Related papers: Spider Evaluation and Representations of Web Group…

Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose…

Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…

We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic…

The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The…

In this paper, we show the image of rooted tree maps themselves forms a subspace of the kernel of the evaluation map of multiple $L$-values. For its proof, we define the diamond product as a modified harmonic product and find its…

The non-convexity of a smooth and compact connected component of a real algebraic plane curve can be measured by a combinatorial object called the Poincare-Reeb tree associated to the curve and to a direction of projection. In this paper we…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider…

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned…

We address the problem of classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given $\mathbb{C}$-subalgebra. Hence we can…

Deciding whether there is a single tree -a supertree- that summarizes the evolutionary information in a collection of unrooted trees is a fundamental problem in phylogenetics. We consider two versions of this question: agreement and…

We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or…

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of…

The {\em topological symmetry group} of an embedding $\Gamma$ of an abstract graph $\gamma$ in $S^3$ is the group of automorphisms of $\gamma$ which can be realized by homeomorphisms of the pair $(S^3, \Gamma)$. These groups are motivated…

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the…

Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we…

Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer…

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…

Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…