Title: The Stern Sequence
The Stern sequence is defined by s(0) = 0, s(1) = 1, and for k > 0, s(2k) = s(k), s(2k+1) = s(k) + s(k+1). It was defined in 1858 and has many applications to and from number theory, digital representations, graph theory, geometry, analysis and probability. The talk has cameo appearances (at least) by Eisenstein, Minkowski, Einstein, de Rham, Dijkstra and other famous people. Stern himself was Gauss' first PhD student and the first non-baptised Jew to be a full professor at any German university. He proved that every positive rational p/q can be written uniquely as s(n)/s(n+1) well before Cantor. where the binary expansion of n is related to the simple continued fraction representation of p/q. A digital interpretation is that s(n) is the number of ways to write n-1 = \sum a_k 2^k, where a_k is in \{0,1,2\}. De Rham used the Stern sequence to define a convex curve on which points of zero flatness and points of infinite flatness are dense. This talk will be accessible to first-year graduate students.
There will be a reception in 300 Harker preceding the talk (3-4pm)