Related papers: Intersection numbers in the curve graph with a uni…

We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.

We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…

A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…

We study the convexity of the entropy functional along particular interpolating curves defined on the space of finitely supported probability measures on a graph.

The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. Firstly, the connective…

Given two curves in $\PP^3$, either implicitly or by a parameterization, we want to check if they intersect. For that purpose, we present and further develop generalized resultant techniques. Our aim is to provide a closed formula in the…

We consider two incidence problems for integral curves of vector fields. The first is an analogue of the Euclidean joints problem, in which lines are replaced by integral curves of smooth vector fields taken from some finite-dimensional…

This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a…

It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex…

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection…

In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their…

We study topological properties of the graph topology.

Consider two paths $\phi,\psi:[0;1]\to [0;1]^2$ in the unit square such that $\phi(0)=(0,0)$, $\phi(1)=(1,1)$, $\psi(0)=(0,1)$ and $\psi(1)=(1,0)$. By continuity of $\phi$ and $\psi$ there is a point of intersection. We prove that from…

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points…

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…

We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete…

Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while…

This is an exposition of results on the existence problem of $\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are…