Algebraic and Analytic Geometry Summary

Monday - Friday, July 14 - 18, 2008

This week, we continued with Professor Garrity's course, building on our first examination of cubic curves last week to fill in some of the underpinnings of the group structure of the points on a cubic curve. This took us through a variety of fields in mathematics, showing the connections and the dependence of algebraic geometry on multivariable calculus, linear algebra, and complex analysis (as well as abstract algebra and geometric intuition) that make it a rich and challenging field. At the undergraduate level this makes the course well-suited for a senior-level capstone course; at the secondary level, it suggests that engaging with these ideas can help students make connections among multiple representations and motivate them to connect and apply ideas that may initially seem quite different. On a PCMI-specific note, many of us were happy to see the ideas we had discussed and wrestled with in this course show up during both Clay Lectures, and especially happy that our hard work had helped us keep up with these lectures an epsilon longer.

In the hours we worked together after the course, we tried to make these ideas more concrete by working through examples and sharing both our confusions and questions, and different ways we addressed them, including analogies and visualizations. The group has settled on two major ideas around which to structure projects for secondary students: stereographic projection and nomographs (function diagrams). Both are inspired by the material in the course, and especially by the flavor of the mathematics (using multiple sets of tools and representations), though they make some major modifications (in particular, the use of real rather than complex numbers) to facilitate student understanding. As the ideas and sketches and plans develop and hit our group wiki, we hope our own understanding is growing as well.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.