Independence complexes and incidence graphs

We show that the independence complex of the incidence graph of a hypergraph is homotopy equivalent to the combinatorial Alexander dual of the independence complex of the hypergraph, generalizing a result of Csorba. As an application, we refine and generalize a result of Kawamura on a relation between the homotopy types of the independence complex and the edge covering complex of a graph

Supersymmetric lattice fermions on the triangular lattice: superfrustration and criticality

We study a model for itinerant, strongly interacting fermions where a judicious tuning of the interactions leads to a supersymmetric Hamiltonian. On the triangular lattice this model is known to exhibit a property called superfrustration, which is characterized by an extensive ground state entropy. Using a combination of numerical and analytical methods we study various ladder geometries obtained by imposing doubly periodic boundary conditions on the triangular lattice. We compare our results to various bounds on the ground state degeneracy obtained in the literature. For all systems we find that the number of ground states grows exponentially with system size. For two of the models that we study we obtain the exact number of ground states by solving the cohomology problem. For one of these, we find that via a sequence of mappings the entire spectrum can be understood. It exhibits a gapped phase at 1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate fillings. The gapless phase separates into an exponential number of sectors, where the continuum limit of each sector is described by a superconformal field theory.Comment: 50 pages, 12 figures, 2 appendice

Geometric and Algebraic Combinatorics

The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture

Graded Betti numbers of edge ideals

Esta tesis se centra en el estudio de los números de Betti graduados y otros invariantes homológicos como la regularidad de Castelnuovo-Mumford o la dimensión proyectiva correspondientes a ideales asociados a grafos. Se estudian los valores de los números de Betti graduados así como la estructura de los valores no nulos organizados en su diagrama de Betti. Se aportan fórmulas combinatorias para calcular todos los números de Betti graduados en dos familias particulares y se analiza la forma del diagrama de Betti de ideales de grafos cuya resolución no es lineal. En el caso de grafos bipartitos se caracterizan aquellos cuyo ideal asociado tiene regularidad 3 y se analiza la forma del diagrama de Betti de aquellos cuya regularidad es estrictamente mayor. Esta memoria incluye un apéndice con un algoritmo para la generación de una lista exhaustiva de representantes de cada clase de isomorfía de grafos bipartitos conexos.Departamento de Álgebra, Geometría y Topología2012-11-1

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres