### Linear Algebra - [[MATH307Su25 - Low Rank Approximations of Images| Low Rank Approximations of Images ]] - You'll use Singular Value Decomposition (SVD) to compress digital images by approximating them with fewer mathematical components, discovering how linear algebra enables data compression. The project involves hand calculations of eigenvalues/eigenvectors, coding SVD implementations, and creating animations that show how adding more terms improves image quality—connecting matrix decompositions to real applications in image processing and data storage. - [[MATH307Su25 - Matrix Multiplication Speed Testing]] - You'll implement matrix multiplication from scratch and discover why professional software like MATLAB and NumPy can be thousands of times faster than naive algorithms. This project involves building robust functions with error handling, systematic performance benchmarking across different matrix sizes, and analyzing why optimized libraries excel—especially with sparse matrices where the speedup differences become truly dramatic. - **Pagerank** - [[MATH307Su25 - Collaborative Filtering and Latent Factor Discovery | MATH307Su25 - Collaborative Filtering and Latent Factor Discovery (aka recommendation systems based on user rating data) ]] - You'll replicate Netflix-style recommendation systems using SVD to discover hidden patterns in user-movie rating data, then investigate how these mathematical "concepts" change when new users or movies enter the system. Starting with Stanford's exact 8×5 example, you'll implement collaborative filtering and systematically explore how the latent factor structure responds to different types of data additions. - [[MATH307Su25 - Least Squares Approximation for $(x,y)$ Scatter Data and Applications| Least Squares Approximation for $(x,y)$ Scatter Data and Applications]] - You'll discover powerful linear relationships hidden in a mystery dataset of 7,500+ coordinate pairs, applying ordinary least squares to find the best-fit line through matrix projections. Only after your mathematical analysis will you learn these are height-wingspan measurements used by forensic scientists and anthropologists—demonstrating how linear algebra enables real-world identification and biological research. - [[MATH307Su25 - Fourier Regression and Mean Sea Level Change]] - You'll decompose 30+ years of global sea level data into sine and cosine waves using least squares regression, separating seasonal cycles from long-term climate trends. This project reveals how Fourier analysis is fundamentally a linear algebra problem, enabling you to quantify annual sea level rise rates and detect acceleration patterns crucial for climate change assessment. ### Taylor Series - [[MATH307Su25 - Multivariate Taylor Series, Hessian matrices, and Eigendata | Multivariate Taylor Series, Hessian matrices, and Eigendata]] - You'll extend single-variable Taylor series to functions of two variables, using hand calculations of gradients and Hessian matrices to create local quadratic approximations around critical points. The project connects eigenvalue analysis of Hessian matrices to geometric surface shapes, revealing how this mathematics enables modern optimization algorithms in machine learning and engineering design. - Numerical Taylor series ### Numerical Differentiation and Differential Equations - Round off versus truncation error - [[MATH307Su25 - Chaos, Butterflies, and the Limits of Prediction (Exploring the Lorenz Equations)]] - You'll implement the famous Lorenz equations from first principles and discover the butterfly attractor through single trajectory analysis, then investigate both sides of chaos theory's complete message. Starting with three simple differential equations that model atmospheric convection, you'll explore sensitive dependence that makes weather prediction impossible beyond a few days and strange attractors that reveal hidden mathematical order within apparent randomness—connecting deterministic mathematics to philosophical questions about prediction, free will, and the limits of scientific knowledge. - Hodgkin-Huxley Model of Neuronal Firing - [[MATH307Su25 - From Linear Springs to Nonlinear Pendulums (When Small Changes Break Everything)]] - You'll analyze oscillatory systems through energy conservation and phase space techniques, discovering how mathematical assumptions shape the behavior we observe and predict. Starting with linear mass-spring systems that behave predictably regardless of amplitude, you'll contrast this with pendulum motion where frequency depends on swing angle—revealing how crossing from linear to nonlinear mathematics breaks familiar rules and creates fundamentally different system behavior that affects everything from precision timekeeping to structural engineering. ### Numerical Integration - Numerical Fourier Series - [[MATH307Su25 - Building the Fourier Transform from Scratch Using Numerical Integration]] - Students build the discrete Fourier transform from scratch using adaptive quadrature instead of the Fast FFT, experiencing the O(N²) computational reality that plagued 1960s nuclear test detection. They work backwards from frequency domain, analyze synthetic signals, and witness dramatic FFT speedups. This reveals how algorithmic breakthroughs enable technological revolutions, connecting mathematical theory to the digital age through Cold War history and modern applications. - Finding the face - [[MATH307Su25 - Orbital Mechanics and Numerical Integration]] - Students implement numerical integration methods to solve orbital mechanics problems that have no closed-form solutions. Starting with hand calculations of ISS parameters, they build trapezoidal and Simpson's rules step-by-step, verifying accuracy against exact circular orbit solutions. The project culminates in analyzing highly elliptical Molniya satellite orbits, demonstrating how speed variations affect integration challenges. Students discover how computational accuracy directly impacts mission-critical space operations, from GPS satellites to communication coverage planning. - [[MATH307Su25 - Fat Tails and Black Swans (When Mathematical Models Meet Financial Reality)]] - You'll implement numerical integration to verify the normal distribution's famous properties, then watch these same algorithms fail spectacularly when applied to fat-tailed distributions like the Cauchy distribution. Starting with trapezoid and Simpson's rule verification, you'll document how MATLAB's built-in functions break down trying to calculate basic statistics (mean, variance) for distributions with heavy tails, connecting these computational failures to real financial catastrophes like the 1987 market crash that normal models predicted as essentially impossible. - Illumination integral - Nearly impossible integral and comparisons ### Interpolation and Splines - *Dog running interpolation* - [[MATH307Su25 - Agricultural Interpolation in the Research Literature]] - You'll apply Lagrange interpolation to real Turkish crop data and discover how elegant mathematical tools can be catastrophically inappropriate for real-world problems. Using five years of agricultural records from Konya province, you'll implement the same polynomial approach proposed in a published research paper, then document how degree-4 polynomials produce physically impossible predictions like negative crop yields—connecting mathematical sophistication to practical consequences in food security planning and revealing when simpler approaches are more appropriate than complex mathematics. ### Approximating Solutions to Nonlinear Algebraic Equations - [[MATH307Su25 - Physics, AI, and Interactive Game Development]] - You'll step into the role of both physicist and game designer as you build a tank-targeting game from the ground up, starting with the mathematical modeling of projectile motion. After deriving the equations that govern every shot, you'll implement a text-based game where players adjust launch velocity and angle to hit randomly placed targets—discovering firsthand how small parameter changes affect outcomes. With AI collaboration, you'll animate your game and design a creative feature of your own, documenting your prompting strategies and iterative development process. This project connects elegant mathematics to engaging user experiences, revealing how human-AI partnerships are transforming scientific computing, engineering, and interactive entertainment. - [[MATH307Su25 - Root Finding Methods and the Mathematics of Vibrating Membranes]] - You'll explore how numerical root-finding algorithms—like Newton's method, Secant, and Bisection—reveal the hidden structure of vibrating systems, from guitar strings to your own eardrum. Starting with convergence analysis and error tracking, you'll apply your methods to the zeros of Bessel functions, which govern the resonant frequencies of circular membranes. With AI-assisted visualizations, you'll watch your algorithms converge in real time and connect each mathematical root to a physical vibration mode. This project bridges numerical analysis and acoustics, showing how the mathematics of roots underpins everything from musical instruments to medical imaging and architectural acoustics. - **Lagrange multipliers with/ radio telescope** ### Misc - **Machine number line (some data - w/ videos)** - Rates of convergence and superconvergence ### Student Created Project Resources - Stochastic matrices and the [Gambler's Ruin](https://en.wikipedia.org/wiki/Gambler%27s_ruin) - [[MATH310_WS_StochasticDynamics_F22_v1.0.pdf]] - See section 1 for Markov chain work/examples - Nice webpage w/ python code windows and concept development - https://web.pdx.edu/~gjay/teaching/mth271_2020/html/13_GamblersRuin.html - Simulation and analysis of random walks on $\mathbb{Z}$: Gambler’s Ruin - https://github.com/scottstrong/Introduction-to-Mathematical-Modeling/tree/2e8fe3e88a4f0c897d938328b3d04f642aca4618/Fall2022