Related papers: On the closed geodesics problem

We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology with compact supports H^i_c(X,E) are finite…

Two extension problems are solved. First, the class of locally matricial algebras over an arbitrary field is closed under extensions. Second, the class of locally finite dimensional semisimple algebras over a fixed field is closed under…

We study stable finiteness of extensions of 2-graph C*-algebras determined by saturated hereditary sets of vertices. We use two iterations of the Pimsner-Voiculescu sequence to calculate the map in K-theory induced by the inclusion of a…

We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the…

We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many…

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not…

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we…

In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we…

For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each…

We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of arithmetic sequences is unbounded.

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

Let $\Delta \subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_{\Delta}^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |k\Delta \cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients.…

We study the convolution algebra $H_{*}(Z^{\theta}_{W})$ of homology on diagram automorphism fixed point quiver variety and prove that there exists an algebra homomorphism from the universal enveloping algebra of the diagram automorphism…

A finite connected CW complex which is a co-H-space is shown to have the homotopy type of a wedge of a bunch of circles and a simply-connected finite complex after almost $p$-completion at a prime $p$.

In the present paper we indicate some Leibniz algebras whose closures of orbits under the natural action of $\GL_n$ form an irreducible component of the variety of complex $n$-dimensional Leibniz algebras. Moreover, for these algebras we…

Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only…

The Ricci curvature equations are a central subject of study in geometry. However, in the smooth real case, their linear analysis is often confined to settings in which the background metric is Einstein. In this paper, we establish…

Suppose $G$ is a connected complex Lie group and $\Gamma$ is a discrete subgroup such that $X := G/\Gamma$ is K\"ahler and the codimension of the top non--vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or…

Bell and Zhang have shown that if $A$ and $B$ are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the…