PUBLICATION LIST (Anar Dosi)

Transcription

1 PUBLICATION LIST (Anar Dosi) Research areas: Noncommutative analysis, Banach and topological algebras, multivariable operator theory, topological homology, noncommutative holomorphic functional calculus, quantum functional analysis, noncommutative algebraic geometry 41. Dosi A. A., Noncommutative localizatıons of Lie-complete rings, Communications in Algebra 44 (2016) Dosi A. A., Noncommutative affi ne spaces and Lie-complete rings, C. R. Acad. Sci. Paris, Ser I, 353 (2015) Dosi A.A. Functional calculus on noetherian schemes, C. R. Acad. Sci. Paris, Ser I, 353 (2015) Dosi A. A., Injectivity in the quantum space framework, Operators and matrices, 8 (4) (2014) Dosi A. A., A representation theorem for quantum systems, Funct. Anal. and its Appl., 47 (3) (2013) Dosi A. A., Quantum cones and their duality, Houston J. Math. 39 (3) (2013) ( ). 35. Dosi A. A., Quantum systems and representation theorem, Positivity. 17 (3) (2013) Dosi A. A., Multinormed W -algebras and unbounded operators, Proc. Amer. Math. Soc. 140 (2012) Dosi A. A., Bipolar theorem for quantum cones, Funct. Anal. and its Appl., 46 (3) (2012) Dosi A. A., Noncommutative Mackey theorem, Intern. J. Math. 22 (4) (2011) Dosi A. A., Local operator algebras, fractional positivity and quantum moment problem, Trans. Amer. Math. Soc. 363 (2011), Dosi A.A., Quotients of quantum bornological space, Taiwanese J. Math. 15 (3) (2011) Dosi A. A., Formally radical functions in elements of a nilpotent Lie algebra and noncommutative localizations, Algebra Colloquium 17 (Spec 1) (2010) Dosi A. A., Quantum duality, unbounded operators and inductive limits, J. Mathematical Physics 51 (6) (2010)

2 27. Dosi A. A., Taylor functional calculus for supernilpotent Lie algebra of operators, J. Operator Theory 63 (1) (2010) Dosi A.A., Taylor spectrum and transversality for a Heisenberg algebra of operators, Matem. Sbornik, 201 (3) (2010) Dosi A. A., Noncommutative holomorphic functions in elements of a Lie algebra and noncommutative localizations, Izvestiya Math. RAN, 73 (06) (2009) Dosiev A. A., Local left invertibility for operator tuples and noncommutative localizations, J. K-theory, 4 (01) (2009) Dosiev A.A., Perturbations of nonassociative Banach algebras, Rocky Mountain J. Math., 39 (2) (2009) Dosiev A.A., Frechet sheaves and Taylor spectrum for supernilpotent Lie algebra of operators, Mediterr. J. Math. 6 (2009) Dosiev A. A., Regularities in noncommutative Banach algebras, Integ. Eq. Oper. Th., 61 (3) (2008) Dosiev A. A., Local operator spaces, unbounded operators and multinormed C*- algebras, J. Funct. Anal., 255 (7) (2008) Dosiev A.A., Quantized moment problem, C. R. Acad. Sci. Paris, Ser I, 344 (10) (2007) Dosiev A.A., The representation theorem for local operator spaces, Funct. Anal. and its Appl., 41 (4) (2007) Dosiev A.A., Quasispectra of solvable Lie algebra homomorphisms into Banach algebras, Studia. Math. 174 (1) (2006) Dosiev A.A., Cartan-Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems, J. Funct. Anal., 230 (2) (2006) Dosiev A.A., Spectral mapping framework, Proceedings of the Conference on Topological Algebras, Banach Center Publications, 67 (2005) Dosiev A.A., Cohomology of sheaves of Frechet algebras and spectral theory, Funct. Anal. and its Appl., 39 (3) (2005) Dosiev A.A., Algebras of power series in elements of a Lie algebra and Slodkowski spectra, J. Math. Sciences (translated from: Zapiski POMI, St-Petersburg), 124 (2) (2004) Dosiev A.A., Homological dimensions of the algebra formed by entire functions of elements of a nilpotent Lie algebra, Funct. Anal. and its Appl., 37 (1) (2003)

3 11. Dosiev A.A., Spectra of infinite parametrized Banach complexes, J. Operator Theory, 48 (3) (2002) Dosiev A.A., Ultraspectra of a representation of a Banach Lie algebra, Funct. Anal. and its Appl., 35 (3) (2001) Dosiev A.A., Holomorphic functions of a basis of a nilpotent Lie algebra, Funct. Anal., and its Appl. 34 (4) (2000) National type papers: 8. Dosi A.A. A survey of spectra of parametrized Banach space complexes, Azerb. J. Math., 1 (1) (2011) Dosiev A.A., Note on algebras of power series in elements of a Lie algebra, Trans. NAS Azerb., 12 (7) (2004) Dosiev A.A., Projection property of spectra for solvable Lie algebra representations, Trans. NAS Azerb., 23 (1) (2003) Dosiev A.A., On fredholm pairs of operators, Proc. IMM NAS Azerb., 13 (2000) Dosiev A.A., Cartan type criterion for solvability of subalgebras of an associative Banach algebra, Proc. IMM NAS Azerb., 10 (1999) Dosiev A.A., Holomorphic functions on polydisk of Heisenberg algebra generators, Trans. NAS Azerb., 19 (1999) Dosiev A.A., On Arens-Michael hull of the universal enveloping algebra of a nilpotent Lie algebra, Trans. NAS Azerb., 18 (1997) (in russian). 1. Dosiev A.A., The Taylor joint spectrum of noncommuting variables, Proc. IMM NAS Azerb., 5 (1996) (in russian). International conference presentations: 1. Dosi A.A., Lie-operator rings and noncommutative affi ne schemes, International Conference dedicated to the 65th birthday of Askold Khovanskii 2012, Moscow, Russia ( 2. Dosi A.A., Quantum duality and noncommutative Arens-Mackey theorem, Banach algebras 2009, Bedlewo, Poland ( 3. Dosiev A.A. Local operator spaces and noncommutative rational functions, Operator algebras and topology, Moscow (2007), Russia ( 3

4 4. Dosiev A.A., Local operator systems and multinormed C*-algebras, 21-th International Conference on Operator Theory, Timisoara (2006) Romania ( 5. Dosiev A.A., Frechet algebra sheaf cohomology and spectral theory (talk I), Perturbations of nonassociative Banach algebras (talk II), 20-th International Conference on Operator Theory, Timisoara (2004) Romania, Dosiev A.A., Multi-operator functional calculus for supernilpotent operator Lie subalgebras, Conference on Topological Algebras, their Applications, and Related Topics, Bedlewo 2003, Poland ( 7. Dosiev A.A., Taylor spectrum and holomorphic functions of noncommuting variables, 3-th Linear algebra workshop, Bled 2002, Slovenia, 8. Dosiev A.A., Holomorphic functions of noncommuting variables, 15-th International Conference on Banach algebras, Balticon 2001, Odense, Denmark ( 9. Dosiev A.A., Noncommutative spectral mapping theorem, 18-th International Conference on Operator Theory, Timisoara (2000) Romania ( Education: Institute of Mathematics and Mechanics NAS Azerbaijan, Habilitation in Mathematics (Leading organization: Moscow State University, Russia) Institute of Mathematics and Mechanics NAS Azerbaijan, Ph.D. in Mathematics ( ) (Leading organization: Moscow State University, Russia) Baku State University M.S. Mathematics, ( ). Novosibirsk State University, Russia B.S. Mathematics ( ) Title and year of Habilitation D.: Noncommutative holomorphic functional calculus for representations of a Lie algebra (Baku 2012). Title and year of Ph.D.: Taylor spectrum of Banach-Lie algebra representations (Baku 1998) Advisor: Yu. V. Turovskii (Baku), coadvisor A. Ya. Helemskii (Moscow) Languages spoken: English, Russian, Turkish. Editorship: Turkish Journal of Mathematics. 4

5 References: (Avaiable upon request) D. Blecher, Department of Mathematics, University of Houston, Houston, TX ( ) A. Ya. Helemskii, Moscow State University, Russia F.-H. Vasilescu, Universite de Lille 1, France ((33)(0) ); V. I. Lomonosov, Kent State University, USA M. Putinar, University of California, Santa Barbara, USA ((805) ); Research Proposal: Quantum Cones and their Applications to Quantum Information Theory, Noncommutative geometry background of Lie-nilpotent rings. My research interests are mainly in noncommutative analysis. This is a modern branch of mathematical, more precisely functional analysis, whose classical foundation goes back to the known calculus that we have been teaching at the universities. The central idea of noncommutative analysis is to develop a calculus over elements of a certain noncommutative base algebra. Classical multivariable calculus can be quantized on by replacing commuting variables x 1,..., x n by noncommuting (for instance 2 2) matrices A 1,..., A n. These are so called "quantum" or "noncommutative variables". This type of model certainly involves lots of abstract algebra, topology, geometry and functional analysis techniques. In order to have a strong and unique mathematical language of quantization there was recently ( ) created so called quantum functional analysis. The latter in turn can be treated as the basic language of quantum physics. To have a comprehensive mathematical model of quantum physics, it is also necessary to consider the linear spaces of unbounded Hilbert space operators or "noncommutative variable spaces". It is well known that if we interpret the time dependent variables as infinite matrices rather than functions, then quantum phenomena can be deduced from the equations of Newtonian physics. This is the basic idea of Heisenberg s "matrix mechanics". These "quantum variables" have provided functional analysis with new constructions, methods and problems. Being a new and modern branch of functional analysis, quantum functional analysis is covered by normed and polynormed topics, as in the classical theory. Within the framework of this theory one can develop the noncommutative holomorphic functional calculus for linear operators. Taylor s program on multi-operator holomorphic functional calculus for noncommutative operators has been implemented in my habilitation dissertation (2012). In the past three years I got various results on quantum polynormed spaces, namely, representation theorems, duality theory, quantum systems, local operator algebras, quantum bornology. The results have been successfully applied to quantum moment problems. My future plan is to investigate quantum cones in their general setting and provide applications to quantum information theory. Quantum cones probably will be the main fundamental concept of the whole quantum functional analysis and they will play the crucial role in the applications. 5

6 The last direction of interest deals with a new approach to noncommutative algebraic geometry. One of the principal foundations of noncommutative algebraic geometry is to extend the concept of affi ne schemes to noncommutative rings. As in the commutative case the main motivation for this is to represent a non- commutative ring as the ring of "functions" over its spectrum. The classical result, which is due to I. M. Gelfand, asserts that a commutative Banach algebra can be realized as the algebra of continuous functions over the space of all its maximal ideals modulo its Jacobson radical. For the commutative rings this result has been generalized in the following way, which is due to A. Grothendieck. A commutative ring A is the ring of all global sections Γ (Spec (A), O) of the structure sheaf O of the ring A over its scheme Spec (A) (the space of all prime ideals of A) up to an isomorphism. The construction of the relevant schemes, and "functions" over them, for the noncommutative ring requires extraordinary efforts. The problem can not be solved in a unique framework based on a certain special category of objects and morphisms. All this diversity reflects in various methods and constructions picked up in noncommutative geometry. Lie algebra methods allow us to handle the problem. As shown in many investigations the most reliable case for the relevant scheme theory to be built up is the class of Lie-nilpotent rings. This proposal has been partially supported in Kapranov s theory (1998) of NC-schemes. Another motivation for this direction is to use an operator approach to noncommutative schemes, without any sheaf construction as classically done. The operator realization of many noncommutative algebras is the well known fact. In this direction I intend to develop a noncommutative regularity in the general purely algebraic case, and propose a new approach to noncommutative spectra which is based on concrete operator realization of an abstract regularity in a Lie-complete ring. The approach proposed presents an interest in the commutative case as well. Teaching Experience: I have been teaching many different courses mainly oriented to engineers in the last 10 years. In the last Spring semester. I had some experience teaching undergraduate courses: Abstract algebra, Real Analysis, Functional analysis, Homology of Banach algebras. Full teaching responsibilities of 2 courses per semester. Service courses taught: Calculus (Math 119, 120), Differential equations (Math 219), Complex analysis. Upper level undergraduate courses taught: Abstract algebra, Functional analysis, Homology of Banach algebras. The average of student evaluation points (accepted in METU) for service courses is about 4 within 5 points (available upon request). Almost all universities in Turkey have Calculus based mathematical programs: Calculus I, Calculus II, Linear algebra (in a very non-serious level) and finally hard Differential Equations course. But times are changing and we come up with new demands. For example how to teach our Calculus based students quantum mechanics or quantum information theory, which all are based on high level functional analysis, or at least on a serious level of Linear algebra courses. The Calculus based mathematics is a stumbling block in our future evolutionary progress. I suggested a new course: Quantum Calculus (devised by me). Ta) Quantum Calculus for Engineers, MAT 3XX (a course for EEE and CNG students of METU NCC) Frequency: Fall/Spring Terms; Prerequisite: MAT 260 (corequisite MAT 219); Credit: 3 Catalog description: Vector spaces and linear constructions, duality 6

7 of vector spaces, Jordan normal form of a linear operator, normed spaces and functional calculus for a linear operator, linear systems of differential equations, finite dimensional Hilbert spaces, normal operators and the spectral theorem, quadratic forms and eigenvalues of self-adjoint operators, quantum systems, quantum probability, the average values and dispersions, Heisenberg uncertainty principle, tensor calculus, tensor algebra, exterior forms, Pauli principle, quantum channels, quantum information theory, quantum positivity. Tb) Justification for the Course Proposal: The theory of quantum information aims to study the general regularity of transmission, storage and transformation of information in the systems which obey the laws of quantum mechanics. For investigation of potential capability of such systems and engineering of their rational synthesis, the quantum information theory actively absorbs analytic methods of matrices and linear operators on a Hilbert space, operator algebras and systems, and the theory of quantum spaces. The quantum information theory has been included into the list of main modern themes. The main ideology of this attention is the possibility of quantum computers to be pushed forward, whose mathematics would stand head and shoulders above the classical calculus that we are teaching now. The proposed course is designed to fill this gap by directing our engineering students towards the so called quantum calculus. That would prepare them for possible future global changes in science and technology. It is a fundamental course designed for all science and engineering students who deal with information theory. Tc) Course Objectives: The aim of this course is to prepare students to have necessary skills and abilities to handle mathematical methods of quantum information theory and quantum mechanics. All details of the theory are explained over finite dimensional Hilbert spaces which requires just possession of linear algebra skills. It is desirable to have some elementary knowledge of linear systems of differential equations as well. Td) Textbook: L. Smith, Linear algebra (1998); A. I. Kostrikin, Yu. I. Manin, Linear algebra and geometry (1986); V. Pauslen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003, E. Effros, Z.-J., Ruan, Operator spaces, Oxford (2000). A.C. Holevo, Quantum systems, channels, information (2010); A. Ya. Helemskii, Quantum functional analysis, Moscow (2009). Te) Lectures: Week 1: Vector spaces, subspaces, quotient spaces, and duality. Week 2: Jordan normal form, functional calculus, normed spaces, applications. Week 3: Linear system of differential equations in C n, fundamental matrices. Week 4: Finite dimensional Hilbert spaces, adjoint of an operator. Week 5: Self-adjoint and normal operators. Week 6: Spectral theorem, the Principal Axis Theorem for Quadratic Forms. Week 7: The average values, states, observables, quantum probability and, the Heisenberg uncertainty principle. Week 8: Tensor product of vector spaces, Grassmanians, tensor products of Hilbert spaces. Week 9: Tensor algebra and exterior algebra of a vector space. Composed states. Week 10: Pauli principle, Naimark extension theorem, quantum system as an information carrier. Week 11: Matrix norms, Schatten matrix norms. Week 12: Matrices of operators, matrix positivity Week 13: Quantum channels and matrix positivity. 7

8 This is a completely new course (I believe it has no analog in the world) which aims to bring forward students to the frontline of quantum analysis. This is the way that I can contribute to teaching programs at the universities. Finally, I can teach the following special courses with great pleasure: Functional analysis on subdifferentials; Uniform spaces and filtered rings; Topological homology; C*-algebras and K-theory; Radon integrals and vector lattices; Operator spaces. Work address: Middle East Technical University Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey ( dosiev@metu.edu.tr, dosiev@yahoo.com) work phone: ( ). 8