Highschooler looking for math research/projects

Hi, I am in Highschool and I am interested in pursuing mathematics as a career. I am particularly interested in doing some math research/projects (even if simple) to boost my maturity of the subject. How would I go about doing research/projects? Are there any good ideas?
Math background:
- Calculus I, II, III
- Real analysis (Rudin)
- Linear algebra (Axler)
- Proofs
Currently studying:
- Abstract algebra (Artin)
- Topology (Munkres)
Interested in:
- Differential Geometry
- Statistics
Other skills:
- Programming with python, etc. (also web dev but not really applicable here)
Thanks in advance!

Hello and !

I have a good idea and could give you a subject nobody else (but me) is working on. It is not too difficult. However, you should study linear algebra and
https://www.amazon.com/Introduction...-Mathematics/dp/0387900535/?tag=pfamazon01-20
first.

Linear algebra is no problem since you will need it in every other STEM field anyway, i.e. you won't waste any time. You can find cheap sources on university servers around the world if you google Linear Algebra I + pdf and Linear Algebra II + pdf if you're done with part one. Eventually, add lecture notes to the search key.

The book I quoted is an easy read, too, once you get accustomed to the new mathematical language. But before you start with it, you should reconsider if Abstract Algebra is your thing; maybe you like Calculus or something in Applied Mathematics (Computer science, Statistics, Probability Theory, Numerical Analysis) more.

The first semesters at college are usually filled with Linear Algebra (I + II) and Calculus (real I, multidimensional II, complex III). So anything in one of these areas would be useful.

Doing research in the sense of new research is normally difficult. Will say: the easy stuff has been solved and researched in the last four centuries. The example I spoke about above is already an easy way to research something new and it still requires profound knowledge in some areas.

I have written many insight articles here which could give you a first impression of what it is all about. (I have no idea how to find them. Ask me about them. I just checked a search by my username and found not one article. I guess, the search key requires something from the title, not the author.)

If you are asking for a doable high school project, then things are different. In that case, I would recommend having a look at the problems and the solution manual here:
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
There are problems of almost any kind of level. Unfortunately, they are ordered by date and not by difficulty, search the document for "HS" to find specifically the high school challenges - and ignore the others! Frustration is a very bad advisor!

Have fun and stay curious!

Quite an impressive list of courses under your belt. You're tracking an undergrad in applied mathematics suitable for physics.

I followed a similar route while in college having learned differential calculus the summer before my freshman year. It allowed me to skip past that course once I proved to my prof that I understood the definition of a limit.

I also took differential geometry, studied the Frenet-Serret formulation, and learned about tensor notation. Tensors make more sense once you have linear algebra and vector analysis.

Math, though, is a very large field of study. This video by Domain of Science shows graphically the various fields of math:



Another excellent video is this NOVA one on math:



and this one on the math and art of origami featuring math Prof DeMaine:



His specialty is computational origami, which combines geometry, paper folding, and computers into interesting research projects.

https://en.wikipedia.org/wiki/Erik_Demaine

I would explore these fields and see whether you are a visual thinker (geometry) or an analytical thinker (abstract algebra, number theory, modular forms...)

I've always liked visual things and was attracted to differential geometry (General Relativity...) and, more recently, geometric calculus featuring the geometric product of vectors, bivectors...

POST SCRIPT:

Look into Srinivasan Ramanujan, John Conway, John Nash, Terence Tao and other great mathematicians for inspiration on what they have worked on:
,
https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

https://en.wikipedia.org/wiki/John_Horton_Conway

https://en.wikipedia.org/wiki/John_Forbes_Nash_Jr.

https://en.wikipedia.org/wiki/Terence_Tao

Bookwise, check out Prof Richard Elwes's book:

Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations

https://www.amazon.com/Mathematics-...-Explanations/dp/1554077192?tag=pfamazon01-20

for further inspiration. It covers a wide range of current math problems in various math fields with brief reaable summaries.

For more in-depth coverage there are the Princeton Companion books:

https://www.amazon.com/Princeton-Co...imothy-Gowers/dp/0691118809?tag=pfamazon01-20

https://www.amazon.com/Princeton-Companion-Applied-Mathematics/dp/0691150397?tag=pfamazon01-20

My suggestion to you is to explore all of math, trying out different problems, moving on when you get too stuck, and always stay curious. There is just so much to see and do and so much to marvel at and enjoy.

There's a Feynman book that encapsulates his view of things: The Pleasure of Finding Things Out

https://www.amazon.com/Pleasure-Finding-Things-Out-Richard/dp/0465023959?tag=pfamazon01-20

https://topdocumentaryfilms.com/pleasure-finding-things-out/

You certainly have an impressive list of studies for a high school student, congratulations!

But, real research as an individual is tough in a purely academic sense. The chance that whatever you research is novel is nearly zero. Plus time and finances might be a big issue.

What I see missing, compared to my version of "normal" curriculum, is lab work (i.e. experimental physics). The point here wouldn't to do something that's never been done. It would be to just go through the process of independently choosing and designing an experiment to do and writing up a short report about theory/methods/results, maybe with some error analysis. There could be a million ideas about what to do. I'll just toss out a few:

- Verify the Brachistochrone solution.
- Measure the frequency stability of a green laser pointer with a diffraction grating (or Iodine cell?).
- Build and analyze a small trebuchet.
- Design and build a telescope or radio antenna and analyze it's performance.

In each case compare your results to theory.

On a related note, why not create a journal of your mathematical explorations as your project.

You find something interesting, research it on Wikipedia and YouTube, then branch out to whatever further sources are listed and go as deep as you can or in whatever direction interests you all while keeping a journal.

Some Youtube resources are:
- 3brown1blue
- mathologer
- numberphile

with more listed here:

https://www.educatorstechnology.com/2023/01/20-excellent-youtube-channels-for-math.html

Some website resources are:
- quantamagazine.org
- bigthink.com
- khanacademy.com
- mathispower4u.com

Ramanujan did a similar thing, tracking what he learned and his discoveries. It's been said that mathematicians are still perusing his notebooks. One of the most recent findings was his taxicab number research.

The story goes that he was sick in bed with pneumonia and stayed at the hospital. His advisor, Prof Hardy, took a cab to visit him. Hardy remarked how dreary the day was, and even the taxicab number 1729 was rather dull, to which Ramanujan replied: Oh no, Professor, 1729 is the sum of two cubes two different ways.

##1729 = Ta(2) = 1^3 + 12^3 = 9^3 + 10^3##

https://en.wikipedia.org/wiki/Taxicab_number

The researchers found a list of related numbers in his notebooks and realized that Ramanujan was investigating Fermat's little theorem.

https://plus.maths.org/content/ramanujan

@vaibhavvenkat what do you think of our suggestions? what are you planning to do?

vaibhavvenkat said:

Hi, I am in Highschool and I am interested in pursuing mathematics as a career. I am particularly interested in doing some math research/projects (even if simple) to boost my maturity of the subject. How would I go about doing research/projects? Are there any good ideas?

What exactly do you mean by research/project? Most textbooks have exercises some of which can be hard and long. Why not work on those?