University of Durham --- Department of Mathematical Sciences

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Project III topics 2023-24

Area: Algebra and Number Theory

Topic: Non-archimedean analysis.

Let p be a prime number. The p-adic numbers appear in a very similar way to real numbers as the completion of the field of rational numbers. In the case of real numbers we use the usual metric given by the modulus of distance between two given rational numbers. In the case of p-adic numbers we must use another, so-called non-archimedean, metric given by the p-adic distance: two integral rational numbers are p-adically close if their difference is divisible by a large power of prime number p. The theory of p-adic numbers is based on elementary number theory and at the same time contains many results which are similar to the corresponding properties of usual real numbers.

In particular, the p-adic numbers can be used for developing all basic concepts of the traditional calculus: functions, limits, derivatives, integrals, infinite series and so on. For example, in the field of 3-adic numbers the series 1+3+9+27+81+... converges and its sum equals(!) minus 1/2. One can introduce an analogue of the usual exponential function exp(x), which satisfies many usual properties, but the corresponding infinite Taylor series does not converge for all x. There is a very interesting theory of p-adic integration. The project will be concerned with the study of basic properties of p-adic numbers and functions, p-adic differential equations, the construction of p-adic integration and so on. Apart from traditional applications to Number Theory, there appeared recently unexpected and interesting applications of p-adic numbers to Mathematical Physics and Probability.

Prerequisites:

Algebra and Number Theory II

Basic resources:

p-adic analysis compared with real, by S. Katok

p-adic numbers, p-adic analysis, and zeta-functions, by N. Koblitz

p-adic analysis: a short course on recent work, by N. Koblitz

p-adic analysis and mathematical physics, by V. S. Vladimirov, I. V.Volovich, and E. I.Zelenov

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