R}} \times S^1$ or the band ${\mathbb {R}} \times I$, where I stands for the closed interval. We consider 2-moderate immersions of closed curves (“doodles”) and compact surfaces (“blobs”) in $A$, up to cobordisms that also are 2-moderate immersions in $A \times [0, 1]$ of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order $\ge 3$ to the fibers of the obvious projections $A \rightarrow S^1$, $A \times [0, 1] \rightarrow S^1 \times [0, 1]$ or $A \rightarrow I$, $A \times [0, 1] \rightarrow I \times [0, 1]$. These bordisms come in different flavors: in particular, we consider one flavor based on regular embeddings of doodles and blobs in $A$. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on $A= {\mathbb {R}} \times I$, our computations of 2-moderate immersion bordisms $\textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)$ are near complete: we show that they can be described by an exact sequence of abelian groups $$\begin{aligned} 0 \rightarrow {\textbf{K}} \rightarrow \textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)\big /\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) {\mathop {\longrightarrow }\limits ^{{\mathcal {I}} \rho }} {\mathbb {Z}} \times {\mathbb {Z}} \rightarrow 0, \end{aligned}$$ where $\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) \approx {\mathbb {Z}} \times {\mathbb {Z}}$, the epimorphism ${\mathcal {I}} \rho $ counts different types of crossings of immersed doodles, and the kernel ${\textbf{K}}$ contains the group $({\mathbb {Z}})^\infty $ whose generators are described explicitly.
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  2. An Observation About Conformal Points on Surfaces
    Peter Albers & Gabriele Benedetti
    Arnold Mathematical Journal (2024)
    Published: 12 April 2024
    Abstract
    We study the existence of points on a compact oriented surface at which a symmetric bilinear two-tensor field is conformal to a Riemannian metric. We give applications to the existence of conformal points of surface diffeomorphisms and vector fields.
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  3. Kirillov Polynomials for the Exceptional Lie Algebra $\mathfrak g_{2}$
    Martin T. Luu
    Arnold Mathematical Journal (2024)
    Published: 04 April 2024
    Abstract
    As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of $q$ elements and fixed Jordan type. One obtains polynomials with respect to $q$ with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra $\mathfrak g_2$. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.
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  4. Euler Characteristics of Collapsing Alexandrov Spaces
    Tadashi Fujioka
    Arnold Mathematical Journal (2024)
    Published: 18 March 2024
    Abstract
    We prove that the Euler characteristic of a collapsing Alexandrov space (in particular, a Riemannian manifold) is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the fibers over the strata. This was conjectured by Semyon Alesker.
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  5. Polynomial Superpotential for Grassmannian ${\text {Gr}}(k,n)$ from a Limit of Vertex Function
    Andrey Smirnov & Alexander Varchenko
    Arnold Mathematical Journal (2024)
    Published: 15 March 2024
    Abstract
    In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $X=T^{*}{\text {Gr}}(k,n)$. This integral representation can be used to compute the $\hbar\rightarrow \infty$ limit of the vertex function, where $\hbar$ denotes the equivariant parameter of a torus acting on $X$ by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian ${\text {Gr}}(k,n)$ with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.
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  6. A Note on Contact Manifolds with Infinite Fillings
    Zhengyi Zhou
    Arnold Mathematical Journal (2024)
    Published: 01 March 2024
    Abstract
    We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension $> 1,$ which were previously unknown for dimensions equal to $4n+1$. The argument does not involve understanding factorizations in the symplectic mapping class group.
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  7. Higher Dimensional Versions of Theorems of Euler and Fuss
    Peter Gibson, Nicolau Saldanha & Carlos Tomei
    Arnold Mathematical Journal (2024)
    Published: 22 January 2024
    Abstract
    We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If ${\text{tr}}\, A = 1$, there exists a bijection between the orthogonal group $O(n)$ and the set of such labeled simplices. Similarly, if ${\text {tr}}\,A^2 = 1$, there are families of parallelotopes and of cross polytopes, also indexed by $O(n)$.
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  8. Toric Orbit Spaces Which are Manifolds
    Anton Ayzenberg & Vladimir Gorchakov
    Arnold Mathematical Journal (2024)
    Published: 03 January 2024
    Abstract
    We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.
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  9. Newton Polyhedra and Stratified Resolution of Singularities in the Class of Generalized Power Series
    Palma-Márquez, Jesús
    Arnold Mathematical Journal (2023)
    Published: 08 December 2023
    Abstract
    We generalize the construction of a toric variety associated with an integer convex polyhedron to construct generalized analytic varieties associated with polyhedra with not necessarily rational vertices. For germs of generalized analytic functions with a given Newton polyhedron $\Gamma $, the generalized analytic variety associated with $\Gamma $ provides a stratified resolution of singularities of these functions; also ensuring a full resolution for almost all of them. Thus, this constructive and elementary approach replaces the non-effective previous proof of this result based on consecutive blow-ups.
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  10. The Probabilistic Method in Real Singularity Theory
    Antonio Lerario & Michele Stecconi
    Arnold Mathematical Journal (2023)
    Published: 28 November 2023
    Abstract
    We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.
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  11. Cohomology of Spaces of Complex Knots
    V. A. Vassiliev
    Arnold Mathematical Journal (2023)
    Published: 24 October 2023
    Abstract
    We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in ${{\mathbb {C}}}^k$, $k \ge 3$, and obtain these groups of low dimensions.
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  12. Existence of a Conjugate Point in the Incompressible Euler Flow on a Three-Dimensional Ellipsoid
    L. A. Lichtenfelz, T. Tauchi & T. Yoneda
    Arnold Mathematical Journal (2023)
    Published: 13 October 2023
    Abstract
    The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold M is related to the Lagrangian stability of a solution of the incompressible Euler equation on M. The Misiołek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiołek curvature and give a positivity result of the Misiołek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary.
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  13. On Groups with Few Subgroups not in the Chermak-Delgado Lattice
    David Burrell, William Cocke & Ryan McCulloch
    Arnold Mathematical Journal (2023)
    Published: 20 September 2023
    Abstract
    We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is $S_3$.
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  14. Normality of Two Families of Meromorphic Functions Concerning Partially Shared Values
    Manish Kumar
    Arnold Mathematical Journal (2023)
    13 September 2023
    Abstract
    In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let ${\mathcal {F}}$ and ${\mathcal {G}}$ be two families of meromorphic functions on a domain $D$, and $a,\ b,\ c$ be three finite complex numbers such that $a\ne 0$ and $b\ne c$. Suppose that ${\mathcal {G}}$ is normal in $D$ such that no sequence in ${\mathcal {G}}$ converges locally uniformly to infinity in $D$. If $n\ge 3$ and for each function $f\in {\mathcal {F}}$ there exists $g\in {\mathcal {G}}$ such that $f^{'}-af^{n}$ and $g^{'}-ag^{n}$ partially share the values $b$ and $c$, then ${\mathcal {F}}$ is normal in $D$. Further, examples are given to establish the sharpness of the result.
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  15. Approximate Real Symmetric Tensor Rank
    Alperen A. Ergür, Jesus Rebollo Bueno & Petros Valettas
    Arnold Mathematical Journal (2023)
    Published: 22 August 2023
    Abstract
    We investigate the effect of an $\varepsilon $-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $\left\Vert \cdot \right\Vert $ on the space of symmetric $d$-tensors, and $\varepsilon \gt 0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon $-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon $-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind, we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.
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  16. Steady-State Flows of Ideal Incompressible Fluid with Velocity Pointwise Orthogonal to the Pressure Gradient
    Vladimir Yu. Rovenski & Vladimir A. Sharafutdinov
    Arnold Mathematical Journal (2023)
    Published: 01 August 2023
    Abstract
    A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. We describe the local structure of Gavrilov flows in terms of the geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for axisymmetric Gavrilov flows and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.
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  17. On Affine Real Cubic Surfaces
    S. Finashin & V. Kharlamov
    Arnold Mathematical Journal (2023)
    Published: 25 July 2023
    Abstract
    We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.
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  18. Supertraces on Queerified Algebras
    Dimitry Leites & Irina Shchepochkina
    Arnold Mathematical Journal (2023)
    Published: 06 July 2023
    Abstract
    We describe supertraces on “queerifications” (see arXiv:2203.06917 ) of the algebras of matrices of “complex size”, algebras of observables of Calogero–Moser model, Vasiliev higher spin algebras, and (super)algebras of pseudo-differential operators. In the latter case, the supertraces establish complete integrability of the analogs of Euler equations to be written (this is one of several open problems and conjectures offered).
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  19. On Partial Differential Operators Which Annihilate the Roots of the Universal Equation of Degree $k$
    Daniel Barlet
    Arnold Mathematical Journal (2023)
    Published: 21 June 2023
    Abstract
    The aim of this paper is to study in details the regular holonomic D-module introduced in Barlet (Math Z 302 $n^03$ : 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface $\{\Delta (\sigma ).\sigma _k = 0 \}$ are given by the local system generated by the power $\lambda $ of the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0 $. We show that for $\lambda \in \mathbb {C} {\setminus } \mathbb {Z}$ this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set $\{\sigma _k\Delta (\sigma ) \not = 0\}$. We then study the structure of these D-modules in the cases where $\lambda = 0, 1, -1$ which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when $\lambda $ is in $\mathbb {Z}$. As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near $-1$ of the equation: $\begin{aligned} z^k + \sum _{h=-1}^k (-1)^h\sigma _hz^{k-h} - (-1)^k = 0. \end{aligned}$ near $z^k - (-1)^k = 0$ .
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  20. Sandpile Solitons in Higher Dimensions
    Nikita Kalinin
    Arnold Mathematical Journal volume 9, pages 435-454 (2023)
    Published: 30 January 2023
    Abstract
    Let $p\in {\mathbb {Z}}^n$ be a primitive vector and $\Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min (p\cdot z, 0)$. The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $\Psi $ “at infinity”. We apply this result to sandpile models on ${\mathbb {Z}}^n$. We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope $A$ without lattice points except its vertices. Namely, for each function $$\begin{aligned} \Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min _{p\in A\cap {\mathbb {Z}}^n}(p\cdot z+c_p), c_p\in {\mathbb {Z}}\end{aligned}$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $\Psi $ “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).
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  21. Classification of Schubert Galois Groups in $Gr(4,9)$
    Abraham Martín del Campo, Frank Sottile & Robert Lee Williams
    Arnold Mathematical Journal volume 9, pages 393-433 (2023)
    Published: 17 January 2023
    Abstract
    We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.
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  22. A Generalized Problem Associated to the Kummer-Vandiver Conjecture
    Hiroki Sumida-Takahashi
    Arnold Mathematical Journal volume 9, pages 381-391 (2023)
    Published: 07 November 2022
    Abstract
    To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let $p$ be an odd prime number and $\zeta _p$ a primitive $p$-th root of unity. Using new programs, we compute the Iwasawa invariants of ${\textbf{Q}}(\sqrt{d},\zeta _p)$ in the range $|d|\lt200$ and $200\lt1{,}000{,}000$. From our data, the actual numbers of exceptional cases seem to be near the expected numbers for $p\lt1{,}000{,}000$. Moreover, we find a few rare exceptional cases for $|d|\lt10$ and $p\gt1{,}000{,}000$. We give two partial reasons why it is difficult to find exceptional cases for $d=1$ including counter-examples to the Kummer–Vandiver conjecture.
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  23. A Polyhedral Homotopy Algorithm for Real Zeros
    Alperen A. Ergür & Timo de Wolff
    Arnold Mathematical Journal volume 9, pages 305-338 (2023)
    Published: 27 October 2022
    Abstract
    We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying $A$-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of $A$-discriminant amoeba complements, and computational real algebraic geometry.
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  24. Counting Tripods on the Torus
    Jayadev S. Athreya, David Aulicino & Harry Richman
    Arnold Mathematical Journal volume 9, pages 359-379 (2023)
    Published: 29 August 2022
    Abstract
    Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in ${\mathbb {C}}^2$, and we give an asymptotic counting result using lattice point counting techniques.
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  25. Enumeration of Multi-rooted Plane Trees
    Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle & Vasilisa Shramchenko
    Arnold Mathematical Journal (2023)
    Published: 11 May 2023
    Abstract
    We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.
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  26. Red Sizes of Quivers
    Eric Bucher & John Machacek
    Arnold Mathematical Journal (2023)
    Published: 06 March 2023
    Abstract
    In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper, we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.
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  27. Hochschild Entropy and Categorical Entropy
    Kohei Kikuta & Genki Ouchi
    Arnold Mathematical Journal (2022)
    Published: 18 July 2022
    Abstract
    We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.
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  28. Relations Between Escape Regions in the Parameter Space of Cubic Polynomials
    Araceli Bonifant, Chad Estabrooks & Thomas Sharland
    Arnold Mathematical Journal (2022)
    Published: 22 July 2022
    Abstract
    We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves $\mathcal {S}_{p}$ as the set of all cubic polynomials with a marked critical point of period p. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves $\mathcal {S}_{1}$ and $\mathcal {S}_{2}$.
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  29. Correction to: The Dynamics of Complex Box Mappings
    Trevor Clark, Kostiantyn Drach, Oleg Kozlovski & Sebastian van Strien
    Arnold Mathematical Journal 2022
    Published: 01 August 2022
    Abstract
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  30. Spontaneously Stochastic Arnold’s Cat
    Alexei A. Mailybaev & Artem Raibekas
    Arnold Mathematical Journal 9, pages 339-357 (2023)
    Published: 24 August 2022
    Abstract
    We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, such as non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics.
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  31. Correction to: The Dynamics of Complex Box Mappings
    Trevor Clark, Kostiantyn Drach, Oleg Kozlovski & Sebastian van Strien
    Arnold Mathematical Journal 9, pages 303-304 (2023)
    Published: 19 September 2022
    Abstract
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  32. Revisiting Kepler: New Symmetries of an Old Problem
    Gil Bor & Connor Jackman
    Arnold Mathematical Journal 9, pages 267-299 (2023)
    Published: 12 September 2022
    Abstract
    The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits $\mathrm { PSL}_2(\mathbb {R})$ as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre $y^{\prime \prime }= \omega (x, y, y^{\prime })$, vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.
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  33. Properness of Polynomial Maps with Newton Polyhedra
    Toshizumi Fukui, Takeki Tsuchiya
    Arnold Mathematical Journal (2022)
    Published: 29 June 2022
    Abstract
    We discuss the notion of properness of a polynomial map ${f}:\mathbb {K}^m\rightarrow \mathbb {K}^n$, $\mathbb {K}=\mathbb {C}$ or $\mathbb {R}$, at a point of the target. We present a method to describe the set of non-proper points of ${f}$ with respect to Newton polyhedra of ${f}$. We obtain an explicit precise description of such a set of ${f}$ when ${f}$ satisfies certain condition (1.5). A relative version is also given in Sect. 3. Several tricks to describe the set of non-proper points of ${f}$ without the condition (1.5) is also given in Sect. 5.
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  34. Classification of Generic Spherical Quadrilaterals
    Andrei Gabrielov
    Arnold Mathematical Journal (2022)
    Published: 26 April 2022
    Abstract
    Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.
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  35. Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties
    Grigory Solomadin
    Arnold Mathematical Journal 9:1 105-150 (2023)
    Published: 22 April 2022
    Abstract
    In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.
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  36. Quantitative Uncertainty Principles Related to Lions Transform
    A. Achak, A. Abouelaz, R. Daher, N. Safouane
    Arnold Mathematical Journal (2022)
    Published: 16 April 2022
    Abstract
    We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform.
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  37. Invariant Factors as Limit of Singular Values of a Matrix
    Kiumars Kaveh & Peter Makhnatch
    Arnold Mathematical Journal 2022 8:3
    Published: 16 September 2022
    Abstract
    The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let $A(t)$ be an $n \times n$ matrix whose entries are Laurent series in $t$. We show that, as $t \rightarrow 0$, the logarithms of singular values of $A(t)$ approach the invariant factors of $A(t)$. This leads us to suggest logarithms of singular values of an $n \times n$ complex matrix as an analog of the logarithm map on $(\mathbb {C}^*)^n$ for the matrix group ${\text {GL}}(n, \mathbb {C})$.
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  38. Jordan Types of Triangular Matrices over a Finite Field
    Dmitry Fuchs & Alexandre Kirillov Sr.
    Arnold Mathematical Journal (2022) 8:3
    Published: 13 October 2022
    Abstract
    Let $\lambda $ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_\lambda (q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $\lambda $. It is known that the polynomial $P_\lambda $ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)$, where $R_\lambda (0)\ne 0$ and $R_\lambda (1)\ne 0$. In this article, we study the polynomials $P_\lambda (q)$ and $R_\lambda (q)$. Our main results: an explicit formula for $d(\lambda )$ (an explicit formula for $e(\lambda )$ is known, see Proposition 3.3 below), a recursive formula for $R_\lambda (q)$ (a similar formula for $P_\lambda (q)$ is known, see Proposition 3.1 below), the stabilization of $R_\lambda $ with respect to extending $\lambda $ by adding strings of 1’s, and an explicit formula for the limit series $R_{\lambda 1^\infty }$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.
    Download PDF of the paper .
  39. Nontrivial Topological Quandles
    Boris Tsvelikhovskiy
    Arnold Mathematical Journal (2022) 8:3
    Published: 12 July 2022
    Abstract
    We show that there are infinitely many nonisomorphic quandle structures on any topogical space $X$ of positive dimension. In particular, we disprove Conjecture 5.2 in Cheng et al. (Topology Appl 248:64–74, 2018), asserting that there are no nontrivial quandle structures on the closed unit interval $[0,1]$.
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  40. Generalized Permutahedra and Schubert Calculus
    Avery St. Dizier, Alexander Yong
    Arnold Mathematical Journal (2022) 8:3
    Published: 27 June 2022
    Abstract
    We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.
    Download PDF of the paper .
  41. Maximum Likelihood Degree of Surjective Rational Maps
    Ilya Karzhemanov
    Arnold Mathematical Journal (2022) 8:3
    Published: 25 May 2022
    Abstract
    With any surjective rational map $f: \mathbb {P}^n \dashrightarrow \mathbb {P}^n$ of the projective space, we associate a numerical invariant (ML degree) and compute it in terms of a naturally defined vector bundle $E_f \longrightarrow \mathbb {P}^n$.
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  42. Holomorphic Atiyah–Bott Formula for Correspondences
    Grigory Kondyrev, Artem Prikhodko
    Arnold Mathematical Journal (2022) 8:3
    Published: 05 July 2022
    Abstract
    We show how the formalism of 2-traces can be applied in the setting of derived algebraic geometry to obtain a generalization of the holomorphic Atiyah–Bott formula to the case when an endomorphism is replaced by a correspondence.
    Download PDF of the paper .
  43. The Dynamics of Complex Box Mappings
    Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien
    Arnold Mathematical Journal (2022) 8:2, 319-410
    Published: 27 May 2022
    Abstract
    In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:

    • To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.
    • To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.
    • Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:
      • the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749-841, 2007), referred to below as KSS;
      • the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561-593, 2009);
      • the QC-Criterion and the Spreading Principle from KSS.
      The purpose of this paper is to make these tools more accessible so that they can be used as a "black box", so one does not have to redo the proofs in new settings.
    • To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275-296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:
      • puzzle pieces shrink to points,
      • (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.
    • We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattes maps in this setting.
    • We prove a version of Mane's Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.
    Download PDF of the paper .
  44. Two-Sided Fundamental Theorem of Affine Geometry
    Alexey Gorinov
    Arnold Mathematical Journal (2022)
    Published: 24 March 2022
    Abstract
    The fundamental theorem of affine geometry says that if a self-bijection $f$ of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected type, namely $f$ is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.
    Download PDF of the paper .
  45. Renormalization of Bicritical Circle Maps
    Gabriela Estevez, Pablo Guarino
    Arnold Mathematical Journal 9:1 69-104
    Published: 03 March 2022
    Abstract
    A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377 , 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies $C^{1+\alpha }$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type (Corollary 1.1).
    Download PDF of the paper .
  46. Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations
    Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich
    Arnold Mathematical Journal (2022) 8:2, 285-317
    Published: 22 February 2022
    Abstract
    A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
    Download PDF of the paper .
  47. Catastrophe in Elastic Tensegrity Frameworks
    Alexander Heaton, Sascha Timme
    Arnold Mathematical Journal (2022)
    Published: 18 February 2022
    Abstract
    We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.
    Download PDF of the paper .
  48. The ${{\mathbb {F}}}_p$-Selberg Integral
    Rimányi, Richárd, Alexander Varchenko
    Arnold Mathematical Journal (2022) 8:1
    Published: 24 January 2022
    Abstract
    We prove an ${{\mathbb {F}}}_p$-Selberg integral formula, in which the ${{\mathbb {F}}}_p$-Selberg integral is an element of the finite field ${{\mathbb {F}}}_p$ with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.
    Download PDF of the paper .
  49. Partial Duality of Hypermaps
    S. Chmutov, F. Vignes-Tourneret
    Arnold Mathematical Journal (2022)
    Published: 03 January 2022
    Abstract
    We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or $\tau $-model), or as three permutations on the set of half-edges (rotation system or $\sigma $-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.
    Download PDF of the paper .

Open Problems
  1. Open Problems on Billiards and Geometric Optics
    Misha Bialy, Corentin Fierobe, Alexey Glutsyuk, Mark Levi, Alexander Plakhov, Serge Tabachnikov
    Arnold Mathematical Journal (2022)
    Published: 17 January 2022
    Abstract
    This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021.
    Download PDF of the paper .

Research Expositions
  1. The Equivalence Problem in Analytic Dynamics for 1-Resonance
    Christiane Rousseau
    Arnold Mathematical Journal 9:1 1-39 (2023)
    Published: 20 January 2022
    Abstract
    When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point? The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is k-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic k-parameter families unfolding such singularities.
    Download PDF of the paper .

Archive 2015-2021

Journal Description

This journal intends to present mathematics so that it would be understandable and interesting to mathematicians independently on their narrow research fields. We invite articles exercising all formal and informal approaches to "unhide" the process of mathematical discovery.

The name of the journal is not only a dedication to the memory of Vladimir Igorevich Arnold (1937-2010), one of the most influential mathematicians of the twentieth century, but also a declaration that the journal hopes to maintain and promote the style which makes the best mathematical works by Arnold so enjoyable and which Arnold implemented in the journals where he was an editor-in-chief.

The ArMJ is organized jointly by the Institute for Mathematical Sciences (IMS) at Stony Brook, USA, and Springer Verlag, Germany.


1. Objectives

The journal intends to publish interesting and understandable results in all areas of Mathematics. The following are the most desirable features of publications that will serve as selection criteria:

  • Accessibility

    The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions that are necessary for understanding must be provided but also informal motivations even if they are well-known to the experts in the field. If a general statement is given, then the simplest examples of it are also welcome.

  • Interdisciplinary and multidisciplinary mathematics

    We would like to have many research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, except for the most popular combinations such as algebraic geometry and mathematical physics, analysis and dynamical systems, algebra and combinatorics, and the like. For this reason, this kind of research is often under-represented in specialized mathematical journals. The ArMJ will try to compensate for this.

  • Problems, objectives, work in progress

    Most scholarly publications present results of a research project in their "final" form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned but the very process of mathematical discovery remains hidden. Following Arnold, we will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. The journal intends to publish well-motivated research problems on a rather regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold's principle, a general formulation is less desirable than the simplest partial case that is still unknown.

  • Being interesting

    The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author's responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author's understanding of the overall picture is presented; however, these parts must be clearly indicated. Including motivations, informal parts, descriptions of other lines of research, possibly conducted by other mathematicians, should serve this principal objective: being interesting.

1.1  Types of Journal Articles

  • Research contribution.

    This is the classical format: a short (usually up to 20 pages) account of a research project containing original results and complete proofs of them. However, all of the above applies. Contributions containing very technical arguments may not be suitable for the ArMJ.

  • Research exposition.

    This is an exposition of a broad mathematical subject containing a description of recent results (proofs may be included or omitted), historical overview, motivations, open problems. A research exposition may take 60 pages or more.

  • Problem contribution.

    This is a description of an open problem. The problem must be well-motivated, illustrated by examples, and the importance of the problem must be explained. Alternatively, and closer to the original style of Arnold, a problem contribution may consist of a set of several problems that take very short space to state. Problems do not need to be original, however, the authorship must be carefully acknowledged. A problem contribution is meant to be short (normally, up to 4 pages, but exceptions are possible).

1.2  Comparison with Existing Journals

We feel that the following journals have objectives somewhat similar to those of the ArMJ.

  • Functional Analysis and its Applications
  • Russian Mathematical Surveys
  • American Mathematical Monthly
  • Bulletin of the AMS

However, each of these journals complies with only a part of our objectives list.

1.3  Why the Name

There are many great mathematicians of the twentieth century. The choice of the name may look random (why not, say, "Gelfand Mathematical Journal"? - we are often asked) but we have very specific reasons for using the name of Vladimir Arnold.

  1. The principles, according to which the journal operates, are most accurately associated with Vladimir Arnold. He had been actively promoting these or similar principles.

  2. For many years, V. Arnold had been the Editor-in-Chief of the journal Functional Analysis and its Applications (FAA). In 2006, V. Arnold launched a new journal, Functional Analysis and Other Mathematics (FAOM). The initial composition of the ArMJ Editorial Board consists mostly of former editors of the FAOM.

  3. Despite the close connections with the FAA and the FAOM, we decided to avoid mentioning "Functional Analysis" in the name of the journal. These names have appeared historically, and have nothing to do with scientific principles of the journals. More than that, the names are even confusing: not all mathematicians could guess that, say, Functional Analysis and its Applications welcomes papers in all areas of mathematics, including algebra and number theory. On the other hand, we wanted to have an indication of these connections in the name of the journal. The name of Vladimir Arnold serves as this indication.


2. Submissions

The journal is published quarterly, every issue consists of 100-150 pages. Manuscripts should be submitted online at http://www.editorialmanager.com/armj. Accepted file formats are LaTeX source (preferred) and MS Word.

Submission of a manuscript implies: that the work described has not been published before; that it is not under consideration for publication anywhere else; that its publication has been approved by all co-authors, if any, as well as by the responsible authorities - tacitly or explicitly - at the institute where the work has been carried out.

Authors wishing to include figures, tables, or text passages that have already been published elsewhere are required to obtain permission from the copyright owner(s) for both the print and online format and to include evidence that such permission has been granted when submitting their papers. Any material received without such evidence will be assumed to originate from the authors.

Editors

Editor-in-Chief:

   Sergei Tabachnikov, Pennsylvania State University
e-mail: sot2@psu.edu

Managing Editor:

   Maxim Arnold, University of Texas, Dallas
email: Maxim.Arnold@utdallas.edu

Editors:

Andrei Agrachev, International School for Advanced Studies, Trieste
e-mail: agrachevaa@gmail.com

Peter Albers, Heidelberg
e-mail: palbers@mathi.uni-heidelberg.de

Gal Binyamini, The Weizmann Institute of Science
e-mail: gal.binyamini@weizmann.ac.il

Gil Bor, CIMAT, Mexico
e-mail: gil@cimat.mx

Felix Chernous'ko, Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
e-mail: chern@ipmnet.ru

Bertrand Deroin, UMR CNRS
bertrand.deroin@gmail.com

David Eisenbud, University of California, Berkeley
e-mail: de@msri.org

Uriel Frisch, Observatoire de la Côte d'Azur, Nice
e-mail: uriel@oca.eu; uriel@obs-nice.fr

Dmitry Fuchs, University of California, Davis
e-mail: fuchs@math.ucdavis.edu

Alexander Gaifullin, Steklov Mathematical Institute, Moscow
e-mail: agaif@mi-ras.ru

Victor Goryunov, University of Liverpool
e-mail: Victor.Goryunov@liverpool.ac.uk

Sabir Gusein-Zade, Moscow State University
e-mail: sabirg@list.ru

Yulij Ilyashenko, Higher School of Economics, Moscow and Cornell University
e-mail: yulijs@gmail.com

Oleg Karpenkov, University of Liverpool
e-mail: O.Karpenkov@liverpool.ac.uk

Askold Khovanskii, University of Toronto
e-mail: askold@math.toronto.edu

Evgeny Mukhin, Indiana University-Purdue University
e-mail: emukhin@iupui.edu

Anatoly Neishtadt, Loughborough University
e-mail: A.Neishtadt@lboro.ac.uk

Evita Nestoridi, Stony Brook University
e-mail: Evrydiki.Nestoridi@stonybrook.edu

Greta Panova, University of Southern California
e-mail: gpanova@usc.edu

Dan Romik, UC Davis
e-mail: romik@math.ucdavis.edu

Alexander Shnirelman, Concordia University
e-mail: alexander.shnirelman@concordia.ca

Frank Sottile, Texas A&M University
e-mail: sottile@tamu.edu

Michael Yampolsky, University of Toronto
e-mail: yampol@math.toronto.edu

Alexander Varchenko, University of North Carolina, Chapel Hill
e-mail: anv@email.unc.edu

Oleg Viro, Stony Brook University
e-mail: oleg.viro@gmail.com


Advisors

Artur Avila, University of Zurich and IMPA

Etienne Ghys, École normale supérieure de Lyon

Dennis Sullivan, Stony Brook University and CUNY

Eduard Zehnder, ETH, Zurich


Editorial Council

Sergei Tabachnikov, Pennsylvania State University (Editor-in-Chief)

Maxim Arnold, University of Texas, Dallas (Managing Editor)

Vladlen Timorin, Higher School of Economics, Moscow

Oleg Viro, Stony Brook University (A representative of the IMS)

Sabir Gusein-Zade, Moscow State University

Askold Khovanskii, University of Toronto

Yulij Ilyashenko, Higher School of Economics, Moscow and Cornell University

Alexander Varchenko, University of North Carolina, Chapel Hill

Submission