Mathematics Colloquia and Seminars

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In 1980 Michael Freedman proved that for any simple, smooth, closed space curve with nonvanishing torsion, there are an even number of planes tangent to C at exactly three points. Throughout the 1980s several mathematicians including Banchoff, Gaffney and McCrory furthered Freedman's study of these triply tangent planes. The focus of my talk will be a paper of Ozawa published in 1985. Ozawa expands Freedman's work to consider simple smooth closed space curves with possibly vanishing torsion. A plane P is called a triple tangency of C if the total order of contact of P to C is three; thus a triple tangency can be one of three types depending on whether it is tangent to C at 3, 2 or 1 distinct points. The main concern of the paper is to find what relationships exist among the numbers of each type of triple tangency for a given space curve. I will present Ozawa's paper in the context of Freedman's work which preceded it. The presentation will include the results and a brief sketch of the proofs. In addition, I intend to address Freedman's related question: Does every generic smooth space curve have a triply tangent plane?