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## On Local Weyl Equivalence of Higher Order Fuchsian Equations

### Arnold Mathematical Journal (2015-07-01) 1: 141-170 , July 01, 2015

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.

## Riemannian Geometry of the Contactomorphism Group

### Arnold Mathematical Journal (2015-03-01) 1: 5-36 , March 01, 2015

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 $$L^2$$ 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.

## The Exponential Map Near Conjugate Points In 2D Hydrodynamics

### Arnold Mathematical Journal (2015-09-01) 1: 243-251 , September 01, 2015

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.

## N-Division Points of Hypocycloids

### Arnold Mathematical Journal (2016-06-01) 2: 149-158 , June 01, 2016

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.

## Generalized Plumbings and Murasugi Sums

### Arnold Mathematical Journal (2016-03-01) 2: 69-119 , March 01, 2016

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.

## Bollobás–Riordan and Relative Tutte Polynomials

### Arnold Mathematical Journal (2015-09-01) 1: 283-298 , September 01, 2015

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.

## Sergei Duzhin (June 17, 1956–February 1, 2015)

### Arnold Mathematical Journal (2015-12-01) 1: 473-479 , December 01, 2015

## Betti Posets and the Stanley Depth

### Arnold Mathematical Journal (2016-06-01) 2: 267-276 , June 01, 2016

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.

## Flows in Flatland: A Romance of Few Dimensions

### Arnold Mathematical Journal (2017-06-01) 3: 281-317 , June 01, 2017

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.

## The Number $$\pi $$ π and a Summation by $$SL(2,{\mathbb {Z}})$$ S L ( 2 , Z )

### Arnold Mathematical Journal (2017-11-02): 1-7 , November 02, 2017

The sum (resp. the sum of squares) of the defects in the triangle inequalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression.

Namely, let $$f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$$ . Then, where the sum runs by all $$a,b,c,d\in {\mathbb {Z}}_{\ge 0}$$ such that $$ad-bc=1$$ . We present a proof of these formulae and list several directions for the future studies.