The possible range of projects is completely open, and depends on the mutual interests and backgrounds of the mentor/mentee. The only prerequisites for undergraduates are completion of Math 19A/B and 20A/B. There is also the possibility of applied math/statistics or math education projects. Some suggestions from a subset of possible subject areas are:

Modular Arithmetic/Fermat's Little Theorem
Braid Groups
Quadratic Reciprocity

The Contraction Mapping Theorem
The Lorenz System and dynamics
Poincare-Bendixson Theorem, Recurrence
The Cantor set and Lebesgue measure

Maxwell's Equations and Geometry
Matrix Groups/Lie groups and algebras
Projective Geometry
The Fundamental Group
Knot Polynomials

Applied Math
Persistent Homology
SIR Models in epidemiology
Signal Processing, Fourier Analysis
PageRank algorithm and Markov chains
The Capital Assets Pricing Model
Public-key cryptosystems

For more complete lists of past topics/projects visit the below links: