Elementary particles are subatomic particles which cannot be subdivided into smaller pieces. For example, electrons are elementary particles; however, protons are not (they are actually made from three even smaller particles)! Just like how the periodic table of elements describes how molecules are formed, there is a finite list of elementary particles which form all the matter in the universe. An interesting project would be to explore precisely what these elementary particles are, how they interact with one another, how they were discovered, and other related questions to deepen your understanding of the quantum world.

Introduction to Elementary Particles

Knot theory is a branch of mathematics that studies knots. Knots are pieces of string which can be wrapped into complicated shapes and whose ends are then fused together. There is a rich mathematical structure involving knots. It turns out that you cannot deform any particular knot into another knot (some knots are permanently tangled) - this is called a "topological obstruction." We would learn about topology in the context of knot theory and learn a lot about these very fundamental objects. No formal knowledge of math is required to study knot theory (not even algebra), but the subject is very rich and a good introduction to formal mathematics.