Below is an evolving list of possible topics for your paper.  It's incomplete in many ways, so I'll keep adding to it, and of course you need to clear a paper topic with Prof Dyer as well as with me, but if you see something in the list you're curious about, let me know and I can try to say more about it and point you to some good sources.  

Cardinality
Convergence of Fourier Series
Partial Differential Equations 
Kepler's Laws for planetary motion
Numerical linear algebra
Poincare-Bendixson Theorem
Chaos and the forced pendulum
Sarkovski's Theorem
Matrix Groups and algebras
Aspects of Probability
Basic complex analysis
linear recurrence and rational functions
Rational, algebraic and transcendental numbers
Lebesgue measure/integral
Linear programming
The gamma and zeta functions and the prime number theorem
Curvature of surfaces
Isoperimetric problem for curves