Abstract: We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered \(4n^2\) -inequality for complete intersection singularities. PubDate: 2019-03-19

Abstract: We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field \({{\mathbb {C}}}(\!(t)\!)\) of formal Laurent series at an irregular singular point corresponding to \(t=0\) . The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an “automatic translation” which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. In addition, we draw some (apparently unnoticed) parallels between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations. PubDate: 2019-03-18

Abstract: In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas’s invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time. PubDate: 2019-03-15

Abstract: We study the localization properties of the equal-time electron Green’s function in a Chern insulator in an arbitrary dimension and with an arbitrary number of bands. We prove that the Green’s function cannot decay super-exponentially if the Hamiltonian is finite-range and the quantum Hall response is nonzero. For a general band Hamiltonian (possibly infinite-range), we prove that the Green’s function cannot be finite-range if the quantum Hall response is nonzero. The proofs use methods of algebraic geometry. PubDate: 2019-03-08

Abstract: Modular curves \(X_{1}(N)\) parametrize elliptic curves with a point of order N. They can be identified with connected components of projectivized strata \(\mathbb {P}\mathcal {H}(a,-a)\) of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve \(X_{1}(N)\) . This defines a family of graphs with specific combinatorial properties. This approach provides a geometrico-combinatorial computation of the genus and the number of punctures of modular curves \(X_{1}(N)\) . Although the dimension of a stratum of meromorphic differentials depends only on the genus and the numbers of the singularities, the topological complexity of the stratum crucially depends on the order of the singularities. PubDate: 2019-03-07

Abstract: We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by \(\pi \) -rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant \(p^2+q^2+r^2-pqr\) , where p, q, r are the weights of arrows in a quiver. We also classify skew-symmetric mutation-finite real \(3\times 3\) matrices and explore the structure of acyclic representatives in finite and infinite mutation classes. PubDate: 2019-03-04

Abstract: In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity map with respect to some norm, e.g., \(C^1\) - or \(C^0\) -norm for general diffeomorphisms and the \(\gamma \) -norm in the Hamiltonian case, and the third problem is the Lagrangian Poincaré recurrence conjecture. PubDate: 2019-03-04

Abstract: The class \({\mathcal {B}}\) of lacunary polynomials \(f\,(x)\ :=\ -1\ +\ x\ +\ x^{n}\ +\ x^{m_{1}}\ +\ x^{m_{2}}\ +\ \cdots \ +\ x^{m_{s}}\) , where \(s\ \geqslant \ 0\) , \(m_{1}\ -\ n\ \geqslant \ n\ -\ 1\) , \(m_{q+1}\ -\ m_{q}\ \geqslant \ n\ -\ 1\) for \(1\ \leqslant \ q\ <\ s\) , \(n\ \geqslant \ 3\) is studied. A polynomial having its coefficients in \(\{0,\,1\,\}\) except its constant coefficient equal to \(-1\) is called an almost Newman polynomial. A general theorem of factorization of the almost Newman polynomials of the class \({\mathcal {B}}\) is obtained. Such polynomials possess lenticular roots in the open unit disk off the unit circle in the small angular sector \(-\pi /18\ \leqslant \ \arg \,z\ \leqslant \ \pi /18\) and their nonreciprocal parts are always irreducible. The existence of lenticuli of roots is a peculiarity of the class \({\mathcal {B}}\) . By comparison with the Odlyzko–Poonen Conjecture and its variant Conjecture, an Asymptotic Reducibility Conjecture is formulated aiming at establishing the proportion of irreducible polynomials in this class. This proportion is conjectured to be 3 / 4 and estimated using Monte-Carlo methods. The numerical approximate value \(\approx \ 0.756\) is obtained. The results extend those on trinomials (Selmer) and quadrinomials (Ljunggren, Mills, Finch and Jones). PubDate: 2019-03-04

Abstract: We present an improved construction of the fundamental matrix factorization in the FJRW-theory given in Polishchuk and Vaintrob (J Reine Angew Math 714:1–22, 2016). The revised construction makes the independence on choices more apparent and works for a possibly nonabelian finite group of symmetries. One of the new ingredients is the category of dg-matrix factorizations over a dg-scheme. PubDate: 2019-02-25

Abstract: We investigate some geometric properties of the real algebraic variety \(\Delta \) of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in \(\Delta \) . We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of \(\Delta \) ) and random matrix theory. PubDate: 2019-01-02

Abstract: It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase space and with uniqueness replaced by isolatedness are also constructed. Finally, reversible analogues of all the examples are described. PubDate: 2018-11-19

Abstract: We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines. PubDate: 2018-10-01

Abstract: This note contains a proof of van der Waerden’s theorem, “one of the most elegant pieces of mathematics ever produced,” in nine figures. The proof follows van der Waerden’s original idea to establish the existence of what are now called van der Waerden numbers by using double induction. It also contains ideas and terminology introduced by I. Leader and T. Tao. PubDate: 2018-10-01

Abstract: We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global hyperbolicity of renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer. Furthermore, we prove the global \(C^{1+\beta }\) -rigidity conjecture for such maps, giving the first example of a smooth rigidity theorem for unimodal maps whose critical exponent is not an even integer. PubDate: 2018-10-01

Abstract: Picard–Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order n by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. Liouville in 1839 found an elementary criterium for such solvability for \(n=2\) . Ritt simplified Liouville’s theorem (1948). In 1973 Rosenlicht proved a similar criterium for arbitrary n. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary n and proves the same criterium. PubDate: 2018-10-01

Abstract: This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric spaces of classical types. Here, we determine the generating series the numbers of Borel orbits in \({\mathbf {SO}}_{2n+1}/{\mathbf {S(O}}_{2p}\times {\mathbf {O}}_{2q+1} \mathbf {)}\) (type BI) and in \({\mathbf {Sp}}_n/{\mathbf {Sp}}_p \times {\mathbf {Sp}}_q\) (type CII). In addition, we explore relations to lattice path enumeration. PubDate: 2018-10-01

Abstract: In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014—Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar. PubDate: 2018-10-01

Authors:F. Pakovich Abstract: Using dynamical methods we give a new proof of the theorem saying that if A, B, X are rational functions of complex variable z of degree at least two such that \(A\circ X=X\circ B\) and \({\mathbb C}(B,X)={\mathbb C}(z)\) , then the Galois closure of the field extension \({\mathbb C}(z)/{\mathbb C}(X)\) has genus zero or one. PubDate: 2018-02-12 DOI: 10.1007/s40598-018-0081-6

Authors:Oleg Karpenkov Abstract: In this small paper we bring together some open problems related to the study of the configuration spaces of tensegrities, i.e. graphs with stresses on edges. These problems were announced in Doray et al. (Discrete Comput Geom 43:436–466, 2010), Karpenkov et al. (ARS Math Contemp 6:305–322, 2013), Karpenkov (The combinatorial geometry of stresses in frameworks. arXiv:1512.02563 [math.MG], 2017), and Karpenkov (Geometric Conditions of Rigidity in Nongeneric settings, 2016) (by F. Doray, J. Schepers, B. Servatius, and the author), for more details we refer to the mentioned articles. PubDate: 2018-02-02 DOI: 10.1007/s40598-018-0080-7