A high school student can do publishable math research, and the most reachable paths are computational, applied and experimental math: using computers to test conjectures, building and analyzing mathematical models, and studying the math behind algorithms and machine learning. Proving a brand new theorem in pure math is possible but rare before college, so most students who publish start with a question they can attack with computation and careful analysis.
This guide explains the kinds of math research a student can do, example questions, methods you can carry out, where math papers by students get published, which competitions fit, and the math-heavy papers YRI Fellows have written.
Contest math is not research
Many strong math students come from competitions. Contest problems have known answers and are designed to be solved in hours. Research questions have no answer key, can take months, and often end with a partial result. The skills overlap, but they are different activities.
| Contest math | Math research |
|---|---|
| Answer is known to the problem writer | Nobody knows the answer yet |
| Solved in minutes or hours | Takes weeks or months |
| Judged right or wrong | Judged on originality, rigor and clarity |
| Individual speed matters | Patience, reading and writing matter |
A student with no contest background can do excellent math research, and a contest star still has to learn how research works.
Types of math research a student can do
| Type | What it looks like | Reachable before college? |
|---|---|---|
| Pure math | Prove new theorems in number theory, combinatorics, geometry | Possible with a specialist mentor, slow |
| Experimental math | Compute many cases, find patterns, form and test conjectures | Yes |
| Applied math and modeling | Differential equations, epidemic or population models | Yes |
| Probability and statistics | Simulation, Monte Carlo methods, new statistical tests | Yes |
| Optimization and graph theory | Algorithms for hard problems, approximation quality | Yes |
| Math of machine learning | Why optimizers converge, how networks represent equations | Yes, with coding |
Example research questions
Experimental and discrete math:
- For a family of graphs, how often does a fast heuristic find the true minimum vertex cover, and how far off is it when it fails?
- Does a sequence defined by a simple rule always reach the same cycle for every starting value up to a large bound?
Applied math and modeling:
- How sensitive is a simple epidemic model's peak to small changes in the contact rate?
- Which numerical method solves a given differential equation most accurately for a fixed compute budget?
Probability and simulation:
- Does an analytical formula for a system's success rate match Monte Carlo simulation within confidence bounds?
- How many random trials does a game need before a small strategy edge becomes detectable?
Math of machine learning:
- When a neural network solves the heat equation accurately, are its derivatives clean enough to rediscover the equation itself?
- Do matrix-preconditioned optimizers converge to better solutions than standard optimizers on physics-informed networks?
Methods a student can actually carry out
1. Compute, look for patterns, then prove what you can
Experimental math starts with code. You generate thousands of cases, look for structure, state a conjecture, and then try to prove it or at least prove special cases. The On-Line Encyclopedia of Integer Sequences is a classic first stop to check whether a sequence you found is already known.
2. Build and analyze a model
Applied math turns a situation into equations. You build the model, analyze its behavior, test it against data or simulation, and explain where it breaks.
3. Validate theory with simulation
A strong pattern for student papers: derive a formula, then check it with Monte Carlo simulation and report how closely they agree, with confidence intervals.
4. Study algorithms and their guarantees
Many hard problems, such as vertex cover or scheduling, are solved approximately in practice. A student can measure how good the approximations are, propose a repair step, and prove or test bounds.
5. Analyze the math inside machine learning
Physics-informed neural networks, optimizers and loss functions are math objects. Asking precise questions about them, such as derivative quality or convergence, is a fast-growing area where careful students can contribute.
Math-heavy papers by YRI Fellows
These Fellows did mathematical research with PhD mentors. More are on our published work page.
- Manomay J., a 12th grader, studied graph neural networks on minimum vertex cover, a classic hard problem. Raw network outputs were valid in only 38.67% of cases. His repair-and-prune pipeline restored 100% validity with a mean optimality gap of 2.75% against exact solutions, evaluated on 570 graphs plus 135 larger graphs. First-author paper accepted at an IEEE conference. Read his story.
- Nishanth A., an 11th grader, compared neural network architectures on the heat equation under noise and found what he named the Jitter Paradox: the more accurate networks produced rougher derivatives that corrupted symbolic equation discovery. He also proposed a new metric, the Bias Inheritance Ratio. First-author paper accepted at an IEEE conference. Read his story.
- Raaghav G. built an analytical model of losses in photonic quantum benchmarks and validated it with Monte Carlo simulation. His predicted success rates matched the simulations within 95% confidence intervals across three circuits. First-author paper accepted at an IEEE conference. Read his story.
- Akshara R. wrote "Empirical Solution-Gap and Stability Analysis of Matrix-Preconditioned Optimizers for Physics-Informed Neural Networks," accepted at an IEEE conference.
- Ashwajit W., a Springer-published Fellow, also entered the 2026 Breakthrough Junior Challenge with YRI's guidance, with a short film explaining the Kakeya conjecture. Read his story.
Notice the pattern: each project took a precise mathematical question, attacked it with computation, and reported numbers a reviewer could check.
Where student math papers get published
arXiv. Mathematicians widely share new work on arXiv as preprints. arXiv requires new submitters to be endorsed before their first paper in a category, usually through an established author such as a mentor.
IEEE and computing conferences. Computational, applied and algorithmic math, and the math of machine learning, fit IEEE conferences, where accepted papers appear in IEEE Xplore.
Math journals. Pure math journals have long review times and high bars. A strong result with a specialist mentor can get there, but plan for months or more.
Student journals. Some student research journals accept math. Check each venue's scope and fees first, and avoid any journal that promises acceptance for a payment.
Math at competitions and programs
Regeneron ISEF Mathematics category. Subcategories are Algebra; Analysis; Combinatorics, Graph Theory, and Game Theory; Geometry and Topology; Number Theory; Probability and Statistics; and Other. Applied projects may also fit Physics and Astronomy or computing categories.
Regeneron Science Talent Search. A research competition for high school seniors that judges a full research report. See our Regeneron STS guide.
Selective math research programs. MIT PRIMES and the Ross Mathematics Program are well-known paths into mathematics. See our guides to MIT PRIMES and the Ross Program.
Breakthrough Junior Challenge. Students make a short video explaining a concept in math or science. It rewards explanation rather than new results, and it pairs well with a research paper. See how to win the Breakthrough Junior Challenge.
Common mistakes in student math research
- Choosing a famous unsolved problem. Work on a small, specific question instead.
- Patterns without checks. A pattern in the first 100 cases is a conjecture, not a theorem. Say which it is.
- Skipping the literature. Many "new" results are old. Search before you spend months.
- Hiding the code. For computational math, share enough detail for others to reproduce results.
- Unclear writing. Define every symbol. A reviewer should be able to follow every step.
Frequently asked questions
Can a high school student do original math research?
Yes. The most common route is computational or applied math, where students test conjectures, build models or analyze algorithms. Pure math theorems are possible with a specialist mentor but take longer.
What math do I need to start?
It depends on the area. Discrete math and graph theory projects can start after algebra. Modeling and machine learning projects need calculus and some linear algebra, which you can learn alongside the project.
Do I need to be a math competition winner?
No. Research rewards patience, reading and careful writing more than speed.
Should I learn to code for math research?
Yes, for most paths. Python is enough for experiments, simulation and analysis.
What ISEF category is a math project?
Mathematics, with subcategories in algebra, analysis, combinatorics and graph theory, geometry and topology, number theory, and probability and statistics.
How the YRI Fellowship helps you publish math research
The YRI Fellowship matches you 1:1 with a PhD mentor in mathematics or a math-heavy field for 10 weeks of weekly sessions, about 5 to 8 hours of work a week. Your mentor helps you pick a question small enough to finish, choose the right computation or proof strategy, and check every step. An editing team works through every draft, and a publishing team gets the paper into IEEE, Springer or Elsevier journals and conferences, with free help until it is published.
87% of Fellows publish. ISEF and JSHS preparation and a mentor letter of recommendation are included. No research experience is needed, YRI works with 1,000+ PhD mentors, and decisions come within 48 hours.
Tuition is $2,997, or 3 payments of $999, with scholarships considered case by case. See pricing and more student stories.
Sources
- Society for Science: ISEF categories and subcategories (Mathematics)
- Society for Science: Regeneron Science Talent Search
- arXiv: endorsement policy
- On-Line Encyclopedia of Integer Sequences
- MIT PRIMES
- Breakthrough Junior Challenge
- YRI case study: Manomay J.
- YRI case study: Nishanth A.
- YRI case study: Raaghav G.
- YRI case study: Ashwajit W.
- YRI Published Work
