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Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V)…
The idea that possible configurations of a physical system can be represented as points in a multidimensional configuration space ${\cal C}$ is explored. The notion of spacetime, without ${\cal C}$, does not exist in this theory. Spacetime…
Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant…
By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with…
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and…
In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a…
In a background independent theory without boundary, physical observables may be defined with respect to dynamical reference systems. However, I argue here that there may be a symmetry that exchanges the degrees of freedom of the physical…
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental…
Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we…
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…
We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
The well-known theory of "rational canonical form of an operator" describes the invariant factors, or elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space $\V$ over a…
The quantum Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for 2nd order superintegrable…
We define a Frobenius algebra over fusion categories of the form Rep$(G)\boxtimes$Rep$(G)$ which generalizes the diagonal subgroup of $G\times G$. This allows us to extend field theoretical constructions which depend on the existence of a…
This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to…
We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two…