Zakaria Giunashvili - Academia.edu (original) (raw)

Papers by Zakaria Giunashvili

Research paper thumbnail of Noncommutative Geometry of Phase Space

arXiv (Cornell University), Mar 18, 2002

We investigate the geometric, algebraic and homologic structures related with Poisson structure o... more We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncommutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated manifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Poisson algebra.

Research paper thumbnail of Intercultural competencies of students vs. their civic activities (Case of Georgia)

Cogent Education, 2021

Abstract The research highlighted the correlation of the intercultural competencies of middle sch... more Abstract The research highlighted the correlation of the intercultural competencies of middle school students in Georgia with their civic activities and levels of integration, as well as the factors contributing to the development of relevant competencies of students at the formal educational level. In particular, interactive learning contributes to the development of intercultural competencies in students and the integration of different cultural groups, their involvement in the decision-making process, in taking initiatives, and so on. This experience gained in the learning process helps the student to develop the skills needed for an active citizen. The analysis of the research results also revealed: (a) knowledge of different cultures by students and recognition of diversity are in a positive correlation with their willingness to engage, appreciate and take into account different opinions when making decisions; (b) the advantage of the experience gained through informal communication compared with the formal one over the development of students’ intercultural and civic competencies in a diverse school environment; (c) low benchmark of intercultural competence and civic activism in monocultural settings, especially in non-Georgian-speaking school students, which is caused by solitariness of minority communities, insufficient knowledge of state language, and less access to the media. The research confirmed that the intercultural and civic competencies of students are mostly influenced by the cultural characteristics of the living and school environments. The role of formal teaching is relatively minor in the process of successful civic integration of culturally diverse students.

Research paper thumbnail of The role of colaborative learning in the education for democratic citizenship (case of Georgia)

Research paper thumbnail of 1 Distributions on C ∞-Class Manifolds : General Overview

We investigate the geometric, algebraic and homologic structures related with Poisson structure o... more We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncom-mutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated man-ifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Pois-son algebra.

Research paper thumbnail of 3 Bott Connection and Generalized Functions on Poisson Manifold

We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the ... more We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the problem of construction of Hamiltonianinvariant generalized functions. For this we introduce the notion of generalized center of a Poisson algebra, which is the space of generalized Casimir functions. We study as the case when the set of test-objects for generalized functions is the space of compactly supported smooth functions, so the case when the test-objects are n-forms, where n is the dimension of the Poisson manifold. We describe the relations of this problem with the homological properties of the Poisson structure, with Bott connection for the corresponding symplectic foliation and the modular class.

Research paper thumbnail of Bott Connection and Generalized Functions on Poisson Manifold

We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the ... more We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the problem of construction of Hamiltonian-invariant generalized functions. For this we introduce the notion of generalized center of a Poisson algebra, which is the space of generalized Casimir functions. We study as the case when the set of test-objects for generalized functions is the space of compactly supported smooth functions, so the case when the test-objects are n-forms, where n is the dimension of the Poisson manifold. We describe the relations of this problem with the homological properties of the Poisson structure, with Bott connection for the corresponding symplectic foliation and the modular class.

Research paper thumbnail of Cyclic Evolution on Grassmann Manifold and Berry Phase

For a given k-dimensional subspace V_0 in a Hilbert space and a unitary transformation g_0:V_0 V_... more For a given k-dimensional subspace V_0 in a Hilbert space and a unitary transformation g_0:V_0 V_0, we find a path in the Grassmann manifold the monodromy of which coincides with g_0.

Research paper thumbnail of Holonomic Quantum Computing and Noncommutative Geometry

We investigate the notion of computability from the point of view of Quantum Computation. A quant... more We investigate the notion of computability from the point of view of Quantum Computation. A quantum algorithm for the holonomic Quantum Computation is considered from the point of view of connection in the differential fiber bundle over the parametrized space of controls and the graph (in the case of discrete algorithm). The noncommutative differnetial geometric approach to the quantum computation process is considered as a special case of the holonomic Quantum Computing, which allows to involve in the model of holonomic QC the classical computing (finite state machine) too.

Research paper thumbnail of Differential Complex of Poisson Manifold and Distributions

We study variuos homological structures associated with Poisson algebra, the canonical differenti... more We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure. 1 Lie superalgebra structure on the space of multiderivations of a commutative algebra. Poisson cohomologies Let A be a real or complex vector space. For each integer k let L k (A) be the space of multilinear antisymmetric maps from A k into A. Let L 0 (A) be A and L(A) be ⊕ ∞ k=0 Lk (A).

Research paper thumbnail of The Characteristic Class of a Lie Algebra Ideal, Contact Structures and the Poisson Algebra of Basic Functions

Some cohomology classes associated with an ideal in a Lie algebra, the Poisson structure on the a... more Some cohomology classes associated with an ideal in a Lie algebra, the Poisson structure on the algebra of basic functions for a contact structure, its Poisson cohomologies and geometric (pre)quantization

Research paper thumbnail of Geometric Control Methods for Quantum Computations

arXiv: Quantum Physics, 2004

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the ... more The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint.

Research paper thumbnail of 2 Submanifolds and Quotient Manifolds in Noncommutative Differential Calculus : Definitions and a Brief Review

We study noncommutative generalizations of such notions of the classical symplectic geometry as d... more We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf, for associative Poisson algebras. We give the full description of the family of Poisson structures on the endomorphism algebra of a vector bundle and study the above structures in the case of this algebra. Introduce the notion of generalized center of Poisson algebra as a subspace of the space of generalized functions (distributions) on a Poisson manifold and study its relation with the geometrical and homological properties of a singular Poisson structure.

Research paper thumbnail of The Canonical Differential Complex of Poisson Manifold and Distributions

We study variuos homological structures associated with Poisson algebra, the canonical differenti... more We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure.

Research paper thumbnail of Noncommutative Symplectic Geometry of the Endomorphism Algebra of a Vector Bundle

We study noncommutative generalizations of such notions of the classical symplectic geometry as d... more We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a vector bundle, and give the full description of the family of Poisson structures for this algebra.

Research paper thumbnail of Noncommutative geometry of Poisson structures

Journal of Mathematical Sciences, 2007

Research paper thumbnail of Geometric methods for construction of quantum gates

Journal of Mathematical Sciences, 2008

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the ... more The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the wellknown result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems.

Research paper thumbnail of Hamiltonian Systems on Complex Grassmann Manifold. Holonomy and Schrödinger Equation

Differential geometric structures such as the principal bundle for the canonical vector bundle on... more Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered

Research paper thumbnail of Intercultural competencies of students vs. their civic activities (Case of Georgia)

Cogent Education

The research highlighted the correlation of the intercultural competencies of middle school stude... more The research highlighted the correlation of the intercultural competencies of middle school students in Georgia with their civic activities and levels of integration, as well as the factors contrib...

Research paper thumbnail of Hamiltonian Systems on Complex Grassmann Manifold. Holonomy and Schrodinger Equation

Differential geometric structures such as the principal bundle for the canonical vector bundle on... more Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered and investigated for the integral curves of Hamiltonian dynamical systems.

Research paper thumbnail of Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

Journal of Mathematical Sciences, 2006

Differential geometric structures such as principal bundles for canonical vector bundles on compl... more Differential geometric structures such as principal bundles for canonical vector bundles on complex Grassmann manifolds, canonical connection forms on these bundles, canonical symplectic forms on complex Grassmann manifolds, and the corresponding dynamical systems are investigated. Grassmann manifolds are considered as orbits of the co-adjoint action and symplectic forms are described as the restrictions of the canonical Poisson structure to Lie coalgebras. Holonomies of connections on principal bundles over Grassmannians and their relation with Berry phases is considered and investigated for integral curves of Hamiltonian dynamical systems.

Research paper thumbnail of Noncommutative Geometry of Phase Space

arXiv (Cornell University), Mar 18, 2002

We investigate the geometric, algebraic and homologic structures related with Poisson structure o... more We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncommutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated manifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Poisson algebra.

Research paper thumbnail of Intercultural competencies of students vs. their civic activities (Case of Georgia)

Cogent Education, 2021

Abstract The research highlighted the correlation of the intercultural competencies of middle sch... more Abstract The research highlighted the correlation of the intercultural competencies of middle school students in Georgia with their civic activities and levels of integration, as well as the factors contributing to the development of relevant competencies of students at the formal educational level. In particular, interactive learning contributes to the development of intercultural competencies in students and the integration of different cultural groups, their involvement in the decision-making process, in taking initiatives, and so on. This experience gained in the learning process helps the student to develop the skills needed for an active citizen. The analysis of the research results also revealed: (a) knowledge of different cultures by students and recognition of diversity are in a positive correlation with their willingness to engage, appreciate and take into account different opinions when making decisions; (b) the advantage of the experience gained through informal communication compared with the formal one over the development of students’ intercultural and civic competencies in a diverse school environment; (c) low benchmark of intercultural competence and civic activism in monocultural settings, especially in non-Georgian-speaking school students, which is caused by solitariness of minority communities, insufficient knowledge of state language, and less access to the media. The research confirmed that the intercultural and civic competencies of students are mostly influenced by the cultural characteristics of the living and school environments. The role of formal teaching is relatively minor in the process of successful civic integration of culturally diverse students.

Research paper thumbnail of The role of colaborative learning in the education for democratic citizenship (case of Georgia)

Research paper thumbnail of 1 Distributions on C ∞-Class Manifolds : General Overview

We investigate the geometric, algebraic and homologic structures related with Poisson structure o... more We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncom-mutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated man-ifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Pois-son algebra.

Research paper thumbnail of 3 Bott Connection and Generalized Functions on Poisson Manifold

We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the ... more We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the problem of construction of Hamiltonianinvariant generalized functions. For this we introduce the notion of generalized center of a Poisson algebra, which is the space of generalized Casimir functions. We study as the case when the set of test-objects for generalized functions is the space of compactly supported smooth functions, so the case when the test-objects are n-forms, where n is the dimension of the Poisson manifold. We describe the relations of this problem with the homological properties of the Poisson structure, with Bott connection for the corresponding symplectic foliation and the modular class.

Research paper thumbnail of Bott Connection and Generalized Functions on Poisson Manifold

We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the ... more We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the problem of construction of Hamiltonian-invariant generalized functions. For this we introduce the notion of generalized center of a Poisson algebra, which is the space of generalized Casimir functions. We study as the case when the set of test-objects for generalized functions is the space of compactly supported smooth functions, so the case when the test-objects are n-forms, where n is the dimension of the Poisson manifold. We describe the relations of this problem with the homological properties of the Poisson structure, with Bott connection for the corresponding symplectic foliation and the modular class.

Research paper thumbnail of Cyclic Evolution on Grassmann Manifold and Berry Phase

For a given k-dimensional subspace V_0 in a Hilbert space and a unitary transformation g_0:V_0 V_... more For a given k-dimensional subspace V_0 in a Hilbert space and a unitary transformation g_0:V_0 V_0, we find a path in the Grassmann manifold the monodromy of which coincides with g_0.

Research paper thumbnail of Holonomic Quantum Computing and Noncommutative Geometry

We investigate the notion of computability from the point of view of Quantum Computation. A quant... more We investigate the notion of computability from the point of view of Quantum Computation. A quantum algorithm for the holonomic Quantum Computation is considered from the point of view of connection in the differential fiber bundle over the parametrized space of controls and the graph (in the case of discrete algorithm). The noncommutative differnetial geometric approach to the quantum computation process is considered as a special case of the holonomic Quantum Computing, which allows to involve in the model of holonomic QC the classical computing (finite state machine) too.

Research paper thumbnail of Differential Complex of Poisson Manifold and Distributions

We study variuos homological structures associated with Poisson algebra, the canonical differenti... more We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure. 1 Lie superalgebra structure on the space of multiderivations of a commutative algebra. Poisson cohomologies Let A be a real or complex vector space. For each integer k let L k (A) be the space of multilinear antisymmetric maps from A k into A. Let L 0 (A) be A and L(A) be ⊕ ∞ k=0 Lk (A).

Research paper thumbnail of The Characteristic Class of a Lie Algebra Ideal, Contact Structures and the Poisson Algebra of Basic Functions

Some cohomology classes associated with an ideal in a Lie algebra, the Poisson structure on the a... more Some cohomology classes associated with an ideal in a Lie algebra, the Poisson structure on the algebra of basic functions for a contact structure, its Poisson cohomologies and geometric (pre)quantization

Research paper thumbnail of Geometric Control Methods for Quantum Computations

arXiv: Quantum Physics, 2004

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the ... more The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint.

Research paper thumbnail of 2 Submanifolds and Quotient Manifolds in Noncommutative Differential Calculus : Definitions and a Brief Review

We study noncommutative generalizations of such notions of the classical symplectic geometry as d... more We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf, for associative Poisson algebras. We give the full description of the family of Poisson structures on the endomorphism algebra of a vector bundle and study the above structures in the case of this algebra. Introduce the notion of generalized center of Poisson algebra as a subspace of the space of generalized functions (distributions) on a Poisson manifold and study its relation with the geometrical and homological properties of a singular Poisson structure.

Research paper thumbnail of The Canonical Differential Complex of Poisson Manifold and Distributions

We study variuos homological structures associated with Poisson algebra, the canonical differenti... more We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure.

Research paper thumbnail of Noncommutative Symplectic Geometry of the Endomorphism Algebra of a Vector Bundle

We study noncommutative generalizations of such notions of the classical symplectic geometry as d... more We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a vector bundle, and give the full description of the family of Poisson structures for this algebra.

Research paper thumbnail of Noncommutative geometry of Poisson structures

Journal of Mathematical Sciences, 2007

Research paper thumbnail of Geometric methods for construction of quantum gates

Journal of Mathematical Sciences, 2008

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the ... more The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the wellknown result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems.

Research paper thumbnail of Hamiltonian Systems on Complex Grassmann Manifold. Holonomy and Schrödinger Equation

Differential geometric structures such as the principal bundle for the canonical vector bundle on... more Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered

Research paper thumbnail of Intercultural competencies of students vs. their civic activities (Case of Georgia)

Cogent Education

The research highlighted the correlation of the intercultural competencies of middle school stude... more The research highlighted the correlation of the intercultural competencies of middle school students in Georgia with their civic activities and levels of integration, as well as the factors contrib...

Research paper thumbnail of Hamiltonian Systems on Complex Grassmann Manifold. Holonomy and Schrodinger Equation

Differential geometric structures such as the principal bundle for the canonical vector bundle on... more Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered and investigated for the integral curves of Hamiltonian dynamical systems.

Research paper thumbnail of Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

Journal of Mathematical Sciences, 2006

Differential geometric structures such as principal bundles for canonical vector bundles on compl... more Differential geometric structures such as principal bundles for canonical vector bundles on complex Grassmann manifolds, canonical connection forms on these bundles, canonical symplectic forms on complex Grassmann manifolds, and the corresponding dynamical systems are investigated. Grassmann manifolds are considered as orbits of the co-adjoint action and symplectic forms are described as the restrictions of the canonical Poisson structure to Lie coalgebras. Holonomies of connections on principal bundles over Grassmannians and their relation with Berry phases is considered and investigated for integral curves of Hamiltonian dynamical systems.