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!
Best Answer
Ok, here is my sincere suggestion. First: +1 for your question. Since you have done multi variable calculus and differential equations, how about studying the Laplace Transform? This is (I think) new to you but yet, with your back ground, accessible. It is a cool topic (my opinion) with applications within the framework of calculus you have studied. You can solve (systems of) differential equations with it, as well as certain types of convergent improper integrals. I believe this will be a doable challenge for you!