Related papers: On the Arnold's conjecture on hyperbolic homogeneo…

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…

In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we…

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by…

This paper is subsequent to [5]. In this paper, we extend the classification of hyperbolic Dehn fillings with sufficiently large coefficients by addressing the remaining case not covered in [5]. Specifically, by considering the case in…

We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $\delta_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module…

The Vol-Det Conjecture, formulated by Champanerkar, Kofman and Purcell, states that there exists a specific inequality connecting the hyperbolic volume of an alternating link and its determinant. Among the classes of links for which this…

For $n \ge 2$, we prove that a finite volume complex hyperbolic $n$-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two is arithmetic, paralleling our previous work for real…

Using an approach to the Jacobian Conjecture by L.M. Dru\.zkowski and K. Rusek 12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev[27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of…

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result…

A quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms is proved. We also give an application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux.

We explore the connection between the notion of Hopf category and the categorification of the infinite dimensional Heisenberg algebra via graphical calculus proposed by M.Khovanov. We show that the existence of a Hopf structure on a…

We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones…

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of…

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to…

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous…

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…

The relationship between associative composition algebras of dimensions 2 and 4 within the context of homogeneous spaces, with a particular focus on Hamiltonian quaternions, is explored. In the special case of Hamiltonian quaternions, the…

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of M. Sapir.

This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of…

Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical…