Polynomial approximation in homotopy theory

Michael Ching

Amherst College

Brandeis University

Thursday, November 8, 2012

Talk at 4:30 p.m. in 317 Goldsmith Hall

Tea at 4:00 p.m. in 300 Goldsmith Hall

Abstract: One of the central problems of homotopy theory is to calculate homotopy groups: these consist of the homotopy classes of maps from a sphere to a particular topological space. This is a very difficult problem. Here is another (less difficult) problem: find (part of) the decimal expansion of the square-root of 2. One approach to the second problem is to consider the square-root function, approximate it with Taylor polynomials, calculate their coefficients and add up sufficiently many terms.

I will talk about a way to apply the same approach to the problem of calculating homotopy groups. (This is Tom Goodwillie's Calculus of Functors.) As you can imagine, this is more challenging than finding the square-root of 2, but the analogy goes further than you might think. There are concepts that play the roles of function, of polynomial, of Taylor series and Taylor coefficient.

But one particular difficulty is that in topology there is more than one way to add up the terms of a power series. This means that extra information is needed beyond the coefficients in order to recover the Taylor polyomials from their pieces. I will describe this information and how it might be used to attack the problem of finding homotopy groups.

Home Web page: Alexandru I. Suciu Posted: October 20, 2012
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