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COSMOS was a gifted math/science student program run several summers at UC Irvine.
It's heavy minority student population was Latinos, though it attracted students world wide. Some students were pre-high school. The mathematics portion was High School Material writ large. I gave the Main Math Lectures Summer 2001 and 2002, just before my wife, youngest daughter and I left for Montana.
Rainbow Line

✺ The Schedule for the Mathematics/Cognitive Sciences Cluster, Summer 2002: #7 of seven clusters. The sessions both summers ended with a "Project Fair", and a Mathematics Competition. At the "Fair," teams of students presented 15 minute talks on their projects (which included a final written report and a poster presentation). The projects were coordinated by faculty advisors: Ludmil Katzarkov (a colleague of mine in the department), Fedor Yotov (who also ran the MathCounts seminar through the year, with Ludmil) and myself.

Roughly 200 students attended each summer. This cluster had 35 students, the limit for the classrooms we used. A final presentation included awards for exceptional projects, and for the Mathematics Competition standouts. 2002lect.pdf

✺ The first summer topic, geometry, was in two parts: Part I: COSMOS: EUCLIDEAN GEOMETRY TRANSFORMATIONS. Started with the phrase "Elements of Euclidean geometry include distances, angles, areas and some basic shapes that we understand using these three measurements." This handout was a synopsis of the lecture topics. The idea was that you could combine translating and rotating – rigid motions – into one "group" of operations, that isn't commutative.
  1. Relation between distance and angles
  2. Translating parallelograms
  3. Rotations and the Galilean group
  4. What some elements of the Galilean group do
  5. The subgroup of the Galiliean group fixing a polygon
coseuclid-01.pdf

✺ Summer 2001, Part II: SPHERICAL, TOROIDAL AND HYPERBOLIC GEOMETRIES: Building on measuring distances, and relating to angles and areas in Euclidean geometry, we discussed how astronomers would have to rethink measuring distances between objects in the "spherical" sky. This lecture mentioned recent developments in modern mathematics, as completing problems left unsolved from hundreds of years ago.
  1. Making convex polyhedrons
  2. Platonic Solids
  3. Triangles and angles on the sphere
  4. Distances on the sphere and rotations as distance preserving operations
  5. Distances on a donut and the analog of rotations
  6. Lessons from spherical geometry
  7. The 3rd geometry – hyperbolic – having a complete set of distance preserving operations
cosspherehyper-01.pdf

✺ Summer 2001: FRACTIONS AND INTEGERS MODULO PRIMES: This course was built around quadratic equations and their solutions. It started with a discussion about the square root of -1. Then, it went to a question in long division: How would you figure out (guess at) the period of the decimal expansion of 1/n for various integers n? The course conclusion was how such calculations relate to keeping data safe at banks (the modern word being cryptography).
  1. Warmup with a famous formula of De Moivre
  2. Long Division
  3. Clock Arithmetic
  4. Quadratic Equations, and solving them in clock arithmetic
  5. Abelian cryptography
cosmos07-15-02ClockArith.pdf

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