Vol 10, No 4 (2018)

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and second-quantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion.We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where p-adic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author’s talk at the 6th International Conference on p-adicMathematical Physics and its Applications,Mexico 2017.

In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p.

For every finite ultrametric space X we can put in correspondence its representing tree T_X. We found conditions under which the isomorphism of representing trees T_X and T_Y implies the isometricity of ultrametric spaces X and Y having the same range of distances.

Yuri Manin’s approach to Zipf’s law (Kolmogorov complexity as energy) is applied to investigation of biological evolution. Model of constructive statistical mechanics where complexity is a contribution to energy is proposed to model genomics. Scaling laws in genomics are discussed in relation to Zipf’s law. This gives a model of Eugene Koonin’s Third Evolutionary Synthesis – physical model which should describe scaling in genomics.

Throughout this paper, using the p-adic wavelet basis together with the help of separation of variables and the Adomian decomposition method (as a scheme in numerical analysis) we initially investigate the solution of Cauchy problem for two classes of the first and second order of pseudo-differential equations involving the pseudo-differential operators such as Taibleson fractional operator in the setting of p-adic field.

This paper contains a brief review of a very diverse and vast scientific work of Andrei Yurievich Khrennikov on the occasion of his 60th birthday.