Let q be a power of a prime, and E be an elliptic curve defined over Fq . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions Fqk for all k ≥ 1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a...

Let A be a supersingular abelian variety deened over a nite eld k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f = Q gi e i for distinct monic irreducible polynomials gi and positive integers ei, we show that there is a group homomorphism ' : A(...

Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups KiR of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, K1R and K2R are characterized, respectively,...

Given a graph covering, there is an induced surjection of critical groups. We will try to determine the structure of the kernel and give results at different levels of generality. A main tool will be dualizing the short exact sequence and working instead with the cokernel. In the case of signed graphs and two-sheeted coverings, we will show that the kernel of the induced surjection can be descr...

For any given graph $G$ consider a $\widetilde{G}$ which is cone over $G.$ In this paper, we study two important invariants of such cone. Namely, complexity (the number spanning trees) and the Jacobian graph. We prove that coincides rooted forests in isomorphic to cokernel operator $I+L(G),$ where $L(G)$ Laplacian $I$ identity matrix. As consequence, one can calculate as $\det(I+L(G)).$

We show that bigrassmannian permutations determine the socle of cokernel an inclusion Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from penultimate two-sided cell. Combinatorially, a module $w\in S_n$ into dominant are shown determined essential set $w$ and their degrees graded picture computable terms associated rank function. As applic...

In this thesis, we explore techniques in statistics and persistent homology, which detect features among data sets such as graphs, triangulations and point cloud. We accompany our theorems with algorithms and experiments, to demonstrate their effectiveness in practice. We start with the derivation of graph scan statistics, a measure useful to assess the statistical significance of a subgraph in...