Preface

Summary

The present book aims to provide an interdisciplinary perspective of the state of the art of the theory of ultrametric equations and its applications, starting from physical motivations and applications of the ultrametric geometry, and covering connections with probability, functional analysis, number theory, etc. in a novel form. In recent years the connections between non-Archimedean mathematics (mainly analysis) and mathematical physics have received a lot of attention, see e.g. [53]–[60], [63], [90], [90], [132]–[137], [164]–[166], [168], [190], [191], [220]–[228], [322]–[328], [336], [346]–[350], [366], [373], [413]–[411], [423]–[435] and the references therein. All these developments have been motivated by two physical ideas. The first is the conjecture (due to Igor Volovich) in particle physics that at Planck distances space-time has a non-Archimedean structure, see e.g. [438]–[435], [413], [412]. The second idea comes from statistical physics, more precisely, in connection with models describing relaxation in glasses, macromolecules, and proteins. It has been proposed that the non-exponential nature of those relaxations is a consequence of a hierarchical structure of the state space which can in turn be related to p-adic structures. Giorgio Parisi introduced the idea of hierarchy for spin glasses (disordered magnetics) in a more precise form in 1979, then the idea was extended to other physical problems and combinatorial optimization problems, see [336]. Then in the 1980s effects of slow non-exponential relaxation and aging were observed in deeply frozen proteins, implying the occurrence of a glass transition similar to that in spin glasses. Thus in the middle of the 1980s the idea of using ultrametric spaces to describe the states of complex biological systems, which naturally possess a hierarchical structure, emerged in the works of Frauenfelder, Parisi, Stain, and among others see e.g. [164]. In protein physics, it is regarded as one of the most profound ideas put forward to explain the nature of distinctive attributes of life.

For replica symmetry breaking in spin glasses the p-adic models were proposed independently by Avetisov et al. [53] and Parisi and Sourlas [373]. The idea of using p-adic diffusion equation to describe protein relaxation was proposed in [53].