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The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of…
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
This expository article presents a self-contained introduction to simplicial homology for finite simplicial complexes, emphasizing concrete computation and geometric intuition. Beginning with orientations of simplices and the construction…
In this paper we study the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization…
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the…
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
In this article, we study the cohomology ring of real moment-angle complexes over a simplicial complex $K$. Combinatorial generators for the cohomology can be given in terms of $K$. For $K$ the boundary of an $n$-gon, we give a full…
Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define - perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which…
We introduce weak morphisms of higher dimensional automata and use them to define preorder relations for HDAs, among which homeomorphic abstraction and trace equivalent abstraction. It is shown that homeomorphic abstraction is essentially…
We demonstrate the possibility of coding parts, features that are higher level than boundaries, using a modified AT field after augmenting the interaction term of the AT energy with a non-local term and weakening the separation into…
We present a novel class of robots belonging to Constrained Collaborative Mobile Agents (CCMA) family which consists of ground mobile bases with non-holonomic constraints. Moreover, these mobile robots are constrained by closed-loop…
We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of…
Convex regression is a promising area for bridging statistical estimation and deterministic convex optimization. New piecewise linear convex regression methods are fast and scalable, but can have instability when used to approximate…
There is a large class of problems in algebraic combinatorics which can be distilled into the same challenge: construct an explicit combinatorial bijection. Traditionally, researchers have solved challenges like these by visually inspecting…
This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a…
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we…
To comprehend the hierarchical organization of large integrated systems, we introduce the hierarchical map equation, which reveals multilevel structures in networks. In this information-theoretic approach, we exploit the duality between…