Related papers: Trends in Topological Combinatorics
This is an extended version of a talk given at the conference "Algebra and Topology in Interaction" on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995…
The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying…
Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this…
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain…
We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…
The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last…
This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories and give the reader the background material needed to understand almost any…
This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of…
Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…
We introduce a new class of extensions of terms that consists in navigation strategies and insertion of contexts. We introduce an operation of combination on this class which is associative, admits a neutral element and so that each…
We present 18 Introductory Lectures on K-Theory covering its basic three branches, namely topological, analytic (K-Homology) and Higher Algebraic K-Theory, 6 lectures on each branch. The skeleton of these notes was provided by the author's…
The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom…
We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure.…
We describe various strengthenings of the concept of topological transitivity. Especially when one departs from the family of invertible systems, a number of interesting properties arise. We present the architecture of implications among…
This paper aims to undertake an exploration of the behavior of the moduli space of line arrangements while establishing its combinatorial interplay with the incidence structure of the arrangement. In the first part, we investigate…
We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical…
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of…
Transportation is an essential area in the nowadays society, both for business sector and citizenry. There are different kinds of transportation systems, each one with its own characteristics. In the same way, various areas of knowledge can…
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy…