Squaring the Circle and Other Impossible Dreams: Periods and Transcendence

Start: 04/13/2016 - 4:15pm
End  : 04/13/2016 - 5:15pm


Paula Tretkoff (TAMU)


The phrase “squaring the circle” is a recognized idiom of the English language that has come to mean “to solve an unusually hard problem”. This idiomatic meaning is very much reflected in the history of the ancient mathematical problem that gave rise to it: roughly speaking, the problem asks if it is possible to construct a square with the same area as the unit circle using compass and straightedge. The impossibility of squaring the circle is equivalent to the transcendence of the number π: that is, to showing there is no non-zero polynomial P(x) with rational coefficients such that P(π) = 0. Proving the transcendence of π was indeed unusually hard and was only achieved in 1882 by Lindemann, following a method due to Hermite, who proved the transcendence of e in 1873. The transcendence of π is equivalent to that of 2πi, a period of the function exp(z) in that exp(z + 2πi) = exp(z). Moreover e is the special value exp(1). The work of Hermite and Lindemann opened the way to a grand modern theory for proving the transcendence of numbers related to periods and values at algebraic points of other classical functions, like elliptic and, more generally, abelian functions, as well as hypergeometric functions. In this talk, we focus on periods in transcendental number theory, the evolution of results, for example, Hilbert’s seventh problem, the breakthroughs of Baker (Fields Medal 1970) and W¨ustholz, as well as some on some of my work, in part joint with M.D. Tretkoff. We also discuss challenges for the future. The talk presumes the intrigue of impossible dreams but does not assume any background in transcendence.

Argue Auditorium, Millikan, Pomona College