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We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (aka. synchronous relations, or regular relations). They are the equivalent of monoids for regular languages, however they conceptually…

We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of…

We study pseudoalgebras from the point of view of pseudo-dual of classical Lie coalgebra structures. We define the notions of Lie H-coalgebra and Lie pseudo-bialgebra. We obtain the analog of the CYBE, the Manin triples and Drinfeld's…

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_\omega$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex)…

The relationship between the sources of physical fields and the fields themselves is investigated with regard to the coupling of topological information between them. A class of field theories that we call topological field theories is…

This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and…

$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose…

Two-field mimetic gravity was recently realized by looking at the singular limit of the conformal transformation between the auxiliary metric and the physical metric with two scalar fields involved. In this paper, we reanalyze the singular…

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the…

The presence of a boundary (or defect) in a conformal field theory allows one to generalize the notion of an exactly marginal deformation. Without a boundary, one must find an operator of protected scaling dimension $\Delta$ equal to the…

Using the superconformal (SC) indices techniques, we construct Seiberg type dualities for $\mathcal{N}=1$ supersymmetric field theories outside the conformal windows. These theories are physically distinguished by the presence of chiral…

This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.

Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed.

This work is divided between two main areas: in the theory of multialgebras, we focus mostly on a new definition of what a freely generated object should be in their category, and on how this category is equivalent to another with partially…

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The…

In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the…

We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.