# Math research ideas for high school student?

bmcphysics
So I've been accepted into a research program at my school, and I'm to come up with a research topic to work on over the school year. My mathematics is limited to geometry, though after this year will encompass trig and pre-calc. Anyway, I'm interested in pure mathematics, but obviously my knowledge limits what I can do. Any original research is surely out of the question, so I've been thinking something along the lines of analyzing a conflicted area of study and coming to my own judgement. Some one suggested the concept of infinity, which certainly could be interesting. Any suggestions? and remember this has to be a YEAR of research.

. Some one suggested the concept of infinity, which certainly could be interesting.

is that a research program in philosophy or in mathematics? - or in some sort of mixture? You could certain study various mathematics that uses the word "infinity" (such as limits, transfinite ordinals and cardinals). However, if you want to study the "concept" of infinity, you might find mathematics very restrictive because the word "infinity" has no independent meaning in mathematics, it is only used as part of a phrase or sentence and thus it has meaning in narrow contexts.

You could study the Calculus Of Finite Differences and matrices and then answer my question!

(Of course I should have written those sums as $\sum_{i=1}^n i^2$ and $\sum_{i=1}^n (2i + 1)$.)

Hey bmcphysics.

One definite requirement of understanding infinity has to be to study Hilbert-Spaces and Operator Algebras on Hilbert-Spaces which is basically looking at infinite-dimensional operators.

Also you might want to look at Projective geometry as well.

Take a course on Hilbert-Spaces and Operator Algebras to really look at the current understandings of infinity, but make sure you have the right pre-requisites so that you can understand how the infinity makes sense (or screws things up from the finite point of view: i.e. the convergence aspect).

Diffy
is that a research program in philosophy or in mathematics? - or in some sort of mixture? You could certain study various mathematics that uses the word "infinity" (such as limits, transfinite ordinals and cardinals). However, if you want to study the "concept" of infinity, you might find mathematics very restrictive because the word "infinity" has no independent meaning in mathematics, it is only used as part of a phrase or sentence and thus it has meaning in narrow contexts.

You could study the Calculus Of Finite Differences and matrices and then answer my question!
(Of course I should have written those sums as $\sum_{i=1}^n i^2$ and $\sum_{i=1}^n (2i + 1)$.)