Extremal forms and rigidity in arithmetic geometry and in dynamics

N. M. Glazunov

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 124-146.

Résumé

Ryshkov S. S. in his papers has investigated extremal forms and extremal lattices. Extremal forms and lattices are connected with hard or rigid (by M. Gromov and other) objects in mathematics. In their work with colleagues S. S. Ryshkov came also to the other hard (or rigid) objects, for instance, to rigidly connected chain. Rigid and soft methods and results already evident in the study of the classical problems in number theory. Let us dwell briefly on the interpretation in terms of hard and soft methods of binary and ternary Goldbach problems. Since the binary (respectively ternary) Goldbach problems in their present formulation there are about equalities of the type $2n = p_1 + p_2 $ (respectively $2n+1 = p_1 + p_2 + p_3 $), where $n$ is a natural number greater than $1$ (respectively $n$ is a natural number greater than $2$), $p_1, p_2, p_3 $ prime numbers, then these are hard (rigid) problems; the results of their studies are also hard. However, the methods of their study include both rigid methods — the exact formula of the method of Hardy–Littlewood–Ramanujan and a combination of hard and soft methods under the investigation by the Vinogradov`s method of trigonometric sums. A number of problems of analytic number theory allow dynamic interpretation. We note in this regard that on connection of methods of analytic number theory and the theory of dynamical systems paid attention and has developed such analogies A. G. Postnikov. The purpose of the paper is not to provide any sort of comprehensive introduction to rigidity in arithmetic and dynamics. Rather, we attempt to convey elementary methods, results and some main ideas of the theory, with a survey of some new results. We do not explore an exhaustive list of possible topics, nor do we go into details in proofs. After giving an elementary number theoretic, algebraic and algebraic geometry introduction to rigid non-Archimedean spaces in the framework of local one dimensional complete regular rings, modules over rings, trees and formal schemes follow to I. R. Shafarevich, J.-P. Serre, J. Tate, D. Mumford, we review some novel results and methods on rigidity. These include (but not exhaust) methods and results by H. Furstenberg, G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, and other. Bibliography: 52 titles.

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