ARNOLD  MATHEMATICAL  JOURNAL

 Editor-in-Chief:     Askold Khovanskii Managing Editor:     Vladlen Timorin

 A Journal of the IMS,   Stony Brook University Published by
 home editors submission

## Open Problems

1. Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals
Faustin Adiceam, David Damanik, Franz Gähler, Uwe Grimm, Alan Haynes, Antoine Julien, Andrés Navas, Lorenzo Sadun, Barak Weiss
Problem Contribution,   Received: 15 January 2016 / Revised: 21 May 2016 / Accepted: 11 June 2016 / Published Online: 11 July 2016
Abstract
This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field.
2. Betti Posets and the Stanley Depth
L. Katthän
Problem Contribution,   Received: 9 October 2015 / Revised: 19 December 2015 / Accepted: 4 February 2016 / Published online: 15 January 2016
Abstract
Let S be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of I determines the Stanley projective dimension of S/I or I. Our main result is that this conjecture implies the Stanley conjecture for I, and it also implies that ${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I - 1.$ Recently, Duval et al. (A non-partitionable Cohen-Macaulay simplicial complex, arXiv:1504.04279, 2015), found a counterexample to the Stanley conjecture, and their counterexample satisfies ${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I - 1$. So if our conjecture is true, then the conclusion is best possible.
3. Volumes of Strata of Abelian Differentials and Siegel-Veech Constants in Large Genera
A. Eskin, A. Zorich
Received: 19 July 2015 / Revised: 16 September 2015 / Accepted: 20 October 2015 / Published online: 05 November 2015
Abstract
We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.
4. Disconjugacy and the Secant Conjecture
A. Eremenko
Received: 5 July 2015 / Accepted: 28 July 2015 / Published online: 4 August 2015
Abstract
We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable.
5. A Few Problems on Monodromy and Discriminants
V. A. Vassiliev
Received: 15 February 2015 / Accepted: 31 March 2015 / Published online: 16 April 2015
Abstract
The article contains several problems concerning local monodromy groups of singularities, Lyashko-Looijenga maps, integral geometry, and topology of spaces of real algebraic manifolds.
6. Problems Around Polynomials: The Good, The Bad and The Ugly...
Boris Shapiro
Received: 7 November 2014 / Accepted: 16 March 2015 / Published online: 25 March 2015
Abstract
The Russian style of formulating mathematical problems means that nobody will be able to simplify your formulation as opposed to the French style which means that nobody will be able to generalize it, - Vladimir Arnold.
7. Space of Smooth 1-Knots in a 4-Manifold: Is Its Algebraic Topology Sensitive to Smooth Structures?
Oleg Viro
Received: 12 December 2014 / Accepted: 12 February 2015
Abstract
We discuss a possibility to get an invariant of a smooth structure on a closed simply connected 4-manifold from homotopy invariants of the space of loops smoothly embedded into the manifold.
8. Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes
A. M. Vershik
Received: 8 November 2014 / Accepted: 31 December 2014
Abstract
We describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.
9. Periods of Pseudo-Integrable Billiards
Received: 10 November 2014 / Accepted: 26 December 2014
Abstract
Consider billiard desks composed of two concentric half-circles connected with two edges. We examine billiard trajectories having a fixed circle concentric with the boundary semicircles as the caustic, such that the rotation numbers with respect to the half-circles are ρ1 and ρ2 respectively. Are such billiard trajectories periodic, and what are all possible periods for given ρ1 and ρ2?
10. A Baker's Dozen of Problems
Serge Tabachnikov
Received: 16 September 2014 / Revised: 23 September 2014 / Accepted: 14 October 2014
Abstract
This article is a collection of open problems, with brief historical and bibliographical comments, somewhat in the spirit of the problem with which V. Arnold opened his famous seminar every semester and that were recently collected and published in a book form.

## Research Papers

1. Convex Shapes and Harmonic Caps
Laura DeMarco, Kathryn Lindsey
Research Contribution,   Received: 4 February 2016 / Revised: 30 November 2016 / Accepted: 23 December 2016
Abstract
Any planar shape $P\subset{\mathbb{C}}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^{3}$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q=S{\setminus}P$ is the associated cap. We study the cap construction when the curvature is harmonic measure on the boundary of $({\hat{{\mathbb{C}}}}{\setminus}P,\infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.
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2. Random Chain Complexes
Viktor L. Ginzburg, Dmitrii V. Pasechnik
Research Contribution,   Received: 16 March 2016 / Revised: 9 December 2016 / Accepted: 23 December 2016
Abstract
We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology.
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3. The $4n^{2}$-Inequality for Complete Intersection Singularities
Aleksandr V. Pukhlikov
Research Contribution,   Received: 11 July 2016 / Revised: 25 October 2016 / Accepted: 17 November 2016
Abstract
The famous $4n^{2}$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than $4n^{2}\mu$, where $\mu$ is the multiplicity of the singular point.
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4. Flows in Flatland: A Romance of Few Dimensions
Gabriel Katz
Research Contribution,   Received: 2 March 2016 / Revised: 15 October 2016 / Accepted: 23 October 2016
Abstract
This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on $n$-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.
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5. The Geometry of Axisymmetric Ideal Fluid Flows with Swirl
Pearce Washabaugh, Stephen C. Preston
Research Contribution,   Received: 3 February 2016 / Revised: 21 August 2016 / Accepted: 15 October 2016
Abstract
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X=u(r)\partial_{\theta}$ iff $\partial_{r}(ru^{2})>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$.
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6. On Postsingularly Finite Exponential Maps
Walter Bergweile
Research Contribution,   Received: 5 December 2015 / Revised: 15 August 2016 / Accepted: 6 September 2016
Abstract
We consider parameters $\lambda$ for which 0 is preperiodic under the map $z\mapsto\lambda e^{z}$. Given $k$ and $l$, let $n(r)$ be the number of $\lambda$ satisfying $0<|\lambda|\leq r$ such that 0 is mapped after $k$ iterations to a periodic point of period $l$. We determine the asymptotic behavior of $n(r)$ as $r$ tends to $\infty$.
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7. Spherical Rectangles
Alexandre Eremenko, Andrei Gabrielov
Research Contribution,   Received: 24 January 2016 / Revised: 9 August 2016 / Accepted: 30 August 2016
Abstract
We study spherical quadrilaterals whose angles are odd multiples of $\pi/2$, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heunâ~@~Ys equations with real parameters, whose exponent differences are odd multiples of $1/2$, with unitary monodromy.
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8. q-Polynomial Invariant of Rooted Trees
Jözef H. Przytycki
Research Contribution,   Received: 7 December 2015 / Revised: 28 July 2016 / Accepted: 2 August 2016
Abstract
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers of Arnold Journal of Mathematics is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology.
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9. On the Roots of a Hyperbolic Polynomial Pencil
Victor Katsnelson
Research Contribution,   Received: 03 May 2016 / Accepted: 20 July 2016 / Published Online: 02 August 2016
Abstract

Let $\nu_0(t),\nu_1(t),\ldots,\nu_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form

 $\displaystyle \begin{eqnarray*} R(z)=z-\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k}, \end{eqnarray*}$

$\mu_k$ are pairwise distinct real numbers, $\alpha_k> 0,\,1\leq{}k\leq{}n$. Then for each real $\xi$, the function $e^{\xi\nu_0(t)}+e^{\xi\nu_1(t)}+\,\cdots\,+e^{\xi\nu_n(t)}$ is exponentially convex on the interval $-\infty< t< \infty$.

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10. A Generalisation of the Cauchy-Kovalevskaïa Theorem
Mauricio Garay
Research Contribution,   Received: 1 July 2015 / Revised: 15 May 2016 / Accepted: 23 June 2016 / Published Online: 09 August 2016
Abstract
We prove that time evolution of a linear analytic initial value problem leadsto sectorial holomorphic solutions in time.
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11. A Classification of Spherical Curves Based on Gauss Diagrams
Guy Valette
Research Contribution,   Received: 28 August 2015 / Revised: 4 May 2016 / Accepted: 23 June 2016 / Published Online: 11 July 2016
Abstract
We consider generic smooth closed curves on the sphere $S^{2}$. These curves (oriented or not) are classified relatively to the group $\mbox{Diff}(S^{2})$ or its subgroup $\mbox{Diff}^{+}(S^{2})$, with the Gauss diagrams as main tool. V. I. Arnold determined the numbers of orbits of curves with $n$ double points when $n<6$. This paper explains how a preliminary classification of the Gauss diagrams of order 5, 6 and 7 allows to draw up the list of the realizable chord diagrams of these orders. For each such diagram $\Gamma$ and for each Arnold symmetry type $T$, we determine the number of orbits of spherical curves of type $T$ realizing $\Gamma$. As a consequence, we obtain the total numbers of curves (oriented or not) with 6 or 7 double points on the sphere (oriented or not) and also the number of curves with special properties (e.g. having no simple loop).
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12. On Malfatti's Marble Problem
Uuganbaatar Ninjbat
Research Contribution,   Received: 3 April 2015 / Revised: 9 April 2016 / Accepted: 20 June 2016 / Published Online: 11 July 2016
Abstract
Consider the problem of finding three non-overlapping circles in a given triangle with the maximum total area. This is Malfatti's marble problem, and it is known that the greedy arrangement is the solution. In this paper, we provide a simpler proof of this result by synthesizing earlier insights with more recent developments. We also discuss some related geometric extremum problems, and show that the greedy arrangement solves the problem of finding two non-overlapping circles in a tangential polygon with the maximum total radii and/or area. In the light of this discussion, we formulate a natural extension of Melissen's conjecture.
13. Volume Polynomials and Duality Algebras of Multi-Fans
Anton Ayzenberg, Mikiya Masuda
Research Contribution,   Received: 17 October 2015 / Revised: 12 November 2015 / Accepted: 23 June 2016 / Published Online: 11 July 2016
Abstract
We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik-Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.
14. Generalizations of Tucker-Fan-Shashkin Lemmas
Oleg R. Musin
Research Contribution,   Received: 26 November 2014 / Revised: 25 April 2016 / Accepted: 27 May 2016 / Published online: 16 June 2016
Abstract
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.
15. Strange Duality Between Hypersurface and Complete Intersection Singularities
Wolfgang Ebeling, Atsushi Takahashi
Research Contribution,   Received: 22 September 2015 / Revised: 9 May 2016 / Accepted: 12 May 2016 / Published online: 24 May 2016
Abstract
W. Ebeling and C. T. C. Wall discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.
16. The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids
James Benn
Research Contribution,   Received: 11 August 2015 / Revised: 25 February 2016 / Accepted: 11 April 2016 / Published online: 18 May 2016
Abstract
We give a new description of the coadjoint operator $Ad^*_{\eta^{-1}(t)}$ along a geodesic $\eta(t)$ of the $L^2$ metric in the group of volume-preserving diffeomorphisms, important in hydrodynamics. When the underlying manifold is two dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations modulo a compact operator; when the manifold is three dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations plus a bounded operator. We give two applications of this result when the underlying manifold is two dimensional: conjugate points along geodesics of the $L^2$ metric are characterized in terms of the coadjoint operator and thus determining the conjugate locus is a purely algebraic question. We also prove that Eulerian and Lagrangian stability of the $2D$ Euler equations are equivalent and that instabilities in the $2D$ Euler equations are contained and small.
17. Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials
Dierk Schleicher
Research Contribution,   Received: 15 October 2015 / Revised: 9 February 2016 / Accepted: 7 April 2016 / Published online: 02 August 2016
Abstract
We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sects.Â 3 and 4). A simple extension, angled internal addresses, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Sect.Â 6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Sect.Â 7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.
18. Non-avoided Crossings for $n$-Body Balanced Configurations in $\mathbb R^3$ Near a Central Configuration
Alain Chenciner
Research Contribution,   Received: 4 September 2015 / Revised: 14 January 2016 / Accepted: 10 March 2016 / Published online: 8 April 2016
Abstract
The balanced configurations are those $n$-body configurations which admit a relative equilibrium motion in a Euclidean space $E$ of high enough dimension $2 p$. They are characterized by the commutation of two symmetric endomorphisms of the $(n-1)$-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism $B$ which encodes the shape and the Wintner-Conley endomorphism $A$ which encodes the forces. In general, $p$ is the dimension $d$ of the configuration, which is also the rank of B. Lowering to $2(d-1)$ the dimension of $E$ occurs when the restriction of $A$ to the (invariant) image of $B$ possesses a double eigenvalue. It is shown that, while in the space of all $d\times d$ symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition $(H)$ is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if $d = n-1$), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of four bodies with no three of the masses equal, of exactly three families of balanced configurations which admit relative equilibrium motion in a four dimensional space.
19. Geodesics on Regular Polyhedra with Endpoints at the Vertices
Dmitry Fuchs
Research Contribution,   Received: 3 October 2015 / Revised: 23 October 2015 / Accepted: 3 March 2016 / Published online: 23 March 2016
Abstract
In a recent work of Davis et al. (2016), the authors consider geodesics on regular polyhedra which begin and end at vertices (and do not touch other vertices). The cases of regular tetrahedra and cubes are considered. The authors prove that (in these cases) a geodesic as above never begins at ends at the same vertex and compute the probabilities with which a geodesic emanating from a given vertex ends at every other vertex. The main observation of the present article is that there exists a close relation between the problem considered in Davis et al. (2016) and the problem of classification of closed geodesics on regular polyhedra considered in articles (Fuchs and Fuchs, Mosc Math J 7:265-279, 2007; Fuchs, Geom Dedic 170:319-333, 2014). This approach yields different proofs of result of Davis et al. (2016) and permits to obtain similar results for regular octahedra and icosahedra (in particular, such a geodesic never ends where it begins).
20. On Foliations in Neighborhoods of Elliptic Curves
M. Mishustin
Research Contribution,   Received: 13 April 2015 / Revised: 24 August 2015 / Accepted: 13 January 2016 / Published online: 26 January 2016
Abstract
A counterexample is given to a conjecture from the comments to Arnold's problem 1989-11 about the existence of a tangent foliation in a zero type neighborhood of an elliptic curve.
21. Skewers
Serge Tabachnikov
Research Contribution,   Received: 19 September 2015 / Revised: 29 December 2015 / Accepted: 11 January 2016 / Published online: 27 January 2016
Abstract
The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen-Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.
22. An Invariant of Colored Links via Skein Relation
Francesca Aicardi
Research Contribution,   Received: 30 May 2015 / Accepted: 14 December 2015 / Published online: 1 March 2016
Abstract
In this note, we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.
23. N-Division Points of Hypocycloids
N. Mani, S. Rubinstein-Salzedo
Research Contribution,   Received: 4 May 2015 / Revised: 19 October 2015 / Accepted: 6 December 2015 / Published online: 04 January 2016
Abstract
We show that the n-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers n, given a pre-drawn hypocycloid. We also consider the question of constructibility of n-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the 1, 2, 3, and 6-division points of a tricuspoid are constructible in this manner.
24. Polynomials Invertible in k-Radicals
Y. Burda, A. Khovanskii
Received: 18 May 2015 / Revised: 22 December 2015 / Accepted: 25 December 2015 / Published online: 09 February 2016 2015
Abstract
A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if $k\le 14$, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Müller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.
25. Generalized Plumbings and Murasugi Sums
B. Ozbagci, P. Popescu-Pampu
Received: 6 July 2015 / Revised: 28 October 2015 / Accepted: 23 November 2015 / Published online: 23 December 2015
Abstract
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory and contact topology.
26. The Gabrielov-Khovanskii Problem for Polynomials
A. V. Pukhlikov
Received: 19 June 2015 / Revised: 24 October 2015 / Accepted: 6 November 2015 / Published online: 27 November 2015
Abstract
We state and consider the Gabrielov-Khovanskii problem of estimating the multiplicity of a common zero for a tuple of polynomials in a subvariety of a given codimension in the space of tuples of polynomials. For a bounded codimension we obtain estimates of the multiplicity of the common zero, which are close to optimal ones. We consider certain generalizations and open questions.
27. Galois Correspondence Theorem for Picard-Vessiot Extensions
T. Crespo, Z. Hajto, E. Sowa-Adamus
Received: 16 April 2015 / Revised: 23 September 2015 / Accepted: 23 October 2015 / Published online: 03 November 2015
Abstract
For a homogeneous linear differential equation defined over a differential field K, a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K-differential isomorphism. In this case the differential Galois group is defined as the group of K-differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.
28. On Maps Taking Lines to Plane Curves
V. Petrushchenko, V. Timorin
Received: 24 March 2015 / Accepted: 16 October 2015 / Published online: 03 November 2015
Abstract
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
29. Solvability of Linear Differential Systems with Small Exponents in the Liouvillian Sense
R. R. Gontsov, I. V. Vyugin
Received: 25 November 2014 / Revised: 20 August 2015 / Accepted: 11 November 2015 / Published online: 26 November 2015
Abstract
The paper is devoted to solvability of linear differential systems by quadratures, one of the classical problems of differential Galois theory. As known, solvability of a system depends entirely on properties of its differential Galois group. However, detecting solvability or non-solvability of a given system is a difficult problem, because the dependence of its differential Galois group on the coefficients of the system remains unknown. We consider systems with regular singular points as well as those with non-resonant irregular ones, whose exponents (respectively, so-called formal exponents in the irregular case) are sufficiently small. It turns out that for systems satisfying such restrictions criteria of solvability can be given in terms of the coefficient matrix.
30. Finite and Infinitesimal Flexibility of Semidiscrete Surfaces
O. Karpenkov
Research Contribution,   Received: 18 April 2015 / Revised: 28 July 2015 / Accepted: 24 August 2015 / Published online: 3 September 2015
Abstract
In this paper we study infinitesimal and finite flexibility for regular semidiscrete surfaces. We prove that regular 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary $n\ge 3$ we prove that every regular n-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces.
31. Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence
N. Selinger, M. Yampolsky
Received: 14 November 2014 / Revised: 3 June 2015 / Accepted: 4 August 2015 / Published online: 7 September 2015
Abstract
The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem Douady and Hubbard (Acta Math 171:263-297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $S^2$ (cf. Bonnot et al. Moscow Math J 12:747-763, 2012).
32. Bollobás – Riordan and Relative Tutte Polynomials
C. Butler, S. Chmutov
Received: 8 December 2014 / Revised: 29 June 2015 / Accepted: 5 July 2015 / Published online: 28 July 2015
Abstract
We establish a relation between the Bollobás – Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.
33. Critical Set of the Master Function and Characteristic Variety of the Associated Gauss-Manin Differential Equations
A. Varchenko
Received: 7 November 2014 / Accepted: 15 June 2015 / Published online: 7 July 2015
Abstract
We consider a weighted family of n parallelly transported hyperplanes in a k-dimensional affine space and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution. These statements generalize (Varchenko, Mathematics 2:218-231, 2014), where such a description was obtained for a weighted generic family of parallelly transported hyperplanes. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss–Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras.
34. The Exponential Map Near Conjugate Points In 2D Hydrodynamics
G. Misiołek
Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
We prove that the weak-Riemannian exponential map of the $L^2$ metric on the group of volume-preserving diffeomorphisms of a compact two-dimensional manifold is not injective in any neighbourhood of its conjugate vectors. This can be viewed as a hydrodynamical analogue of the classical result of Morse and Littauer.
35. Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in ${\mathbb R}^2$ and ${\mathbb C}^2$
V. A. Vassiliev
Received: 7 November 2014 / Accepted: 3 June 2015 / Published online: 11 August 2015
Abstract
The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials ${\mathbb R}^2 \rightarrow {\mathbb R}$, and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in ${\mathbb C}^2$.
36. Local Invariants of Framed Fronts in 3-Manifolds
V. Goryunov, S. Alsaeed
Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for framed fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.
37. A Formula for the HOMFLY Polynomial of rational links
Sergei Duzhin, Mikhail Shkolnikov
Received: 10 November 2014 / Accepted: 7 April 2015 / Published online: 24 April 2015
Abstract
We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link [after this work was accomplished, the authors learned about a paper by Nakabo (J. Knot Theory Ramif 11(4):565-574, 2002) where a similar result was proved. However, Nakabo's formula is different from ours, and his proof is longer and less clear].
38. Abundance of 3-Planes on Real Projective Hypersurfaces
S. Finashin, V. Kharlamov
Received: 7 November 2014 / Accepted: 2 May 2015 / Published online: 2 June 2015
Abstract
We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2) ={{d+3}\choose3}$, contains many'' real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
Erratum to: Abundance of 3-Planes on Real Projective Hypersurfaces
Published online: 31 July 2015
Abstract
When we published this article, there was a typo in the first line of Theorem 5.3.1. Please find the corrected text in the pdf. The publisher apologises for this mistake.
39. On Local Weyl Equivalence of Higher Order Fuchsian Equations
Shira Tanny, Sergei Yakovenko
Received: 26 December 2014 / Accepted: 15 April 2015/ Published online: 08 May 2015
Abstract
We study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the "type of differential equation" introduced by Frobenius. The main source of difficulties is the fact that there is no natural group action generating this classification. We establish a number of results on higher order equations which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations.
40. On an Equivariant Version of the Zeta Function of a Transformation
S. M. Gusein-Zade, I. Luengo, A. Melle-Hernández
Received: 17 December 2014 / Accepted: 4 April 2015 / Published online: 28 April 2015
Abstract
Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W. Lück and J. Rosenberg. Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring itself. We give an A'Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.
41. Vortex Dynamics of Oscillating Flows
V. A. Vladimirov, M. R. E. Proctor, D. W. Hughes
Received: 22 December 2014 / Accepted: 23 March 2015 / Published online: 10 April 2015
Abstract
We employ the method of multiple scales (two-timing) to analyse the vortex dynamics of inviscid, incompressible flows that oscillate in time. Consideration of distinguished limits for Euler's equation of hydrodynamics shows the existence of two main asymptotic models for the averaged flows: strong vortex dynamics (SVD) and weak vortex dynamics (WVD). In SVD the averaged vorticity is 'frozen' into the averaged velocity field. By contrast, in WVD the averaged vorticity is 'frozen' into the 'averaged velocity + drift'. The derivation of the WVD recovers the Craik-Leibovich equation in a systematic and quite general manner. We show that the averaged equations and boundary conditions lead to an energy-type integral, with implications for stability.
42. Remarks on the Circumcenter of Mass
Serge Tabachnikov, Emmanuel Tsukerman
Received: 15 December 2014 / Accepted: 23 March 2015 / Published online: 31 March 2015
Abstract
Suppose that to every non-degenerate simplex $\Delta\subset\mathbb R^n$ a 'center' $C(\Delta)$ is assigned so that the following assumptions hold:
1. The map $\Delta\to C(\Delta)$ commutes with similarities and is invariant under the permutations of the vertices of the simplex;
2. The map $\Delta\to \operatorname{Vol}(\Delta)C(\Delta)$ is polynomial in the coordinates of the vertices of the simplex.
Then $C(\Delta)$ is an affine combination of the center of mass $CM(\Delta)$ and the circumcenter $CC(\Delta)$ of the simplex: $$C(\Delta)=tCM(\Delta)+(1-t)CC(\Delta),$$ where the constant $t\in\mathbb R$ depends on the map $\Delta\mapsto C(\Delta)$ (and does not depend on the simplex $\Delta$).
The motivation for this theorem comes from the recent study of the circumcenter of mass of simplicial polytopes by the authors and by A. Akopyan.
A. A. Agrachev
Received: 10 November 2014 / Accepted: 16 December 2014
Abstract
We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions.
44. Riemannian Geometry of the Contactomorphism Group
David G. Ebin, Stephen C. Preston
Received: 11 November 2014 / Accepted: 8 December 2014
Abstract
Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

## Research Expositions

1. Flows in Flatland: A Romance of Few Dimensions
Gabriel Katz
Research Exposition,   Received: 2 March 2016 / Revised: 15 October 2016 / Accepted: 23 October 2016
Abstract
This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on $n$-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.
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2. Some Recent Generalizations of the Classical Rigid Body Systems
Vladimir Dragović, Borislav Gajić
Research Exposition,   Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016 / Published online: 19 September 2016
Abstract
Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations.
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3. Building Thermodynamics for Non-uniformly Hyperbolic Maps
Vaughn Climenhaga, Yakov Pesin
Research Exposition,   Received: 4 February 2016 / Accepted: 20 July 2016 / Published online: 09 August 2016
Abstract
We briefly survey the theory of thermodynamic formalism for uniformly hyperbolic systems, and then describe several recent approaches to the problem of extending this theory to non-uniform hyperbolicity. The first of these approaches involves Markov models such as Young towers, countable-state Markov shifts, and inducing schemes. The other two are less fully developed but have seen significant progress in the last few years: these involve coarse-graining techniques (expansivity and specification) and geometric arguments involving push-forward of densities on admissible manifolds.
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4. Kepler's Laws and Conic Sections
A. Givental
Research Exposition,   Received: 5 July 2015 / Revised: 7 September 2015 / Accepted: 24 October 2015 / Published online: 23 December 2015
Abstract
The geometry of Kepler's problem is elucidated by lifting the motion from the (xy)-plane to the cone $r^2=x^2+y^2$.
5. The Conley Conjecture and Beyond
V. L. Ginzburg, B. Z. Gürel
Research & Survey Contribution,   Received: 25 November 2014 / Accepted: 19 May 2015 / Published online: 4 June 2015
Abstract
This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.

## Volume 2, March 2016 - December 2016

### Issue 4, December 2016, Pages 439-592

1. On the Roots of a Hyperbolic Polynomial Pencil
Victor Katsnelson
Research Contribution,   Received: 03 May 2016 / Accepted: 20 July 2016 / Published Online: 02 August 2016
Abstract

Let $\nu_0(t),\nu_1(t),\ldots,\nu_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form

 $\displaystyle \begin{eqnarray*} R(z)=z-\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k}, \end{eqnarray*}$

$\mu_k$ are pairwise distinct real numbers, $\alpha_k> 0,\,1\leq{}k\leq{}n$. Then for each real $\xi$, the function $e^{\xi\nu_0(t)}+e^{\xi\nu_1(t)}+\,\cdots\,+e^{\xi\nu_n(t)}$ is exponentially convex on the interval $-\infty< t< \infty$.

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2. q-Polynomial Invariant of Rooted Trees
Jözef H. Przytycki
Research Contribution,   Received: 7 December 2015 / Revised: 28 July 2016 / Accepted: 02 August 2016
Abstract
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers of Arnold Journal of Mathematics is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology.
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3. Spherical Rectangles
Alexandre Eremenko, Andrei Gabrielov
Research Contribution,   Received: 24 January 2016 / Revised: 9 August 2016 / Accepted: 30 August 2016
Abstract
We study spherical quadrilaterals whose angles are odd multiples of $\pi/2$, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heunâ€™s equations with real parameters, whose exponent differences are odd multiples of $1/2$, with unitary monodromy.
4. Maximal Green Sequences for Cluster Algebras Associated to Orientable Surfaces with Empty Boundary
Eric Bucher
Research Contribution,   Received: 24 August 2015 / Revised: 12 January 2016 / Accepted: 12 September 2016
Abstract
Given a marked surface $(S,M)$ we can add arcs to the surface to create a triangulation, $T$, of that surface. For each triangulation, $T$, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus $n$ with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work along with previous work by others seeks to establish a base case in answering the question of whether a given finite mutation type cluster algebra exhibits a maximal green sequence.
5. Some Recent Generalizations of the Classical Rigid Body Systems
Vladimir Dragović, Borislav Gajić
Research Exposition,   Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016 / Published online: 19 September 2016
Abstract
Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations.
Download PDF of the paper (1007KB) .   View
6. Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals
Faustin Adiceam, David Damanik, Franz Gähler, Uwe Grimm, Alan Haynes, Antoine Julien, Andrés Navas, Lorenzo Sadun, Barak Weiss
Problem Contribution,   Received: 15 January 2016 / Revised: 21 May 2016 / Accepted: 11 June 2016 / Published Online: 11 July 2016
Abstract
This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field.

### Issue 3, September 2016, Pages 277-438

1. Strange Duality Between Hypersurface and Complete Intersection Singularities
Wolfgang Ebeling, Atsushi Takahashi
Research Contribution,   Received: 22 September 2015 / Revised: 9 May 2016 / Accepted: 12 May 2016 / Published online: 24 May 2016
Abstract
W. Ebeling and C. T. C. Wall discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.
2. Generalizations of Tucker-Fan-Shashkin Lemmas
Oleg R. Musin
Research Contribution,   Received: 26 November 2014 / Revised: 25 April 2016 / Accepted: 27 May 2016 / Published online: 16 June 2016
Abstract
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.
3. On Malfatti's Marble Problem
Uuganbaatar Ninjbat
Research Contribution,   Received: 3 April 2015 / Revised: 9 April 2016 / Accepted: 20 June 2016 / Published Online: 11 July 2016
Abstract
Consider the problem of finding three non-overlapping circles in a given triangle with the maximum total area. This is Malfatti's marble problem, and it is known that the greedy arrangement is the solution. In this paper, we provide a simpler proof of this result by synthesizing earlier insights with more recent developments. We also discuss some related geometric extremum problems, and show that the greedy arrangement solves the problem of finding two non-overlapping circles in a tangential polygon with the maximum total radii and/or area. In the light of this discussion, we formulate a natural extension of Melissen's conjecture.
4. Volume Polynomials and Duality Algebras of Multi-Fans
Anton Ayzenberg, Mikiya Masuda
Research Contribution,   Received: 17 October 2015 / Revised: 12 November 2015 / Accepted: 23 June 2016/ Published Online: 11 July 2016
Abstract
We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_{\Delta}$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^{*}(\Delta)$. We study the structure and properties of $V_{\Delta}$ and $\mathcal{A}^{*}(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novikâ~@~SSwartz theory of face rings of simplicial manifolds, generalizations of Minkowskiâ~@~Ys theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.
5. A Classification of Spherical Curves Based on Gauss Diagrams
Guy Valette
Research Contribution,   Received: 28 August 2015 / Revised: 4 May 2016 / Accepted: 23 June 2016 / Published Online: 11 July 2016
Abstract
We consider generic smooth closed curves on the sphere $S^{2}$. These curves (oriented or not) are classified relatively to the group $\mbox{Diff}(S^{2})$ or its subgroup $\mbox{Diff}^{+}(S^{2})$, with the Gauss diagrams as main tool. V. I. Arnold determined the numbers of orbits of curves with $n$ double points when $n<6$. This paper explains how a preliminary classification of the Gauss diagrams of order 5, 6 and 7 allows to draw up the list of the realizable chord diagrams of these orders. For each such diagram $\Gamma$ and for each Arnold symmetry type $T$, we determine the number of orbits of spherical curves of type $T$ realizing $\Gamma$. As a consequence, we obtain the total numbers of curves (oriented or not) with 6 or 7 double points on the sphere (oriented or not) and also the number of curves with special properties (e.g. having no simple loop).
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6. A Generalisation of the Cauchy-Kovalevskaïa Theorem
Mauricio Garay
Research Contribution,   Received: 1 July 2015 / Revised: 15 May 2016 / Accepted: 23 June 2016 / Published Online: 09 August 2016
Abstract
We prove that time evolution of a linear analytic initial value problem leadsto sectorial holomorphic solutions in time.
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### Issue 2, June 2016, Pages 149-276

1. N-Division Points of Hypocycloids
N. Mani, S. Rubinstein-Salzedo
Research Contribution,   Received: 4 May 2015 / Revised: 19 October 2015 / Accepted: 6 December 2015 / Published online: 04 January 2016
Abstract
We show that the n-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers n, given a pre-drawn hypocycloid. We also consider the question of constructibility of n-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the 1, 2, 3, and 6-division points of a tricuspoid are constructible in this manner.
2. An Invariant of Colored Links via Skein Relation
Francesca Aicardi
Research Contribution,   Received: 30 May 2015 / Accepted: 14 December 2015 / Published online: 1 March 2016
Abstract
In this note, we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.
3. Skewers
Serge Tabachnikov
Research Contribution,   Received: 19 September 2015 / Revised: 29 December 2015 / Accepted: 11 January 2016 / Published online: 27 January 2016
Abstract
The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen-Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.
4. On Foliations in Neighborhoods of Elliptic Curves
M. Mishustin
Research Contribution,   Received: 13 April 2015 / Revised: 24 August 2015 / Accepted: 13 January 2016 / Published online: 26 January 2016
Abstract
A counterexample is given to a conjecture from the comments to Arnold's problem 1989-11 about the existence of a tangent foliation in a zero type neighborhood of an elliptic curve.
5. Geodesics on Regular Polyhedra with Endpoints at the Vertices
Dmitry Fuchs
Research Contribution,   Received: 3 October 2015 / Revised: 23 October 2015 / Accepted: 3 March 2016 / Published online: 23 March 2016
Abstract
In a recent work of Davis et al. (2016), the authors consider geodesics on regular polyhedra which begin and end at vertices (and do not touch other vertices). The cases of regular tetrahedra and cubes are considered. The authors prove that (in these cases) a geodesic as above never begins at ends at the same vertex and compute the probabilities with which a geodesic emanating from a given vertex ends at every other vertex. The main observation of the present article is that there exists a close relation between the problem considered in Davis et al. (2016) and the problem of classification of closed geodesics on regular polyhedra considered in articles (Fuchs and Fuchs, Mosc Math J 7:265-279, 2007; Fuchs, Geom Dedic 170:319-333, 2014). This approach yields different proofs of result of Davis et al. (2016) and permits to obtain similar results for regular octahedra and icosahedra (in particular, such a geodesic never ends where it begins).
6. Non-avoided Crossings for $n$-Body Balanced Configurations in $\mathbb R^3$ Near a Central Configuration
Alain Chenciner
Research Contribution,   Received: 4 September 2015 / Revised: 14 January 2016 / Accepted: 10 March 2016 / Published online: 8 April 2016
Abstract
The balanced configurations are those $n$-body configurations which admit a relative equilibrium motion in a Euclidean space $E$ of high enough dimension $2 p$. They are characterized by the commutation of two symmetric endomorphisms of the $(n-1)$-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism $B$ which encodes the shape and the Wintner-Conley endomorphism $A$ which encodes the forces. In general, $p$ is the dimension $d$ of the configuration, which is also the rank of B. Lowering to $2(d-1)$ the dimension of $E$ occurs when the restriction of $A$ to the (invariant) image of $B$ possesses a double eigenvalue. It is shown that, while in the space of all $d\times d$ symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition $(H)$ is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if $d = n-1$), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of four bodies with no three of the masses equal, of exactly three families of balanced configurations which admit relative equilibrium motion in a four dimensional space.
7. The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids
James Benn
Research Contribution,   Received: 11 August 2015 / Revised: 25 February 2016 / Accepted: 11 April 2016 / Published online: 18 May 2016
Abstract
We give a new description of the coadjoint operator $Ad^*_{\eta^{-1}(t)}$ along a geodesic $\eta(t)$ of the $L^2$ metric in the group of volume-preserving diffeomorphisms, important in hydrodynamics. When the underlying manifold is two dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations modulo a compact operator; when the manifold is three dimensional the coadjoint operator is given by the solution operator to the linearized Euler equations plus a bounded operator. We give two applications of this result when the underlying manifold is two dimensional: conjugate points along geodesics of the $L^2$ metric are characterized in terms of the coadjoint operator and thus determining the conjugate locus is a purely algebraic question. We also prove that Eulerian and Lagrangian stability of the $2D$ Euler equations are equivalent and that instabilities in the $2D$ Euler equations are contained and small.
8. Betti Posets and the Stanley Depth
L. Katthän
Problem Contribution,   Received: 9 October 2015 / Revised: 19 December 2015 / Accepted: 4 February 2016 / Published online: 15 January 2016
Abstract
Let S be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of I determines the Stanley projective dimension of S/I or I. Our main result is that this conjecture implies the Stanley conjecture for I, and it also implies that ${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I - 1.$ Recently, Duval et al. (A non-partitionable Cohen-Macaulay simplicial complex, arXiv:1504.04279, 2015), found a counterexample to the Stanley conjecture, and their counterexample satisfies ${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I - 1$. So if our conjecture is true, then the conclusion is best possible.

### Issue 1, March 2016, Pages 1-148

1. On Maps Taking Lines to Plane Curves
V. Petrushchenko, V. Timorin
Research Contribution,   Received: 24 March 2015 / Accepted: 16 October 2015 / Published online: 03 November 2015
Abstract
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
2. Galois Correspondence Theorem for Picard-Vessiot Extensions
T. Crespo, Z. Hajto, E. Sowa-Adamus
Research Contribution,   Received: 16 April 2015 / Revised: 23 September 2015 / Accepted: 23 October 2015 / Published online: 03 November 2015
Abstract
For a homogeneous linear differential equation defined over a differential field K, a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K-differential isomorphism. In this case the differential Galois group is defined as the group of K-differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.
3. The Gabrielov-Khovanskii Problem for Polynomials
A. V. Pukhlikov
Research Contribution,   Received: 19 June 2015 / Revised: 24 October 2015 / Accepted: 6 November 2015 / Published online: 27 November 2015
Abstract
We state and consider the Gabrielov-Khovanskii problem of estimating the multiplicity of a common zero for a tuple of polynomials in a subvariety of a given codimension in the space of tuples of polynomials. For a bounded codimension we obtain estimates of the multiplicity of the common zero, which are close to optimal ones. We consider certain generalizations and open questions.
4. Generalized Plumbings and Murasugi Sums
B. Ozbagci, P. Popescu-Pampu
Research Contribution,   Received: 6 July 2015 / Revised: 28 October 2015 / Accepted: 23 November 2015 / Published online: 23 December 2015
Abstract
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory and contact topology.
5. Polynomials Invertible in k-Radicals
Y. Burda, A. Khovanskii
Research Contribution,   Received: 18 May 2015 / Revised: 22 December 2015 / Accepted: 25 December 2015 / Published online: 09 February 2016 2015
Abstract
A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if $k\le 14$, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Müller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.
6. Kepler's Laws and Conic Sections
A. Givental
Research Exposition,   Received: 5 July 2015 / Revised: 7 September 2015 / Accepted: 24 October 2015 / Published online: 23 December 2015
Abstract
The geometry of Kepler's problem is elucidated by lifting the motion from the (xy)-plane to the cone $r^2=x^2+y^2$.

## Volume 1, March 2015 - December 2015

### Issue 4, December 2015, Pages 345-488

1. A Formula for the HOMFLY Polynomial of rational links
S. Duzhin, M. Shkolnikov
Research Contribution,   Received: 10 November 2014 / Accepted: 7 April 2015 / Published online: 24 April 2015
Abstract
We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link [after this work was accomplished, the authors learned about a paper by Nakabo (J. Knot Theory Ramif 11(4):565-574, 2002) where a similar result was proved. However, Nakabo's formula is different from ours, and his proof is longer and less clear].
2. Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence
N. Selinger, M. Yampolsky
Research Contribution,   Received: 14 November 2014 / Revised: 3 June 2015 / Accepted: 4 August 2015 / Published online: 7 September 2015
Abstract
The key result in the present paper is a direct analogue of the celebrated Thurston's Theorem Douady and Hubbard (Acta Math 171:263-297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $S^2$ (cf. Bonnot et al. Moscow Math J 12:747-763, 2012).
3. Finite and Infinitesimal Flexibility of Semidiscrete Surfaces
O. Karpenkov
Research Contribution,   Received: 18 April 2015 / Revised: 28 July 2015 / Accepted: 24 August 2015 / Published online: 3 September 2015
Abstract
In this paper we study infinitesimal and finite flexibility for regular semidiscrete surfaces. We prove that regular 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary $n\ge 3$ we prove that every regular n-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces.
4. Solvability of Linear Differential Systems with Small Exponents in the Liouvillian Sense
R. R. Gontsov, I. V. Vyugin
Research Contribution,   Received: 25 November 2014 / Revised: 20 August 2015 / Accepted: 11 November 2015 / Published online: 26 November 2015
Abstract
The paper is devoted to solvability of linear differential systems by quadratures, one of the classical problems of differential Galois theory. As known, solvability of a system depends entirely on properties of its differential Galois group. However, detecting solvability or non-solvability of a given system is a difficult problem, because the dependence of its differential Galois group on the coefficients of the system remains unknown. We consider systems with regular singular points as well as those with non-resonant irregular ones, whose exponents (respectively, so-called formal exponents in the irregular case) are sufficiently small. It turns out that for systems satisfying such restrictions criteria of solvability can be given in terms of the coefficient matrix.
5. Sergei Duzhin (June 17, 1956 – February 1, 2015)
Obituary,   Published online: 10 December 2015
Abstract
Sergei Duzhin, a permanent participant of Arnold's seminar at the MSU for more than 20 years and a member of Arnold's school, unexpectedly passed away on February 1, 2015 because of an acute heart failure. This is our tribute to his memory.
6. Volumes of Strata of Abelian Differentials and Siegel-Veech Constants in Large Genera
A. Eskin, A. Zorich
Problem Contribution,   Received: 19 July 2015 / Revised: 16 September 2015 / Accepted: 20 October 2015 / Published online: 05 November 2015
Abstract
We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.

### Issue 3, September 2015, Pages 211-343

1. Local Invariants of Framed Fronts in 3-Manifolds
V. Goryunov, S. Alsaeed
Research Contribution,   Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for framed fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.
2. Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in ${\mathbb R}^2$ and ${\mathbb C}^2$
V. A. Vassiliev
Research Contribution,   Received: 7 November 2014 / Accepted: 3 June 2015 / Published online: 11 August 2015
Abstract
The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials ${\mathbb R}^2 \rightarrow {\mathbb R}$, and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in ${\mathbb C}^2$.
3. The Exponential Map Near Conjugate Points In 2D Hydrodynamics
G. Misiołek
Research Contribution,   Received: 14 January 2015 / Accepted: 3 May 2015 / Published online: 5 August 2015
Abstract
We prove that the weak-Riemannian exponential map of the $L^2$ metric on the group of volume-preserving diffeomorphisms of a compact two-dimensional manifold is not injective in any neighbourhood of its conjugate vectors. This can be viewed as a hydrodynamical analogue of the classical result of Morse and Littauer.
4. Critical Set of the Master Function and Characteristic Variety of the Associated Gauss-Manin Differential Equations
A. Varchenko
Research Contribution,   Received: 7 November 2014 / Accepted: 15 June 2015 / Published online: 7 July 2015
Abstract
We consider a weighted family of n parallelly transported hyperplanes in a k-dimensional affine space and describe the characteristic variety of the Gauss – Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution. These statements generalize (Varchenko, Mathematics 2:218-231, 2014), where such a description was obtained for a weighted generic family of parallelly transported hyperplanes. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss – Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras.
5. Bollobás – Riordan and Relative Tutte Polynomials
C. Butler, S. Chmutov
Research Contribution,   Received: 8 December 2014 / Revised: 29 June 2015 / Accepted: 5 July 2015 / Published online: 28 July 2015
Abstract
We establish a relation between the Bollobás – Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.
6. The Conley Conjecture and Beyond
V. L. Ginzburg, B. Z. Gürel
Research & Survey Contribution,   Received: 25 November 2014 / Accepted: 19 May 2015 / Published online: 4 June 2015
Abstract
This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.
7. Disconjugacy and the Secant Conjecture
A. Eremenko
Problem Contribution,   Received: 5 July 2015 / Accepted: 28 July 2015 / Published online: 4 August 2015
Abstract
We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable.
8. Erratum to: Abundance of 3-Planes on Real Projective Hypersurfaces
S. Finashin, V. Kharlamov
Erratum,    Published online: 31 July 2015
Abstract
When we published this article, there was a typo in the first line of Theorem 5.3.1. Please find the corrected text in the pdf. The publisher apologises for this mistake.

### Issue 2, July 2015, Pages 101-209

1. Remarks on the Circumcenter of Mass
Serge Tabachnikov, Emmanuel Tsukerman
Research Contribution,   Received: 15 December 2014 / Accepted: 23 March 2015 / Published online: 31 March 2015
Abstract
Suppose that to every non-degenerate simplex $\Delta\subset\mathbb R^n$ a 'center' $C(\Delta)$ is assigned so that the following assumptions hold:
1. The map $\Delta\to C(\Delta)$ commutes with similarities and is invariant under the permutations of the vertices of the simplex;
2. The map $\Delta\to \operatorname{Vol}(\Delta)C(\Delta)$ is polynomial in the coordinates of the vertices of the simplex.
Then $C(\Delta)$ is an affine combination of the center of mass $CM(\Delta)$ and the circumcenter $CC(\Delta)$ of the simplex: $$C(\Delta)=tCM(\Delta)+(1-t)CC(\Delta),$$ where the constant $t\in\mathbb R$ depends on the map $\Delta\mapsto C(\Delta)$ (and does not depend on the simplex $\Delta$).
The motivation for this theorem comes from the recent study of the circumcenter of mass of simplicial polytopes by the authors and by A. Akopyan.
2. Vortex Dynamics of Oscillating Flows
V. A. Vladimirov, M. R. E. Proctor, D. W. Hughes
Research Contribution,   Received: 22 December 2014 / Accepted: 23 March 2015 / Published online: 10 April 2015
Abstract
We employ the method of multiple scales (two-timing) to analyse the vortex dynamics of inviscid, incompressible flows that oscillate in time. Consideration of distinguished limits for Euler's equation of hydrodynamics shows the existence of two main asymptotic models for the averaged flows: strong vortex dynamics (SVD) and weak vortex dynamics (WVD). In SVD the averaged vorticity is 'frozen' into the averaged velocity field. By contrast, in WVD the averaged vorticity is 'frozen' into the 'averaged velocity + drift'. The derivation of the WVD recovers the Craik-Leibovich equation in a systematic and quite general manner. We show that the averaged equations and boundary conditions lead to an energy-type integral, with implications for stability.
3. On an Equivariant Version of the Zeta Function of a Transformation
S. M. Gusein-Zade, I. Luengo, A. Melle-Hernández
Research Contribution,   Received: 17 December 2014 / Accepted: 4 April 2015 / Published online: 28 April 2015
Abstract
Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W. Lück and J. Rosenberg. Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring itself. We give an A'Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.
4. On Local Weyl Equivalence of Higher Order Fuchsian Equations
Shira Tanny, Sergei Yakovenko
Research Contribution,   Received: 26 December 2014 / Accepted: 15 April 2015/ Published online: 08 May 2015
Abstract
We study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the "type of differential equation" introduced by Frobenius. The main source of difficulties is the fact that there is no natural group action generating this classification. We establish a number of results on higher order equations which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations.
5. Abundance of 3-Planes on Real Projective Hypersurfaces
S. Finashin, V. Kharlamov
Research Contribution,   Received: 7 November 2014 / Accepted: 2 May 2015 / Published online: 2 June 2015
Abstract
We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2) ={{d+3}\choose3}$, contains many'' real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
6. A Few Problems on Monodromy and Discriminants
V. A. Vassiliev
Problem Contribution,   Received: 15 February 2015 / Accepted: 31 March 2015 / Published online: 16 April 2015
Abstract
The article contains several problems concerning local monodromy groups of singularities, Lyashko-Looijenga maps, integral geometry, and topology of spaces of real algebraic manifolds.

### Issue 1, March 2015, Pages 1-99

1. Journal Description Arnold Mathematical Journal
Editorial,
2. Riemannian Geometry of the Contactomorphism Group
David G. Ebin, Stephen C. Preston
Research Contribution,   Received: 11 November 2014 / Accepted: 8 December 2014
Abstract
Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.
A. A. Agrachev
Research Contribution,   Received: 10 November 2014 / Accepted: 16 December 2014
Abstract
We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions.
4. A Baker's Dozen of Problems
Serge Tabachnikov
Problem Contribution,   Received: 16 September 2014 / Revised: 23 September 2014 / Accepted: 14 October 2014
Abstract
This article is a collection of open problems, with brief historical and bibliographical comments, somewhat in the spirit of the problem with which V. Arnold opened his famous seminar every semester and that were recently collected and published in a book form.
5. Periods of Pseudo-Integrable Billiards
Problem Contribution,   Received: 10 November 2014 / Accepted: 26 December 2014
Abstract
Consider billiard desks composed of two concentric half-circles connected with two edges. We examine billiard trajectories having a fixed circle concentric with the boundary semicircles as the caustic, such that the rotation numbers with respect to the half-circles are ρ1 and ρ2 respectively. Are such billiard trajectories periodic, and what are all possible periods for given ρ1 and ρ2?
6. Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes
A. M. Vershik
Problem Contribution,   Received: 8 November 2014 / Accepted: 31 December 2014
Abstract
We describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.
7. Space of Smooth 1-Knots in a 4-Manifold: Is Its Algebraic Topology Sensitive to Smooth Structures?
Oleg Viro
Problem Contribution,   Received: 12 December 2014 / Accepted: 12 February 2015
Abstract
We discuss a possibility to get an invariant of a smooth structure on a closed simply connected 4-manifold from homotopy invariants of the space of loops smoothly embedded into the manifold.
8. Problems Around Polynomials: The Good, The Bad and The Ugly...
Boris Shapiro
Problem Contribution,   Received: 7 November 2014 / Accepted: 16 March 2015 / Published online: 25 March 2015
Abstract
The Russian style of formulating mathematical problems means that nobody will be able to simplify your formulation as opposed to the French style which means that nobody will be able to generalize it, - Vladimir Arnold.

## Articles not assigned to an issue

1. Convex Shapes and Harmonic Caps
Laura DeMarco, Kathryn Lindsey
Research Contribution,   Received: 4 February 2016 / Revised: 30 November 2016 / Accepted: 23 December 2016
Abstract
Any planar shape $P\subset{\mathbb{C}}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^{3}$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q=S{\setminus}P$ is the associated cap. We study the cap construction when the curvature is harmonic measure on the boundary of $({\hat{{\mathbb{C}}}}{\setminus}P,\infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.
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2. Random Chain Complexes
Viktor L. Ginzburg, Dmitrii V. Pasechnik
Research Contribution,   Received: 16 March 2016 / Revised: 9 December 2016 / Accepted: 23 December 2016
Abstract
We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology.
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3. The $4n^{2}$-Inequality for Complete Intersection Singularities
Aleksandr V. Pukhlikov
Research Contribution,   Received: 11 July 2016 / Revised: 25 October 2016 / Accepted: 17 November 2016
Abstract
The famous $4n^{2}$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than $4n^{2}\mu$, where $\mu$ is the multiplicity of the singular point.
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4. Flows in Flatland: A Romance of Few Dimensions
Gabriel Katz
Research Exposition,   Received: 2 March 2016 / Revised: 15 October 2016 / Accepted: 23 October 2016
Abstract
This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on $n$-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.
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5. The Geometry of Axisymmetric Ideal Fluid Flows with Swirl
Pearce Washabaugh, Stephen C. Preston
Research Contribution,   Received: 3 February 2016 / Revised: 21 August 2016 / Accepted: 15 October 2016
Abstract
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X=u(r)\partial_{\theta}$ iff $\partial_{r}(ru^{2})>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$.
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6. On Postsingularly Finite Exponential Maps
Walter Bergweile
Research Contribution,   Received: 5 December 2015 / Revised: 15 August 2016 / Accepted: 6 September 2016
Abstract
We consider parameters $\lambda$ for which 0 is preperiodic under the map $z\mapsto\lambda e^{z}$. Given $k$ and $l$, let $n(r)$ be the number of $\lambda$ satisfying $0<|\lambda|\leq r$ such that 0 is mapped after $k$ iterations to a periodic point of period $l$. We determine the asymptotic behavior of $n(r)$ as $r$ tends to $\infty$.
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7. Some Recent Generalizations of the Classical Rigid Body Systems
Vladimir Dragović, Borislav Gajić
Research Exposition,   Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016 / Published online: 19 September 2016
Abstract
Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations.
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8. Building Thermodynamics for Non-uniformly Hyperbolic Maps
Vaughn Climenhaga, Yakov Pesin
Research Exposition,   Received: 4 February 2016 / Accepted: 20 July 2016 / Published online: 09 August 2016
Abstract
We briefly survey the theory of thermodynamic formalism for uniformly hyperbolic systems, and then describe several recent approaches to the problem of extending this theory to non-uniform hyperbolicity. The first of these approaches involves Markov models such as Young towers, countable-state Markov shifts, and inducing schemes. The other two are less fully developed but have seen significant progress in the last few years: these involve coarse-graining techniques (expansivity and specification) and geometric arguments involving push-forward of densities on admissible manifolds.
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9. Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials
Dierk Schleicher
Research Contribution,   Received: 15 October 2015 / Revised: 9 February 2016 / Accepted: 7 April 2016 / Published online: 02 August 2016
Abstract
We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sects.Â 3 and 4). A simple extension, angled internal addresses, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Sect.Â 6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Sect.Â 7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.