High school is a great time to go deeper into mathematics, beyond the standard curriculum. If you’re looking for a challenge, here are ten research project ideas to spark your curiosity and build a real understanding of how math works.
Why research?
Research is how we understand the world and solve hard problems. Doing it builds critical thinking and teaches you to ask meaningful questions. It lets you explore topics you care about and uncover new information. You’re not just learning, you’re adding to a broader understanding of your field and maybe making a difference in it.
If you want to take a project to the next level and win science fairs, in-state, nationally, or internationally, check out this video from past ISEF grand award winner Rishab Jain.
Top 10 math research ideas
1. Fractals and chaos theory
Fractals sit at the intersection of math and visual art. Research famous fractals like the Mandelbrot set or Julia sets and their recursive structure, then extend into chaos theory: how tiny changes in initial conditions can lead to very different outcomes, as in weather or markets. Use computer programs to visualize these and create fractal art or simulations, and discuss what chaos theory means for fields like science and finance.
2. Game theory and strategic decision-making
Game theory is a framework for analyzing strategic interactions. Study classics like the Prisoner’s Dilemma and Nash equilibrium, and apply them to real situations such as business negotiations or political agreements. Explore cooperative versus non-cooperative games, then compare the theory’s predictions with how people actually decide.
3. Cryptography and number theory
Cryptography leans heavily on number theory. Study how prime numbers and modular arithmetic underpin modern encryption like RSA, then build simple encryption and decryption algorithms and test them against different attacks. Analyze the tradeoff between key length and security, and consider how quantum computing might challenge today’s methods.
4. Topology and knot theory
Topology and knot theory offer a fresh take on geometry and structure. Study how knots are classified and the idea of invariants, including the Jones and Alexander polynomials that help tell knots apart. Look at practical applications like DNA replication or untangling cables, and the broader role of topology in molecular chemistry and computer graphics.
5. Probability distributions and their applications
Probability distributions are the foundation of statistical analysis. Research the normal, binomial, and Poisson distributions and their properties, then gather your own data and fit it to each, comparing how well they describe it. Look at how these distributions show up in statistics, engineering, and economics, and how they inform data-driven decisions.
6. Combinatorial optimization
Combinatorial optimization is about solving discrete problems efficiently. Study the Traveling Salesman and Knapsack problems and compare algorithms for them, such as dynamic programming, greedy approaches, and branch-and-bound. Apply these to real cases like optimizing delivery routes or budget allocation, and discuss where they matter across industries.
7. Differential equations in modeling
Differential equations model many natural and engineered systems. Build equations for examples like population growth, radioactive decay, or oscillating systems, then solve them analytically or numerically and compare to real data or simulations. Explore their use across physics, biology, and economics, and how they help you understand complex systems.
8. Graph theory and network analysis
Graph theory is a powerful way to analyze networks. Study graph types like Eulerian and Hamiltonian paths and their properties, then analyze real networks such as social media, transportation, or computer networks. Compare algorithms for traversing or optimizing them, and consider where network analysis applies across fields.
9. Linear algebra in machine learning
Linear algebra is the backbone of many machine-learning algorithms. Study vector spaces, matrix operations, and eigenvalues, then build simple models like linear regression or clustering and test them on real datasets. Discuss how concepts like matrix multiplication and singular value decomposition power these models, and where the field might go next.
10. Fourier analysis and signal processing
Fourier analysis breaks functions into sinusoidal components. Study Fourier transforms and how they apply to signal processing, from sound waves to financial data. Build algorithms to filter or enhance signals and compare how well they work, then look at applications in telecommunications, finance, and other technology.
Bonus: Synthetic DNA Engineering With ICOR
Rishab Jain‘s project is in synthetic biology, focused on improving protein production in E. coli, which matters for vaccine development. The core is codon optimization: choosing the best DNA sequences to boost protein synthesis. Traditional methods often ignore cellular dynamics, leading to inefficiencies. Jain built ICOR, a tool that uses a recurrent neural network (RNN) with a bidirectional long short-term memory (LSTM) architecture, trained on a dataset of high-expression E. coli genes. That lets it optimize DNA sequences in a way that aligns better with the cellular environment and improves protein production. Tested against standard methods, ICOR showed significant gains in protein expression efficiency, with broad implications for biotechnology and vaccine development.
Award*: Regeneron Young Scientist Award (i.e. TOP 3, winning $50,000!) at ISEF 2022
If you want to hear more about Rishab’s work and how he did it, check out his YouTube videos on how to get started in science research and win those gold medals.
Final thoughts
Math goes well beyond textbooks and theorems. These advanced projects let you explore the field and its real-world uses, from the elegance of fractals to the strategy of game theory to the practicality of network analysis. Happy researching.
If you want to elevate your project but aren’t sure where to start, take a look at Rishab’s free STEM student guide, full of practical advice for students aiming to take a project all the way to the international level. It covers planning and conducting research, presenting your project, and finding strong opportunities in STEM.
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