The workshop on 'p-Adic Methods for Modeling of Complex Systems' discussed applications of p-adic analysis to modeling of various phenomena and comparison of these methods with established methods of analysis over real numbers.

In science, after Newton and Leibniz, it is used mathematical analysis based on real numbers. However, irrational real numbers are unobservable. In experiments one can observe only rational numbers. To do mathematical analysis one needs a completion of rationals and due to the Ostrovsky's theorem only real or p-adic number fields can be obtained.

There are applications of ultrametric methods to Planck scale physics, theory of spin glasses, biology (genetics, protein dynamics), cosmology.

These applications can be considered as an example of duality between simple and complex behavior. Simple phenomena can be modeled with the help of real models, and for modeling of complex phenomena one needs p-adic methods. p-adic methods are well suited to description of hierarchy in complex systems.

The goal of the activity presented at the Workshop is to establish p-adic and ultrametric analysis as an alternative to analysis over real numbers. Mathematical methods of ultrametric analysis are already well developed and are able to describe a wide variety of phenomena. There are well known mathematical aspects of p-adic analysis: applications to number theory, algebraic geometry, represention theory, ultrametric analysis, dynamical systems, partial differential equations, wavelets, etc. The main challenge of p-adic modeling is related to effective applications of p-adic analysis.

Already developed applications of p-adic analysis include quantum theory, strings, spin glasses, cosmology, turbulence, renormalization theory, dynamics of proteins, genetics, data mining, economics, psychology. We expect new applications of p-adic methods.

Talks given at the Workshop belong to the following directions:

*Dynamical systems*: B.Diarra, S.Jeong, L.B.Tyapaev, E.Yurova, E.I.Zelenov, V.S.Anashin, F.Vivaldi, L.Liao.

p-Adic dynamical systems with applications in number theory and cryptography is an important field of p-adic analysis.*Pseudodifferential equations*: A.N.Kochubei, W.A.Zuniga-Galindo, A.Bendikov, S.Torba.

p-Adic pseudodifferential operators and equations were considered Vladimirov, Volovich, Zelenov, Kochubei, Khrennikov, Kozyrev, Shelkovich, Zuniga-Galindo, Bendikov, Torba and others. Results on spectral properties of pseudodifferential operators and on existence and uniqueness of solutions of pseudodifferential equations were obtained.*Clustering of data*: F.Murtagh, S.V.Kozyrev, P.E.Bradley.

Clustering is an important method in data analysis with applications in bioinformatics and other applied areas. Clustering procedure allows to construct a hierarchy of cluster sets which gives a brief hierarchical description of some properties of the system. In the conference applications of clustering and classification to analysis of data were discussed. A multidimensional generalization of clustering which is related to Bruhat-Tits affine buildings was obtained.*Wavelets*: M.Skopina, E.King, S.Evdokimov.

Theory of p-adic wavelets in relation to multiresolution analysis, representation theory and spectral analysis of pseudodifferential operators was developed by Kozyrev, Benedetto, Khrennikov, Skopina, Shelkovich, King, Evdokimov and others.*Quantum mechanics*: T.Digernes, A.Vourdas.

p-Adic models of quantum mechanics were considered by Vladimirov, Volovich, Zelenov, Varadarajan, Digernes, Voudras and other authors. At the Workshop new results on spectral theory were presented.*Number theory*: Yu.I.Manin, Pei-Chu Hu.

Number theory is the field where p-adic numbers were initially applied. Ideas of number theory were applied in formulation of p-adic string theory (Volovich, Vladimirov, Freund, Witten) and adelic approach (Manin). At the Workhop numbers as functions (Yu.I.Manin) and discriminats of number fields (Pei-Chu Hu) were considered.*Cosmology*: I.Ya.Aref`eva, A.Koshelev.

Cosmology models related to p-adic string theory were discussed. In these models solutions of special real nonlinear pseudodifferential equations which arise in the low energy limit of p-adic string theory are investigated.*Cognitive sciences*: A.Yu.Khrennikov, B.Tirozzi.

Cognitive models based on p-adic dynamical systems were presented. Hierarchical structures in neural networks were considered.*Theory of functions*: A.Escassut, E.Nagel.

p-adic meromorphic functions were considered. Differentiability of p-adic valued functions of p-adic arguments was discussed.*Complex systems and random walks*: S.Albeverio.

Theory of random walks and spectral theory is considered. Applications to complex systems were discussed.*Genetics*: B.Dragovich.

p-adic parametrization of the genetic code was proposed by Dragovich and Dragovich. An alternative 2-dimensional parametrization of the code was found by Kozyrev and Khrennikov.*Renormalization theory*: M.D.Missarov.

Renormalization theory of hierarchical fermion model was discussed.*Representation theory*: A.Kosyak.

The orbit method for infinite dimensional groups was considered.*Non-Newtonian mechanics*: I.V.Volovich.

Non-Newtonian functional mechanics is a probabilistic approach to mechanics in which instead of trajectories of single particles the ensembles of trajectories is investigated. Corrections to the Newton equation for the mean values of observables are calculated.

To conclude, the workshop was very successful, many new important and exciting results were reported. The workshop will stimulate new progress in the field of applications of p-adic methods to modeling of complex systems.

Vladimir Anashin (Moskau, RUS), Irina Ya. Aref'eva (Moskau, RUS), Ekatarina Yurova Axelsson (Växjö, SWE), Erik Makino Bakken (Trondheim, NOR), Alexander Bendikov (Wroclaw, POL), Patrick E. Bradley (Karlsruhe, GER), Bertin Diarra (Aubière, FRA), Trond Digernes (Trondheim, NOR), Alain Escassut (Aubière, FRA), Sergei Evdokimov (St. Petersburg, RUS), Alexander Grigor'yan (Bielefeld, GER), Pei-Chu Hu (Jinan, TPE), Emily King (Berlin, GER), Anatoly N. Kochubei (Kiew, UKR), Alexei Koshelev (Brüssel, BEL), Alexander Kosyak (Bonn, GER), Ligmin Liao (Créteil, FRA), Yuri Manin (Bonn, GER), Mukadas D. Missarov (Kazan, RUS), Fionn Murtagh (Egham, GBR), Enno Nagel (Maceió, BRA), Zoran Rakic (Belgrad, SRB), Oded Shor (Rehovot, ISR), Maria Skopina (St. Petersburg, RUS), Brunello Tirozzi (Rom, ITA), Sergii Torba (Mexico City, MEX), Livat Tyapaev (Saratov, RUS), Franco Vivaldi (London, GBR), Apostolos Vourdas (Bradford, GBR), Sang Tae Jeong (Incheon, KOR), Evgeny Zelenov (Moskau, RUS), Wilson A. Zúñiga-Galindo (Mexico City, MEX)

- Conference program (PDF)