Algebraic and Analytic Geometry Summary

Monday - Friday, July 7 - 11, 2008

The Algebraic Geometry working group is attending Tom Garrity's afternoon lectures to the Undergraduate Faculty Program, where he is outlining (at approximately double speed) the topics of a senior-level "capstone" course targeted at math majors who are already familiar with abstract algebra and some basic ideas about geometry and the complex numbers. (He also meets with the undergraduate faculty in the afternoon to develop problem sets and supporting material, and they sometimes attend the Undergraduate Summer School course meetings, which are taking a different, more algebraic approach to similar material.) So far we have attended four course meetings and had three sessions of our own to work through the material, discuss questions, and begin thinking about ways to connect and adapt these topics to the secondary classroom.

In this first week we got a general sense of the objects and relationships that algebraic geometers look at, especially curves and their intersections. We began with conics (the solution sets of quadratic polynomial equations), and our first significant result was showing the equivalence of all ellipses and hyperbolas, as long as one's transformations used the complex numbers (rather than the real numbers). We then expanded this equivalence from ellipses and hyperbolas to include parabolas as well, by introducing the idea of "points at infinity" to define the projective plane. (We are taking an approach that's more geometric and gets us some cool results quickly, but sometimes demands flexibility both between algebraic and geometric representations and between real and complex cases that's been a bit challenging.) At the end of the week, having established the equivalence of conic sections (in the complex projective plane), we introduced some basic ideas for cubics, which we will be working with for the rest of the course.

Overall, our experience has been challenging but positive. We are pretty comfortable with the idea of projective space, and after working with some examples, are getting more fluent at working with points and lines in projective space and are more confident about the motivation for using it. We have learned how to work through being lost (which is always a valuable experience that gives you more empathy for students, and some ideas on how to support them). We've discussed some pedagogical choices that are interesting, and which of them would translate effectively to the secondary classroom. We hope that we've asked some good questions in lectures, and we're starting to explore ideas for projects that could get some important aspects of the thinking that's emphasized in algebraic geometry into the secondary math classroom.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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