Math Colloquium @ Bryn Mawr: "A Little p-adic Analysis"Math Colloquium @ Bryn Mawr: "A Little p-adic Analysis"http://www.haverford.edu/calendar/details/254401Bryn Mawr, Park 338.2013-11-04T16:00:002013-11-04T17:00:00
November 4,2013 4:00PM–5:00PM
Bryn Mawr, Park 338.
John Boller (University of Chicago). Bryn Mawr, Park 338.
Bi-College Math Colloquium
"A Little p-adic Analysis"
John Boller, University of Chicago.
Talk at 4 in Park 338
Tea at 3:30 in the Math Lounge, Park 355.
Abstract: What are the real numbers? In real analysis, we learn that they are the unique ordered field in which the least upper bound property holds. But where do they come from? One means of constructing the real numbers R is a process known as (analytic) completion, where we take equivalence classes of Cauchy sequences of rational numbers and endow them with the arithmetic of a field and an ordering.
This process of completion requires us to have an absolute value on the rational numbers Q in the first place. But it turns out that there are infinitely many different absolute values on Q! In addition to the usual one, there is one for every prime p. When we complete Q with respect to the absolute value associated with the prime p, we get the p-adic field Q_p. These p-adic fields are quite different in character from the real numbers, but they are still complete, so we can ask all of the usual questions from Calculus and analysis about differentiation, integration, power series, Fourier series, and more. Some of the results will surprise!
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