Related papers: On the Atiyah problem on hyperbolic configurations…
To apply Morse's critical point theory, we use mutual distances as coordinates to discuss a kind of central configuration of the planar Newtonian 5-body problem with a trapezoidal convex hull, i.e., four of the five bodies are located at…
We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex,…
The Atiyah-Hitchin manifold arises in many different contexts, ranging from its original occurrence as the moduli space of two SU(2) 't Hooft-Polyakov monopoles in 3+1 dimensions, to supersymmetric backgrounds of string theory. In all these…
A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…
The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean…
In this paper we continue the analysis of non-diagonalisable hyperbolic systems initiated in \cite{GarJRuz, GarJRuz2}. Here we assume that the system has discontinuous coefficients or more in general distributional coefficients.…
We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar…
We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…
We have solved completely the problem of the description of quasi-linear hyperbolic differential equations in two independent variables that are invariant under three-parameter Lie groups.
A central question in arrangement theory is to determine whether the characteristic polynomial $\Delta_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane…
In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.
We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…
In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah…
We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional…
We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in…
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…
Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that…
A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints…
In all dimensions $n \ge 4$ not of the form $4m+3$, we show that there exists a closed hyperbolic $n$-manifold which is not the boundary of a compact $(n+1)$-manifold. The proof relies on the relationship between the cobordism class and the…
We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…