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Speaker:
Alf van der Poorten, (Centre for Number Theory Research, Sydney)
Title:
Paperfolding, automata, and rational functions
Abstract:
The act of folding a sheet of paper in half, and iterating the operation, places in that sheet a sequence of creases appearing as valleys or ridges. Coding these as 1 and 0 respectively yields a sequence (f_h), the paper folding sequence, with generating function f(X) = \sum_{h >= 1}f_h X^h, the paperfolding function. It turns out to be easy to notice that f(X) satisfies a functional equation of a kind first studied by Mahler nearly seventy years ago. Moreover, viewed as defined over F_2, the field of two elements, the paperfolding function is algebraic --- it satisfies a polynomial equation over F_2(X). It's also easy to see that the paperfolding sequence is `automatic'; it is generated by binary substitutions. These phenomena are not unique to paperfolding. They are shared by the good reductions modulo a prime p of arbitrary diagonals of arbitrary rationals functions in many variables, equivalently by the reductions modulo p of a wide class of series in one variable satisfying linear differential equations with polynomial coefficients. I will tell the necessary stories to explain all this [and will show some relevant pictures from Michael Crichton's novel Jurassic Park. Audience members should bring note paper along, not to write on, of course, but to fold].
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