Mathematics Colloquium
Wild topology deserves a place in our hearts; it is necessary to understand 4-manifolds. Two landmarks (eventually crucial to the 4D theory) are the Alexander horned sphere 1924 and the Bing involution on S3 (1952). I'll talk about the history and recent work with Mike Starbird quantifying how analytically wild Bing's involution needs to be. We show that that any involution topologically conjugate to Bing's needs to have a (weakly) exponential modulus of continuity. [That is, in the usual epsilon, delta definition of continuity, up to log factors, delta-1 must be an exponential (up to log factors) in epsilon-1.] Furthermore, given any function f, no matter how rapidly growing, there is a "ramification" of Bing's construction If so that the modulus of continuity of any conjugate (If)h must be worse (grow faster) than f. There is an old subject within dynamics, of "inherent differentiability" studying the maximum differentiability of a diffeomorphism within its topological conjugacy class. Now we have a theory of "inherent modulus".
The talk is 60 minutes, and there will be 30 minutes of details afterwards for any interested party.