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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.
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