I am a high school senior and I am interested in doing a math research. I hope someone can recommend areas or topics of research that are challenging, rewarding, and yet do not exceed my capability. (I acknowledge this is quite hard)
My math background:
a. I have done competition math (Elementary number theory and combinatorics, Euclidean Geometry, and Algebraic manipulation) and I'm fairly comfortable with proofs.
b. I had my first courses in Multivariable Calculus, Differential Equation, and Linear Algebra (Familiar with fundamental concepts, basic techniques and motivations)
c. I have learned a portion of Abstract Algebra on my own and in summer programs including topics like Lagrange theorem, Vector spaces, Polynomial Rings, and Morphisms.
d. I don't have a good background in statistics and probability
e. I have been exposed to Knot theory and Chaos theory
f. I do have basic programming skills in python and Mathematica, and I can work with LaTeX.
I really appreciate your help!