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p-adic numbers: an elementary introduction and overview of modern applicationsучебный курс
- Авторы: Архипова Людмила Геннадьевна, Строева Е.Н., Снурницын П.В.
- Год создания: 2024
- Организация: МГУ имени М.В. Ломоносова
- Курс на иностранном языке
- Описание: In number theory p-adic numbers offer a powerful toolkit for working with equations and congruences in the ring of integers, employing techniques from calculus and analysis. While traditionally viewed as a subset of modern number theory, p-adics have increasingly surfaced in different areas of pure and applied mathematics, and even in physics. Acquaintance with p-adic numbers proves beneficial even for those not pursuing future studies in number theory and its applications, as this field provides rich sources of non-standard examples of topological and algebraic structures. In our mini-course, we will introduce the concept of p-adic numbers, prove first non trivial results including Ostrowski's theorem and Hensel’s lemma, explore examples using the SageMath computer algebra system and briefly cover modern applications of p-adics both in number theory, coding theory and neural networks. The course is mostly self-contained, assuming some basic knowledge of concepts in number theory (divisibility, congruences and ring of integers modulo N), analysis (convergence of sequence and series) and topology (open and closed sets).
- Добавил в систему: Архипова Людмила Геннадьевна