Algebraic and Analytic Geometry Summary

Monday - Friday, July 21 - 25, 2008

In this final week of the PCMI summer session, we attended Professor Garrity's course for the first half of the week, where we worked to generalize our results on the topology of points on cubic curves to curves of higher degree. We went through an intuitive argument for a formula relating the genus of these spaces to the degree of the curve, and discussed how to define a genus for cases that involve curves over algebraically complete fields other than the complex numbers. Professor Garrity concluded the course with some statements, intuition, definitions and lemmas on the way to proving the Riemann-Roch Theorem, which relates function theory (specifically, the dimensions of vector spaces defined by specific sets of poles and zeros, with multiplicity) to topology (specifically, the genus of the topological spaces defined by polynomial curves). This is a big, non-intuitive result, and though we got that, the pace of the course made it challenging to appreciate the import of the theorem. Still, we've come a long way since the beginning of the course.

Most of us did not attend Thursday's lecture, and none of us attended Friday's lecture, because we were working hard to bring together our work on our projects. Our nomographs (function diagram) group explored the linear and quadratic cases more carefully, and used these examples to help motivate the real projective line and illuminate its structure, as well as making some interesting connections between Mobius transformations and conics. They also created some lesson materials and a Sketchpad tool to help look at both the linear and quadratic cases. Our projective geometry group created a structure for a sequence of lessons that examined the invariants and changes to shapes when projected, including some opportunities to use either manipulatives (especially the Línárt sphere) or technology (especially Cabri 3D) to make these investigations more rich, leading up to some amazing animations from the Dimensions video series. Finally, we assembled some ideas from the lectures and summaries of our projects into a coherent presentation for our fellow SSTP participants.

Overall, we enjoyed attending the lectures, working through the course material, and working together to devise worthwhile experiences for secondary students. A+++ would do again!

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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.