Andrei Khrennikov
Professor of Applied Mathematics, International Center for Mathematical Modeling in Physics, Engineering, Economics and Cognitive Science, Linnaeus University, Vaxjo-Kalmar, Sweden.
Education:
- 1980 - M.Sc., Department of Mechanics and Mathematics, Moscow State University.
- 1983 - Ph.D. Thesis, Department of Mechanics and Mathematics, Moscow State University.
- 1989 - Dr. (Mathematical Physics), Steklov Mathematical Institute of Academy of Science.
Appointments:
- 1983-1990 - Assistant, Moscow University of Electronic Engineering.
- 1988-90 - Assistant Professor, Moscow University of Electronic Engineering.
- 1990-1997 - Professor, Moscow University of Electronic Engineering.
- 1994-1996 - Alexander von Humboldt Fellowship, Bochum University, Germany.
- Since 1997 - Professor of Applied Mathematics, Linnaeus University.
- Since 2000 - The Director of International Center for Mathematical Modeling
in Physics, Engineering, Economy and Cognitive Sciences, Linnaeus University.
Visiting professor fellowships: 1992,1993,1996: Genova University, Italy;1992: Peking and Wuhan Universities (China); 1996,1998, 2003: University of Clermont-Ferrand (France); 1996-97: Rikkyo University (Tokyo); 1998, 2008, 2009, 2010: Tokyo Science University; 1999: University of Rome, "La Sapienza"; 2000: Bremen University; 2000, 2002: University of Santiago; 2003: Universities of Denver and Boulder (Colorado); 2004: University of Urbana-Champain and Kavli Institute for Theoretical Physics (Santa Baraba); 2004, 2007, 2008: University of Rome 2; 2005, 2007: University of Bonn.
He is an organizer of a series of international conferences in foundations of probability and physics, quantum probability, quantum information theory and computing, p-adic mathematical physics.
He is a Member of Council of International Quantum Structures Associations.
Grants:
- 2002-2006 - EU grant: Network "Quantum probability with applications to physics, biology and economy".
- 1997-2007 - Grants of Swedish Royal Academy of Sciences for collaboration with states of former Soviet Union.
- 2002-2009 - Grants of Swedish Science Research Council for a series of conferences in foundations of probability, quantum physics, quantum information.
- 2009-2011 - Grant of Swedish Science Research Council (joint with South-African Research Council) "Non-Archimedean mathematics and Applications".
Editorial Boards of Professional Journals:
- Open Systems and Information Dynamics.
- P-adic Numbers, Ultrametric Analysis and Applications.
- Advance Studies in Contemporary Mathematics.
- Surveys in Mathematics and Applications.
- European Journal of Pure and Applied Mathematics.
- Information.
- Problems and Perspectives in Management.
Publications:
- Books: 16.
- Papers in refereed journals: 320.
- Books Editor: 13.
- Chapters in books and conference proceedings: 87.
Citations:
- More than 1500 citations according to ISI Web of Science.
- Mathematical works were cited 417 times by 102 authors according MathSciNet.
Organization of meetings, schools:
- Series of International Conferences "Quantum Theory: Reconsideration of Foundations", 2001, 2003, 2005, 2007, 2009 (Co-Organizer).
- 2001 - Workshop "Dynamical Days in Växjö" (Co-Organizer).
- 2002 - Workshop "Dynamical systems, number theory and probability" (joint with Lund University), Växjö (Co-Organizer).
- 2005 - "Harmonic, p-adic and wavelet analysis", Vietnam (Co-Organizer).
- 2006 - "Beyond Quantum", Lorentz Center, the Netherlands (Co-Organizer).
- Series of International Conferences "P-adic mathematical physics", 2003, Moscow, 2005, Belgrad, 2007, Moscow, 2009, Grodno (Belorus) (Co-Organizer).
- 2009 - "The Nature of Light: What are Photons?", SPIE Conference, San-Diego, USA (Co-Organizer).
- Series of International Conferences "Foundations of Probability and Physics", 2000, 2002, 2004, 2006, 2008, Växjö, Sweden (Co-Organizer).
- 2010 - International Conference "Advances in Quantum Theory", Växjö, Sweden (Co-Organizer).
Educational activities (lecturing):
- Since 1997 once a year - Measure Theory and Integration, Växjö and then Linnaeus University.
- Since 1997 once a year - Functional Analysis, Växjö and then Linnaeus University.
- Since 1997 each two years - Distribution Theory, Växjö and then Linnaeus University.
- Since 1997 each two years - Mathematical Foundations of QM, Växjö and then Linnaeus University.
- Since 1997 each two years - Topology, Växjö and then Linnaeus University.
- Quantum Information, computing and cryptography, Växjö and then Linnaeus, University, Moscow State University, Tokyo University of Science.
- P-adic dynamical systems with applications to cognitive science, Växjö and then Linnaeus University, University of Concepcion, Chili.
- Infinite Dimensional Analysis and Quantum Field Theory, University of Nagoya (Japan) and Wuhan (China).
- Probability Theory, Moscow University of Electronic Engineering.
- Statistics, Moscow University of Electronic Engineering.
- Fourier Analysis, Moscow University of Electronic Engineering.
Scientific interests:
- Mathematical modeling of complex systems in physics, biology, cognitive science, psychology, social sciences, economics and finances; multidisciplinary projects in collaboration with physicists, biologists, psychologists and at the same time intensive research in pure mathematics and quantum physics.
- Pure and Applied Mathematics: Ultrametric (in particular, p-adic) analysis, functional analysis, theory of partial differential and pseudo-differential equations, theory of distributions, asymptotic analysis for distributions, nonlinear differential and pseudo-differential equations, dynamical systems, wavelets on ultrametric spaces.
- Applications to Biology, Psychology and Cognitive and Social Science: Ultrametric models in cognitive science, psychology, genetics, quantum information modeling of processing of information by biological and social systems (quantum-like models).
- Mathematical Physics: Applications to quantum physics (Feynman integration and operator quantization, deformation quantization, spectral theory), quantum foundations, quantum information theory of spin glasses, superanalysis and quantization of superfields.
- Cryptography and image analysis: Applications of p-adic dynamical systems in cryptography and image analysis
Professor Khrennikov worked actively on applications of p-adic numbers in quantum physics - the activity started in 1987 at Steklov Mathematical Institute of RAN (by academician V.S. Vladimirov and Professor I. V. Volovich). He developed the corresponding formalism of the operator quantization based on the calculus of pseudo-differential operators, including a kind of correspondence principle; in particular, he and Professor S. Albeverio constructed the first p-adic representation of canonical commutation relations for quantization with p-adic valued wave functions. They performed detailed analysis of the spectra of position and momentum operators. Professor Khrennikov in collaboration with professor Albeverio and Cianci developed the Feynman path integral approach to quantization of non-Archimedean systems. Then they applied the non-Archimedean path integral to representation of the solutions of evolutionary equations (with respect to non-Archimedean space and time variables). They also developed essentials of non-Archimedean quantum field theory, including quantum fields with polynomial self-interaction.
Professor Khrennikov is the author of three monographs devoted to non-Archimedean mathematical physics (two in English, Kluwer, and one in Russian, Fizmatlit).
He was one of pioneers in applications of ultrametric methods in cognitive science, psychology, sociology, genetics. In particular, he proposed a model of brain's functioning based on the ultrametric encoding of hierarchic mental information; information is processed by ultrametric (in the simplest case p-adic) dynamical systems. The tree-like structure of neuronal trees couples the ultrametric brain's model with the basic paradigm of modern cognitive science, the neuronal paradigm. This approach was also applied (in collaboration with professors Albeverio, Kloeden, Gundlach and graduate students Lindahl and Nyqvist) to model human's psychology, especially dynamics of depression. Recently 2-adic analysis was applied to clarification of the structure of the genetic code (Professors Dragovich, Khrennikov, Kozyrev). He is the author of four monographs devoted to applications of ultrametric methods (three in English, two by Kluwer, and one in Russian, Fizmatlit).
The intensive work on applications was successfully combined with research in non-Archimedean functional analysis, differential and pseudo-differential equations (linear and nonlinear), wavelets (in collaboration with Professors Albeverio, Kozyrev, Shelkovich and Skopina), and measure theory including p-adic generalization of probability theory (in collaboration with Professors Yamada, Van Rooj, and Huang). The latter was created to serve p-adic quantum physics and modeling of hierarchic biological systems. Recently Albeverio, Khrennikov and Shelkovich published a monograph on p-adic distributions, wavelets and pseudo-differential equations, published by Oxford University Press.
Professor Khrennikov was one of pioneers in theory of p-adic and more general non-Archimedean dynamical systems. He created a scientific school in this domain formed of his former Ph.D. students (employed at Linnaeus University and other universities in Sweden). He and his graduate student M. Nilsson published a monograph of deterministic and random p-adic dynamical systems, by Springer.
He contributed to development of quantum foundations (analysis of the probabilistic structures the EPR and GHZ experiments, detailed analysis of mathematical assumptions for derivation of Bell's theorem), quantum probability (coupling of the formula of total probability and interference), development of generalized quantum-like probabilistic models, especially the model based on the hyperbolic generalization of Hilbert space), classical wave models reproducing quantum averages and correlations (classical statistical mechanics with infinite-dimensional phase space of the L_2-type), accounting of the contribution of the background field, vacuum fluctuations, to quantum correlations, analysis of the von Neumann projection postulate and its role in quantum information theory and especially quantum teleportation. He (with graduate student P. Nyman) constructed an algorithm representing probabilistic data of any origin in the quantum-like way, by complex or more general quantum-like amplitudes (quantum-like representation algorithm). Professor Khrennikov developed a general contextual approach to quantum formalism which was presented in a monograph, Springer (2009).
This contextual approach provides a possibility to apply the mathematical methods of quantum mechanics outside of physics. Professor Khrennikov made a really important contribution to this domain - quantum-like models in biology, psychology, economics and finances (in collaboration with Professor of Psychology E, Conte, Professor in management and bank accounting E. Haven and with graduate student O. Choustova); in particular, Mathematical works of Khrennikov on coupling of interference with the formula of total probability were used as the theoretical basis for experiments (performed at University of Bari, Italy) to find evidences of quantum-like functioning of the brain. Later similar evidences were found in well known statistical data from cognitive economics. These data collected by Shafir and Tversky demonstrated the irrational behavior of traders at the market. It was found that these data violates the classical law of total probability and it can be described by the quantum-like contextual formalism. This study has a close coupling with the game theory, of the Prisoner's Dilemma type. Professor Khrennikov proposed (in collaboration with Prof. Ohya and Dr. Asano) a quantum-like model of decision making reproducing data from Shafir-Tversky experiments and describing "irrational decision making" in games of the Prisoner's Dilemma type. In collaboration with Professor A. Grib, Khrennikov developed general theory of quantum-like games, macroscopic games exhibiting quantum probabilistic behavior. Recently all these studies on quantum-like model were published in a monograph by Springer.
Professor Khrennikov contributed to the development of the mathematical foundations of supersymmetric models by formulating these models in the framework of so called functional superanalysis. In this framework he developed theory of distributions including Fourier and Laplace transform, theory of partial-differential and pseudo-differential equation (linear and nonlinear), mathematical theory of Feynman integral for fermionic and supersymmetric fields, operator quantization based on the calculus of pseudo-differential operators, a supersymmetric version of the correspondence principle.
Professor Khrennikov is one world's leading experts in foundations of probability theory.
He developed p-adic generalization of probability theory, including analogs of Martin-Löf theory of statistical tests (in collaboration with Yamada and Van Rooj) and Kolmogorov's algorithmic complexity; he contributed with novel ideas to the probabilistic foundations of quantum mechanics; in particular, he created a new quantum model based on the hyperbolic Hilbert space. The latter has already found its applications in psychology.
