Joel Langer, Case Western Reserve University

Abstract: When we teach calculus, we may try to convince our students of the value of arc length parameterizations of curves – but will they be on the exam? For algebraic plane curves, at least, explicit parameterizations by unit speed are quite exceptional, and exceptionally interesting. In this talk, pictures will mostly suffice for a brief review of the history of such examples, and their connections to number theory, elliptic functions and conformal mapping. The remainder of the talk will outline a geometric explanation for the scarcity of such curves, based on the natural polyhedral geometry on the compact Riemann surface of an algebraic curve.