1. Representations of the group of two-diagonal triangular matrices
   Dmitry Fuchs, Alexandre Kirillov
   Arnold Mathematical Journal, Volume 12, Issue 1, 2026
   Received: 08 Jul 2025; Accepted: 25 Jan 2026
   DOI: 10.56994/ARMJ.012.005.001
   Abstract
   
   Let \(G\) be a Lie group, \({\mathfrak{g}}=\mathop{\rm Lie}(G)\) – its Lie algebra, \({\mathfrak{g}}^{\ast}\) – the dual vector space and \(\widehat{G}\) – the set of equivalence classes of unitary irreducible representations of \(G\). The orbit method [1] establishes a correspondence between points of \(\widehat{G}\) and \(G\)-orbits in \({\mathfrak{g}}^{\ast}\). For many Lie groups it gives the answers to all major problems of representation theory in terms of coadjoint orbits. Formally, the notions and statements of the orbit method make sense when \(G\) is infinite-dimensional Lie group, or an algebraic group over a topological field or ring \(K\), whose additive group is self dual (e.g., \(p\)-adic or finite).

   In this paper, we introduce a big family of finite groups \(G_{n}\), for which the orbit method works perfectly well. Namely, let \(N_{n}({\mathbb{K}})\) be the algebraic group of upper unitriangular \((n+1)\times(n+1)\) matrices with entries from \({\mathbb{K}}\), and \({\mathbb{F}}_{q}\) be the finite field with \(q\) elements. We define \(G_{n}\) as the quotient of of the group \(N_{n+1}({\mathbb{F}}_{q})\) over its second commutator subgroup.

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2. Integrable geodesic flows with simultaneously diagonalisable quadratic integrals
   Sergey I. Agafonov, Vladimir S. Matveev
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 09 Nov 2024; Accepted 27 Nov 2024.
   DOI: 10.56994/ARMJ.011.004.001
   Abstract
   
   We show that if \(n\) functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an \(n\)-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the Stäckel construction, so the metric admits orthogonal separation of variables.
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3. The Flapping Birds in the Pentagram Zoo
   Richard Evan Schwartz
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 24 Dec 2024; Accepted 7 May 2025.
   DOI: 10.56994/ARMJ.011.004.002
   Abstract
   
   We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a generalization. $\Delta_k$ does not preserve convexity, but we prove that $\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-{\it birds\/}. The action of $\Delta_k$ on $B_k$ seems similar to the action of $\Delta_1$ on the space of convex polygons. We show that some classic geometric results about $\Delta_1$ generalize to this setting.
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4. A Novel Geometric Realization of the Yajima-Oikawa Equations
   Annalisa Calini and Thomas Ivey
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 17 Dec 2024; Accepted 23 May 2025
   DOI: 10.56994/ARMJ.011.004.003
   Abstract
   
   We show that the Yajima-Oikawa (YO) equations, a model of short wave-long wave interaction, arise from a simple geometric flow on curves in the 3-dimensional sphere $S^{3}$ that are transverse to the standard contact structure. For the family of periodic plane wave solutions of the YO equations studied by Wright, we construct the associated transverse curves, derive their closure condition, and exhibit several examples with non-trivial topology.
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5. On dual-projectively equivalent connections associated to second order superintegrable systems
   Andreas Vollmer
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 27 Dec 2024; Accepted 28 May 2025
   DOI: 10.56994/ARMJ.011.004.004
   Abstract
   
   Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in Kähler geome- try is known as 𝐽-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along which the 1-forms associated to the velocity are preserved after a reparametrization.

   Superintegrable systems are Hamiltonian systems with a large number of independent constants of the motion. They are said to be second order if the constants of the motion can be chosen to be quadratic polynomials in the momenta. Famous examples include the Kepler-Coulomb system and the isotropic harmonic oscillator.

   We show that certain torsion-free affine connections which are naturally associated to certain second order superintegrable systems share the same dual-geodesics.

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6. Folding of quadrilaterals and Arnold-Liouville integrability
   Anton Izosimov
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 04 Apr 2025; Accepted 09 Jul 2025
   DOI: 10.56994/ARMJ.011.004.005
   Abstract
   
   We put Darboux’s porism on folding of quadrilaterals, as well as closely related Bottema’s zigzag porism, in the context of Arnold-Liouville integrability.
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7. Spirals, tic-tac-toe partition, and deep diagonal maps
   Zhengyu Zou
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 25 Dec 2024; Accepted 9 Aug 2025
   DOI: 10.56994/ARMJ.011.004.006
   Abstract
   
   The deep diagonal map $T_k$ acts on planar polygons by connecting the $k$-th diagonals and intersecting them successively. The map $T_2$ is the pentagram map, and $T_k$ is a generalization.

   We study the action of $T_k$ on two subsets of the so-called twisted polygons, which we term \textit{type-$\alpha$ and type-$\beta$ $k$-spirals}. For $k \geq 2$, $T_{k}$ preserves both types of $k$-spirals. In particular, we show that for $k = 2$ and $k = 3$, both types of $k$-spirals have precompact forward and backward $T_k$-orbits modulo projective transformations. We derive a rational formula for $T_3$, which generalizes the $y$-variables transformation formula of the corresponding quiver mutation by M. Glick and P. Pylyavskyy. We also present four algebraic invariants of $T_3$. These special orbits in the moduli space are partitioned into cells of a $3 \times 3$ tic-tac-toe grid. This establishes the action of $T_k$ on $k$-spirals as a geometric generalization of $T_2$ on convex polygons.

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8. On symplectic linearizable actions
   Eva Miranda
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 27 Dec 2024; Accepted 24 Aug 2025
   DOI: 10.56994/ARMJ.011.004.007
   Abstract
   
   We prove that linearizable actions are also symplectically linearizable (either smoothly or analytically) in a neighborhood of a fixed point. Specifically, the fundamental vector fields associated with the action can be simultaneously linearized in Darboux coordinates. This result extends {equivariant} symplectic local normal forms to non-compact group actions.

   In both formal and analytic frameworks, the existence of linearizing coordinates is tied to a cohomological equation, which admits a solution for semisimple actions [9,8]. Consequently, an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates, enabling the simultaneous analytic linearization of Hamiltonian vector fields near a shared zero. However, in the smooth setting, this result is restricted to semisimple Lie algebras of compact type. We construct an explicit example of a smooth, non-linearizable Hamiltonian action with a semisimple linear part, thereby answering in the negative a question posed by Eliasson [5].

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9. Discrete Painlevé equations from pencils of quadrics in $\mathbb P^3$ with branching generators
   Jaume Alonso, Yuri B. Suris
   Arnold Mathematical Journal, Volume 11, Issue 4, 2025
   Received 02 Jun 2025; Accepted 07 Jan 2026
   DOI: 10.56994/ARMJ.011.004.008
   Abstract
   
   In this paper we extend the novel approach to discrete Painlev\'e equations initiated in our previous work [2]. A classification scheme for discrete Painlev\'e equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlev\'e equation is viewed as an autonomous transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial $\Delta(\lambda)$ of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlev\'e map corresponds to a translation on the universal cover of the Riemann surface of $\sqrt{\Delta(\lambda)}$, rather than to a M\"obius transformation of the pencil parameter $\lambda$ as in [2].
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10. Bouncing Outer Billiards
    Andrey Gogolev, Levi Keck, Kevin Lewis
    Arnold Mathematical Journal, Volume 11, Issue 3, 2025
    Received 09 Mar 2025; Accepted 22 Jul 2025
    DOI: 10.56994/ARMJ.011.003.005
    Abstract
    
    We introduce a new class of billiard-like system, ``bouncing outer billiards", which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that the bouncing outer billiards system on a smooth convex body has at least four 1-parameter families of fixed points. We also fully describe the dynamics of bouncing outer billiards on a line segment. Finally, we carry out numerical experiments suggesting very complicated (non-ergodic) behavior for several shapes, including the square and an ellipse.
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11. Circumscribed Circles in Integer Geometry
    Oleg Karpenkov, Anna Pratoussevitch, Rebecca Sheppard
    Arnold Mathematical Journal, Volume 11, Issue 3, 2025
    Received 17 Dec 2024; Accepted 7 May 2025
    DOI: 10.56994/ARMJ.011.003.002
    Abstract
    
    Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles in integer geometry. We introduce the notions of integer and rational circumscribed circles of integer sets. We determine the conditions for a finite integer set to admit an integer circumscribed circle and describe the spectra of radii for integer and rational circumscribed circles.
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12. Ancient curve shortening flow in the disc with mixed boundary condition
    Mat Langford, Yuxing Liu, George McNamara
    Arnold Mathematical Journal, Volume 11, Issue 3, 2025
    Received: 4 September 2024; Accepted: 10 March 2025
    DOI: 10.56994/ARMJ.011.003.004
    Abstract
    
    Given any non-central interior point $o$ of the unit disc $D$, the diameter $L$ through $o$ is the union of two linear arcs emanating from $o$ which meet $\partial D$ orthogonally, the shorter of them stable and the longer unstable (under these boundary conditions). In each of the two half discs bounded by $L$, we construct a convex eternal solution to curve shortening flow which fixes $o$ and meets $\partial D$ orthogonally, and evolves out of the unstable critical arc at $t=-\infty$ and into the stable one at $t=+\infty$. We then prove that these two (congruent) solutions are the only non-flat convex ancient solutions to the curve shortening flow satisfying the specified boundary conditions. We obtain analogous conclusions in the ``degenerate'' case $o\in\partial D$ as well, although in this case the solution contracts to the point $o$ at a finite time with asymptotic shape that of a half Grim Reaper, thus providing an interesting example for which an embedded flow develops a collapsing singularity.
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13. Discretization of the sub-Riemannian Heisenberg group
    Evgeny G. Malkovich
    Arnold Mathematical Journal, Volume 11, Issue 3, 2025
    Received: 22 September 2024; Accepted: 9 February 2025.
    DOI: 10.56994/ARMJ.011.003.001
    Abstract
    
    In this article, we present a discrete model of the sub-Riemannian Heisenberg group $\mathcal{H}$, which serves as an analog of a triangulation of a two-dimensional surface embedded in $\mathbb{R}^3$. The constructed discrete model is represented by a spatial graph $\Gamma_r$ with weighted edges. The shortest paths within $\Gamma_r$ approximate geodesics in $\mathcal{H}$.
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14. On fields of meromorphic functions on neighborhoods of rational curves
    Serge Lvovski
    Arnold Mathematical Journal, Volume 11, Issue 2, 2025
    Received: 3 September 2024; Accepted: 9 February 2025.
    DOI: 10.56994/ARMJ.011.002.003
    Abstract
    
    Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the field of meromorphic functions on $F$ is isomorphic to the field of rational functions in one or two variables over $\mathbb C$.
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15. Arithmetic on $q$-deformed rational numbers
    Takeyoshi Kogiso, Kengo Miyamoto, Xin Ren, Michihisa Wakui and Kohji Yanagawa
    Arnold Mathematical Journal, Volume 11, Issue 3, 2025
    Received: 28 October 2024, Accepted: 21 November 2024.
    DOI: 10.56994/ARMJ.011.003.003
    Abstract
    
    Recently, Morier-Genoud and Ovsienko introduced a $q$-{deformation} of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal R_{\frac{r}s}(q), ~\mathcal S_{\frac{r}s}(q) \in \mathbb Z[q]$ with $\mathcal R_{\frac{r}s}(1)=r, \mathcal S_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal S_{\frac{r}s}(q)=\mathcal S_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal S_{\frac{r}s}(q)=\mathcal S_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal S_{\frac{r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{-3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.
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16. Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
    Vladislav Galkin, Olga Pochinka and Danila Shubin
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: 10 March 2024; Accepted: 15 October 2024.
    DOI: 10.56994/ARMJ.011.001.006
    Abstract
    
    Topological equivalence of Morse-Smale flows without fixed points (NMS-flows) under assumptions of different generalities was studied in a number of papers. In some cases when the number of periodic orbits is small, it is possible to give exhaustive classification, namely to provide the list of all manifolds that admit flows of considered class, find complete invariant for topological equivalence and introduce each equivalence class with some representative flow. This work continues the series of such articles. We consider the class of NMS-flows with unique saddle orbit, under the assumption that it is twisted, on closed orientable 4-manifolds and prove that the only 4-manifold admitting the considered flows is the manifold $\mathbb S^3\times\mathbb S^1$. Also, it is established that such flows are split into exactly eight equivalence classes and construction of a representative for each equivalence class is provided.
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17. A simple construction of the field of Witt vectors
    Vladimir Fock
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: March 10 2024; Accepted: October 10 2024.
    DOI: 10.56994/ARMJ.011.001.001
    Abstract
    
    We present a short, hopefully pedagogical construction of the field and ring of Witt vectors. It uses a natural binary operation on polynomials of one variable, which we call convolution.
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18. Contact geometry of Hill's approximation in a spatial restricted four-body problem
    Cengiz Aydin
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: 10 March 2024; Accepted: 16 October 2024.
    DOI: 10.56994/ARMJ.011.001.005
    Abstract
    
    It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-García and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can ``contact'' the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.
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19. Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards
    Airi Takeuchi and Lei Zhao
    Arnold Mathematical Journal, Volume 11, Issue 2, 2025
    Received: 28 October 2024; Accepted: 21 November 2024.
    DOI: 10.56994/ARMJ.011.002.002
    Abstract
    
    The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel transformation. In the first part of this paper, we study how certain integrable mechanical billiards are related by this transformation. This in part illustrates the rotation-invariance of integrable reflection walls in the three-dimensional Kepler billiards found so far. The second part of this paper deals with the Birkhoff-Waldvogel Transformation of the three-dimensional problem wiht two Kepler centers. In particular, we establish an analogous theory of Levi-Civita planes for the Birkhoff-Waldvogel Transformation and show the integrability of certain three-dimensional two-center billiards via a different approach.
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20. On intersection of lemniscates of rational functions
    Stepan Orevkov and Fedor Pakovich
    Arnold Mathematical Journal, Volume 11, issue 1, 2025
    Received: 10 March 2024; Accepted 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.002
    Abstract
    
    For a non-constant complex rational function $P$, the {\it lemniscate} of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve given by the equation $L_P(x,y)=0$, where $L_P(x,y)$ is the numerator of the rational function $P(x+iy)\overline{ P}(x-iy)-1.$ In this paper, we study the following two questions: under what conditions two lemniscates have a common component, and under what conditions the algebraic curve $L_P(x,y)=0$ is irreducible. In particular, we provide a sharp bound for the number of complex solutions of the system $\vert P_1(z)\vert =\vert P_2(z)\vert =1$, where $P_1$ and $P_2$ are rational functions.
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21. Gravitational billiard - bouncing in a paraboloid cavity
    Daniel Jaud
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received 10 March 2024. Accepted: 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.004
    Abstract
    
    In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of a common sphere of radius $R$. The main results are illustrated in various limiting cases and are compared to its one-dimensional counterpart. In the maximum angular momentum configuration we explicitly state the coordinates of the particle at any time within the cavity.
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22. Hodge theory on tropical curves
    Yury V. Eliyashev
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received 10 January 2024. Accepted 10 September 2024.
    DOI: 10.56994/ARMJ.011.001.003
    Abstract
    
    We construct an analog of the Hodge theory on complex manifolds in the case of tropical curves. We use the analytical approach to the problem, it is based on language of tropical differential forms and methods of $L^2-$cohomologies. In particular, the cohomology groups of a tropical curve can be defined via the de Rham complex of tropical differential forms. We translate standard notions of the complex Hodge theory: the Kähler form, the Hodge star operator, the Laplace-Beltrami operator to the tropical case. The main result of the article is that the tropical Laplace-Beltrami operator is a self-adjoint unbounded operator and the cohomology groups of a tropical curve are isomorphic to the spaces of harmonic forms on this curve.
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23. Non-fillability of overtwisted contact manifolds via polyfolds
    Wolfgang Schmaltz, Stefan Suhr and Kai Zehmisch
    Arnold Mathematical Journal, Volume 11, Issue 2, 2025
    Received: 3 October 2024. Accepted: 15 October 2024.
    DOI: 10.56994/ARMJ.011.002.001
    Abstract
    
    We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive boundary satisfies the weak-filling condition and is overtwisted. Similar results are obtained in the presence of bordered Legendrian open books whose binding–complement has vanishing second Stiefel–Whitney class. The results are obtained via polyfolds.
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Articles of issue 11:4

Articles of issue 11:3

Articles of issue 11:2

Articles of issue 11:1

Editor-in-Chief:

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

Managing Editor:

   Maxim Arnold, University of Texas at Dallas (USA)
email: Maxim.Arnold@utdallas.edu

Editors:

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

Peter Albers, Heidelberg University (Germany)
e-mail: palbers@mathi.uni-heidelberg.de

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

Gil Bor, Centro de Investigaci\'on en Matem\'aticas (Mexico)
e-mail: gil@cimat.mx

Felix Chernous'ko, Institute for Problems in Mechanics, RAS (Russia)
e-mail: chern@ipmnet.ru

Bertrand Deroin, Cergy Paris Universit\'e (France)
bertrand.deroin@gmail.com

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

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

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

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

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

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

Yulij Ilyashenko, Higher School of Economics, Moscow (Russia)
e-mail: yulijs@gmail.com

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

Boris Khesin, University of Toronto (Canada)
e-mail: boris.khesin@math.toronto.edu

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

Evgeny Mukhin, IUPI, Indianapolis (USA)
e-mail: emukhin@iupui.edu

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

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

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

Dan Romik, University of California, Davis (USA)
e-mail: romik@math.ucdavis.edu

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

Vladlen Timorin, Higher School of Economics, Moscow (Russia)
e-mail: vtimorin@hotmail.com

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

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

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

Advisors

Artur Avila, University of Zurich (Switzerland) and IMPA (Brasil)

Etienne Ghys, École normale supérieure de Lyon (France)

Dennis Sullivan, Stony Brook University and Graduate Center, CUNY (USA)

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