Documents

The following are some mathematical documents I’ve prepared for various reasons that could be of general interest.

  • The Christoffel-Schwarz Formula — this short article provides a completely rigorous and self-contained proof of the Christoffel-Schwarz formula, which gives an explicit integral representation of a conformal mapping from the unit disc onto an arbitrary polygon.
  • Distances in the Hyperbolic Plane — this article derives several formulas for computing distances in the hyperbolic plane, and it then uses these formulas to prove the Pythagorean theorem in hyperbolic geometry.
  • The Sylvester-Gallai Theorem — a short article developing a remarkable proof of the Sylvester-Gallai theorem from classical geometry using Euler’s formula for connected planar graphs.
  • Periodic billiard orbits in convex polytopes — a LaTeX-beamer presentation that I gave with Nell and Ray at the Penn State 2010 Math REU. This is a completely self-contained presentation introducing the most basic ideas in the theory of mathematical billiards and outlining some of our research that summer.
  • Topology, Groups, and Knots — a light-weight and self-contained introduction to the most basic ideas of knot theory, including the foundational material from group theory and topology. It culminates with the significant and non-trivial Wirtinger presentation of the knot group.
  • Linking numbers in two-fold branched covers of the 3-sphere — a poster covering some minor research I did on how linking numbers lift to two-fold branched covers of S^3. (Yeah, I know there are two boxes numbered 7. I don’t have the TeX source for this poster anymore. :D)

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