math projects

## 1. Applied Mathematics & Modeling
These projects use math to explain how the world works.

*   **Epidemiological Modeling:** Build a variation of the **SIR (Susceptible-Infectious-Recovered)** model to simulate the spread of a specific disease. You could add variables for vaccination rates or seasonal changes.
*   **The Physics of Traffic Flow:** Use differential equations to model how "phantom traffic jams" occur. You can explore the **Lighthill-Whitham-Richards (LWR)** model.
*   **Optimal Strategy in Game Theory:** Analyze a specific game (like Poker, Chess, or even Monopoly) using Nash Equilibria to determine if an unbeatable strategy exists.

## 2. Finance & Economics
Great for those looking to move into actuarial science or quantitative finance.

*   **The Black-Scholes Model:** Conduct a deep dive into the stochastic calculus behind option pricing. You could test the model against real historical stock market data to see where it fails.
*   **Portfolio Optimization:** Use the **Markowitz Bullet** (Mean-Variance Analysis) to create an "optimal" investment portfolio based on a set of chosen stocks.
*   **Predicting Bankruptcy:** Use **Logistic Regression** or **Discriminant Analysis** on financial ratios to predict the likelihood of a company failing.

## 3. Cryptography & Number Theory
The "pure" side of math with massive security implications.

*   **Elliptic Curve Cryptography (ECC):** Explore the algebraic structure of elliptic curves over finite fields and explain why they provide stronger security with smaller keys than RSA.
*   **Prime Number Distribution:** Investigate the **Riemann Zeta Function** and the Prime Number Theorem. You could write a program to visualize the density of primes.
*   **RSA Encryption Walkthrough:** Build a functional (but simple) RSA encryption/decryption tool from scratch, explaining the modular arithmetic and the difficulty of integer factorization.

## 4. Statistics & Data Science
Perfect for showing off your ability to handle "messy" data.

*   **Sports Analytics (Sabermetrics):** Use Bayesian statistics to predict the outcome of a sports season or evaluate the "true value" of a player beyond standard box scores.
*   **Election Forecasting:** Create a model that aggregates polling data, accounting for "house effects" (biases of specific pollsters) using weightings.
*   **The Mathematics of Music:** Use **Fourier Transforms** to decompose musical chords into their constituent frequencies and explain why certain intervals sound "consonant" or "dissonant" mathematically.

## 5. Pure Mathematics (Theoretical)
For those who love the beauty of the proof.

*   **Fractal Geometry:** Explore the **Mandelbrot Set** or **Sierpinski Triangle**. Calculate their Hausdorff dimensions—which are often non-integers (e.g., $\log_2 3 \approx 1.585$).
*   **Graph Theory and Social Networks:** Use the **Small World Phenomenon** (six degrees of separation) to analyze how information spreads through a network graph.
*   **Topology and "The Hairy Ball Theorem":** A fun, rigorous look at algebraic topology, proving why there must always be at least one point on a hairy ball where the hair cannot be combed flat.