Related papers: A Topological Perspective on Interacting Algebraic…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
The algebraic analysis of social systems, or algebraic social network analysis, refers to a collection of methods designed to extract information about the structure of a social system represented as a directed graph. Central among these…
We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy…
Motivated by applications in modelling quantum systems using coalgebraic techniques, we introduce a fibred coalgebraic logic. Our approach extends the conventional predicate lifting semantics with additional modalities relating conditions…
The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's…
Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master equations, which can be interpreted as…
We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is…
In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which…
Recent developments in applied algebraic topology can simplify and extend results in graphic statics - the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in…
We introduce the classical theory of the interplay between group theory and topology into the context of operads and explore some applications to homotopy theory. We first propose a notion of a group operad and then develop a theory of…
Modalities in homotopy type theory are used to create and access subuniverses of a given type universe. These have significant applications throughout mathematics and computer science, and in particular can be used to create universes in…
Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…
Evaluating the quality of reasoning traces from large language models remains understudied, labor-intensive, and unreliable: current practice relies on expert rubrics, manual annotation, and slow pairwise judgments. Automated efforts are…
We introduce $\omega$-catoids as generalisations of (strict) $\omega$-categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce $\omega$-quantales that…
We present a computational implementation of diagrammatic sets, a model of higher-dimensional diagram rewriting that is "topologically sound": diagrams admit a functorial interpretation as homotopies in cell complexes. This has potential…
This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional…