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Quantum gauge theories with finite-dimensional representation spaces are constructed that can have canonical gauge field theories as singular limits. They describe nature as a recursive quantum assembly by iterating Fermi-Dirac…
We develop a mathematical theory of quantization of multidimensional variational principles, and compare it with traditional constructions of quantum field theory. We conjecture that mathematical realization of quantum field theory axioms,…
Using a almost product structure defined by a spray, we give a necessary and sufficient condition, for a linear connection with vanishing torsion to be Riemannian and, for the semi-simplicity of Lie algebra of projectable vector fields…
Goal of this review is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, a…
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.
We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent…
Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible.
We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic…
We describe how physical universes that are composed of gauge and gravitationally interacting bosonic and fermionic quantum fields arise from the generic discrete distribution of many quantifiable properties of arbitrary static entities.…
We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra $kG$ of a finite group $G$ of Lie type defined over a finite field of odd characteristic $p$, where $k$ is an arbitrary field of…
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this…
Our concern is a nonsingular plane curve defined over a finite field of q elements which includes all the rational points of the projective plane over the field. The possible degree of such a curve is at least q+2. We prove that nonsingular…
The paper continues the author's research in the problem of quantitative investigation of basic curvelinear quasiinvariants of quasiconformal curves. It concerns polygons with infinite number of vertices and provides various distortion…
Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which…
We study the three-point quantum $\mathfrak{sl_2}$-Gaudin model. In this case the compactification of the parameter space is $\overline{M_{0,4}(\mathbb{C})}$, which is the Riemann sphere. We analyze sphere coverings by the joint spectrum of…
In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.
Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…
This article extends the study of cyclic ramified covers of the projective line defined by Kummer equations. We consider the most general case of such covers, allowing arbitrary orders in the roots of the generating radicant. The primary…
Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a…