Lucian M Ionescu
Illinois State University, Mathematics, Faculty Member
- I am a mathematician, scientist and artist in my spare time. Welcome!edit
Page 1. ON A GENERALIZED LORENTZ FORCE LUCIAN M. IONESCU Abstract. The generalized Lorentz force is investigated as a possible avenue to counter gravity. Contents 1. Introduction 1 2. Two-body Systems 3 2.1. Coriolis and Centripetal... more
Page 1. ON A GENERALIZED LORENTZ FORCE LUCIAN M. IONESCU Abstract. The generalized Lorentz force is investigated as a possible avenue to counter gravity. Contents 1. Introduction 1 2. Two-body Systems 3 2.1. Coriolis and Centripetal Forces 4 2.2. ...
Research Interests:
Research Interests:
The principle of equivalence implies the inertial mass equals to gravitational mass. Gravity is understood in terms of the quark model, amended by Platonic symmetry. This allows to comment on the origin of inertial mass and how it can be... more
The principle of equivalence implies the inertial mass equals to gravitational mass. Gravity is understood in terms of the quark model, amended by Platonic symmetry. This allows to comment on the origin of inertial mass and how it can be controlled when controlling gravity.
Research Interests:
Research Interests:
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are... more
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.
Research Interests:
Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics SL(2,C)_h-gauge theory of dynamical lattices, with unifying gauge... more
Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics SL(2,C)_h-gauge theory of dynamical lattices, with unifying gauge "group" the quantum bundle obtained from the Hopf monopole bundle underlying the quaternionic algebra and Dirac-Weyl spinors. The deformation parameter is the inverse of light speed 1/c, in duality with Planck's constant h. Then mass and electric charge form a complex coupling constant (m,q), for which the quantum determinant of the quantum group SL(2,C)_h expresses the interaction strength as a linking number 2-form. There is room for both Coulomb constant k_C and Newton's gravitational constant G_N, exponentially weaker then the reciprocal of the fine structure constant α. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized q...
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Ideals are used to define homological functors for additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology... more
Ideals are used to define homological functors for additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. Applications are considered: derived categories and functors.
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We prove the equivalence of several different definitions of higher order differential operators and define differential operators of lower (negative) orders. We then study derived Lie and sh-Lie brackets on an abelian subalgebra of a Lie... more
We prove the equivalence of several different definitions of higher order differential operators and define differential operators of lower (negative) orders. We then study derived Lie and sh-Lie brackets on an abelian subalgebra of a Lie algebra as well as the cohomology of a certain type of
There are numerous indications that Physics, at its foundations, is algebraic Number Theory. The Bohr’s Model for the Hydrogen atom is the starting point of a quantum computing model on serial-parallel graphs is provided as the quantum... more
There are numerous indications that Physics, at its foundations, is algebraic Number Theory. The Bohr’s Model for the Hydrogen atom is the starting point of a quantum computing model on serial-parallel graphs is provided as the quantum system affording the partition function of the Riemann Gas / Primon model. The propagator of the corresponding discrete Path Integral formalism is a fermionic zeta value “closely” related to the experimental value of the fine structure constant corresponding to the continuum Path Integral formalism of Feynmann. The duality of multiplicative number theory, as a theory of the graded Hopf module of integers, and the Kleinian geometry of the primary finite fields underlying its base of primitive elements, are briefly mentioned in this framework (“Integer CFT”).
Some algebraic properties of integralsover configuration spaces are investigated in order to better understandquantization and the Connes-Kreimer algebraic approach to renormalization. <BR>In order to isolate the... more
Some algebraic properties of integralsover configuration spaces are investigated in order to better understandquantization and the Connes-Kreimer algebraic approach to renormalization. <BR>In order to isolate the mathematical-physics interface toquantum field theory independent from the specifics of the variousimplementations, the sigma model of Kontsevich is investigated in moredetail. Due to the convergence of the configuration space integrals, themodel allows to study the Feynman rules independently, from an axiomaticpoint of view, avoiding the intricacies of renormalization, unavoidablewithin the traditional quantum field theory. <BR>As an application, a combinatorial approach to constructingthe coefficients of formality morphisms is suggested, as an alternative tothe analytical approach used by Kontsevich. These coefficients are "Feynman integrals", although not quite typical since they do converge. <BR>A second example of "Feynman integrals&quo...
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The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the... more
The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information.
Research Interests: Mathematics and Physics
We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential... more
We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.
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Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values... more
Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values from Number Theory. But what about finite characteristic, via the global-to-local principle? We include some considerations regarding periods and Jacobi sums, the analog of Veneziano amplitudes in String Theory.
Research Interests: Mathematics and Physics
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to... more
Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience. Such lattice models of finite fields provide a good basis for later developing the theory in a more concrete way, including Frobenius elements, all the way to Artin reciprocity law. Examples are provided, intended for an undergraduate audience in the first place.
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The interface between classical physics and quantum physics is explained from the point of view of quantum information theory (Feynman Processes). The interpretation depends on a hefty sacrifice: the classical determinism or the arrow of... more
The interface between classical physics and quantum physics is explained from the point of view of quantum information theory (Feynman Processes). The interpretation depends on a hefty sacrifice: the classical determinism or the arrow of time. The wave-particle duality steams from the qubit model, as the root of creation and annihilation of possibilities. A few key experiments are briefly reviewed from the above perspective: quantum erasure, delayed-choice and wave-particle correlation. The CPT-Theorem is interpreted in the framework of categories with duality and a timeless interpretation of the Feynman Processes is proposed. A connection between the fine-structure constant and algebraic number theory is suggested.
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We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are... more
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.
Research Interests:
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via... more
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.
Research Interests:
We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and... more
We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is suggested that the QFTs obtained via deformation quantization and renormalization correspond to each other in the sense of Kontsevich/Cattaneo-Felder.
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L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra... more
L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra structure (Forest Formula), the weights of the corresponding expansions are proved to be cycles of the DG-coalgebra of Feynman graphs. The properties of integrals over configuration spaces (Feynman integrals) are investigated. The aim is to develop a cohomological approach in order to construct the coefficients of formality morphisms using an algebraic machinery, as an alternative to the analytical approach using integrals over configuration spaces. The connection with a related TQFT is mentioned, supplementing the Feynman path integral interpretation of Kontsevich formula.
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The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order... more
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operato...
Research Interests:
Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the... more
Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the ``Kontsevich representation'' of a DGLA of graphs coming from a pre-Lie algebra structure defined by graph insertions. Properties of the dual of its UEA (an odd parity analog of Connes-Kreimer Hopf algebra), are investigated in order to find solutions of the deformation equation. The solution of the initial value deformation problem, at tree-level, is unique. For linear coefficients the resulting formulas are relevant to the Hausdorff series.
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Operads and PROPs are presented, together with examples and applications to quantum physics suggesting the structure of Feynman categories/PROPs and the corresponding algebras.
Research Interests:
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order... more
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operato...
We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controling the deformation theory of associa-tive algebras and prove that there is a canonical solution within the class of graphs without circuits, without assuming the... more
We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controling the deformation theory of associa-tive algebras and prove that there is a canonical solution within the class of graphs without circuits, without assuming the Jacobi identity. The proof is based on the unique factorization property of graph insertions.
Research Interests:
The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the... more
The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information. 1
A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted... more
A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless ” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie alg...
Research Interests:
Research Interests:
The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the non-abelian homology of parity quasi-complexes. Parity contracting homotopies are defined, determining the parity free... more
The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the non-abelian homology of parity quasi-complexes. Parity contracting homotopies are defined, determining the parity free resolutions in a canonical way, enabling the non-abelian bar construction. In this way, the even/odd grouping of the simplicial maps in the cocycle conditions of nonabelian cohomology is explained.
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Abstract. We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf... more
Abstract. We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices. It is suggested that the QFTs obtained via deformation quantization and renormalization correspond to each other in the sense of Kontsevich/Cattaneo-Felder [12, 13]. Contents
Research Interests:
Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology... more
Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. The applications considered in this paper are: derived categories and functors. 2000 Mathematics Subject Classification: 18G50, 18A05.
A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted... more
A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless ” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie alg...
Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the... more
Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the “Kontsevich representation ” of a DGLA of graphs coming from a pre-Lie algebra structure defined by graph insertions (Gerstenhaber composition with Leibniz rule). Properties of the dual of its UEA (an odd parity analog of Connes-Kreimer Hopf algebra), are investigated in order to find solutions of the deformation equation. The solution of the initial value deformation problem, at tree-level, is unique. For linear coefficients the resulting formulas are relevant to the Hausdorff series.
The new emerging quantum physics - quantum computing conceptual bridge, mandates a ``grand unification'' of space-time-matter and quantum information (all quantized), with deep implications for science in general. The major... more
The new emerging quantum physics - quantum computing conceptual bridge, mandates a ``grand unification'' of space-time-matter and quantum information (all quantized), with deep implications for science in general. The major physics revolutions in our understanding of the universe are briefly reviewed and a ``missing'' equivalence principle is identified and its nature explained. An implementation as an external super-symmetry $\C{E}=ic\C{P}$ is suggested, generalizing the Wick rotation ``trick''. Taking advantage of the interpretation of entropy as a measure of symmetry, it is naturally asimilated within the present Feynman Path Integral algebraic formalism.
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A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a... more
A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.
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Quantum Relativity is supposed to be a new theory, which locally is a deformation of Special Relativity, and globally it is a background independent theory including the main ideas of General Relativity, with hindsight from Quantum... more
Quantum Relativity is supposed to be a new theory, which locally is a deformation of Special Relativity, and globally it is a background independent theory including the main ideas of General Relativity, with hindsight from Quantum Theory. The qubit viewed as a Hopf monopole bundle is considered as a unifying gauge “group”. Breaking its chiral symmetry is conjectured to yield gravity as a deformation of electromagnetism. It is already a quantum theory in the context of Quantum Information Dynamics as a discrete, background independent theory, unifying classical and quantum physics. Based on the above, Quantum Gravity is sketched as an open project.
Research Interests: Mathematics and Physics
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via... more
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.
Research Interests:
Some algebraic properties of integrals over configuration spaces are investigated in order to better understand quantization and the Connes-Kreimer algebraic approach to renormal- ization. In order to isolate the mathematical-physics... more
Some algebraic properties of integrals over configuration spaces are investigated in order to better understand quantization and the Connes-Kreimer algebraic approach to renormal- ization. In order to isolate the mathematical-physics interface to quantum field theory independent from the specifics of the various implementations, the sigma model of Kontsevich is investigated in more detail. Due to the convergence of the configuration space integrals, the model allows to study the Feynman rules independently, from an axiomatic point of view, avoiding the intricacies of renormalization, unavoidable within the traditional quantum field theory. As an application, a combinatorial approach to constructing the coecients of formality mor- phisms is suggested, as an alternative to the analytical approach used by Kontsevich. These coe- cients are "Feynman integrals", although not quite typical since they do converge. A second example of "Feynman integrals", defined as state...
Research Interests:
Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups. The article focuses on two basic... more
Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups. The article focuses on two basic constructions of deformation theory: the universal solution of Maurer-Cartan Equation (MCE), which plays the role of the exponential of Lie Theory, and its inverse, the Kuranishi functor, as the logarithm. The deformation functor is the gauge reduction of MCE, corresponding to a Hodge decomposition associated to the strong deformation retract data. The above comparison with Lie Theory leads to a better understanding of Deformation Theory and its applications, e.g. the relation between quantization and Connes-Kreimer renormalization, quantum doubles and Birkhoff decomposition.
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Some stages of development of Manifold Theory are inspected, and how they evolved into the modern discrete frameworks of lattice and spin networks, with help from Topology and Homological Algebra. Recalling experimental evidence that... more
Some stages of development of Manifold Theory are inspected, and how they evolved into the modern discrete frameworks of lattice and spin networks, with help from Topology and Homological Algebra. Recalling experimental evidence that reality is discrete, notably quantum Hall effect, includes more recent findings of quantum knots and spin-net condensates. Thus Pythagoras, Zeno and Plato were right after all: “Number rules the Uni- verse”, perhaps explaining the “unreasonable effectiveness of mathematics”, but not quite, why Quantum Physics’ scattering amplitudes are often Number Theory’s multiple zeta values.
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We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are... more
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly mentioned.
Research Interests:
The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers... more
The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers an "Abstract Galois Theory", as a separated abstract structure from, yet compatible with, the Theory of Schemes, which has its historical origin in Commutative Algebra and motivation in the early stages of Algebraic Topology. The approach to Motives via Deformation Theory was suggested by Kontsevich as early as 1999, and suggests Formal Manifolds, with local models formal pointed manifolds, as the source of motives, and perhaps a substitute for a "universal Weil cohomology". The proposed research aims to gain additional understanding of periods via a concrete project, the discrete algebraic de Rham cohomology, a follow-up of author's previous work. The connection with Arithmetic Gauge Theory should provide additional intuiti...
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The theory of parity quasi-complexes (PQC) is developed, preparing a set up for defining derived functors using resolutions in the nonabelian case. A homotopy structure on the category of PQC is defined, yielding a 2-category structure.... more
The theory of parity quasi-complexes (PQC) is developed, preparing a set up for defining derived functors using resolutions in the nonabelian case. A homotopy structure on the category of PQC is defined, yielding a 2-category structure. The nonabelian homology functor factors through the corresponding homotopy category. Following the relative homological algebra approach, resolutions are defined as PQC having parity contracting homotopies in a suitable category. A canonical non-abelian PQC resolution for groups is defined.