## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Sergei Kozyrev

### $p$-Adic numbers and complex systems

Abstract

Hierarchical and ultrametric mathematical methods were applied for the investigation of complex systems. This talk gives a revew of several applications of ultrametric methods. We will discuss the following examples -- trees, buildings, wavelets, applications to spin glasses, proteins and genetic code.
In particular:
1. Wavelets can be used for hierarchical function representations.
2. Clustering is a method of classification with a help of trees. Multiclustering is a generalisation of clustering related to buildings.
3. Application to spin glasses: we will discuss $p$-adic parameters for the Parisi matrix.
4. Another application of hierarchical methods is to description of protein dynamics by $p$-adic diffusion. Hierarchy in protein structure and DNA packing can be also discussed.
5. Genetic code can be described by 2-adic plane -- 2-dimensional 2-adic structure.
Further applications include hierarchical models in machine learning, in particular, in deep learning.