We should review our day 6 program elements. - [[Teaching/MATH310/MATH310S26/Lecture Notes/MATH310S26-Day6-Work]] Note: The fifth graph in LSF#4 was made from an iterative map. Here are the associated eigenlines and first $\sigma$ ellipse ![[Henon.png]] ## Common guidance ### Scope - Start with a 1–2 week literature/idea sweep, then narrow to a single testable, semester-sized question with clear hypotheses, success metrics, milestones, and feasibility constraints. See: [Scientific method](https://en.wikipedia.org/wiki/Scientific_method), [Hypothesis](https://en.wikipedia.org/wiki/Hypothesis), [Operationalization](https://en.wikipedia.org/wiki/Operationalization). ### Symbolic/problem definition - Define variables, parameters, states, and observables in words and symbols; select a mathematical backbone early (e.g., [ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation), [partial differential equations](https://en.wikipedia.org/wiki/Partial_differential_equation), [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process), [graph theory](https://en.wikipedia.org/wiki/Graph_theory), [optimization](https://en.wikipedia.org/wiki/Mathematical_optimization), [statistical inference](https://en.wikipedia.org/wiki/Statistical_inference)), and state objectives/constraints explicitly. - [Stochastic Oscillators](https://drive.google.com/file/d/1MDFDDnMjTjBJTFt8wOUE83nvg5QbCp20/view) ### Data acquisition - List candidate datasets or experimental designs up front, note licenses/access steps, and predefine metrics/cutoffs/scales for measurement and evaluation; include immediate cross-project actions like sharing climate resources, Dartmouth personal-signals data, metronome synchronization papers, derivatives/stochastic finance materials, graph-theory references, and tone/timbre resources. See: [Data collection](https://en.wikipedia.org/wiki/Data_collection), [Measurement](https://en.wikipedia.org/wiki/Measurement), [Evaluation](https://en.wikipedia.org/wiki/Evaluation). ## Finance, behavioral economics, and stochastic markets ### Behavioral Finance Decisions Under Stress #### Original responses - Understand how external events and personal circumstances impact financial decisions - Study effects of natural disasters or recessions on money-related choices - Analyze how personality, values, and religious affiliations influence financial behavior - Use computational data analysis with real values, cutoffs, scales, and metrics #### Summary of guidance - Scope to measurable behavioral effects within a concrete pricing/decision framework; connect to [behavioral economics](https://en.wikipedia.org/wiki/Behavioral_economics) via [prospect theory](https://en.wikipedia.org/wiki/Prospect_theory), [loss aversion](https://en.wikipedia.org/wiki/Loss_aversion), disaster-driven [market sentiment](https://en.wikipedia.org/wiki/Market_sentiment), and [risk perception](https://en.wikipedia.org/wiki/Risk_perception); specify datasets, metrics, and an empirical design. Consider interfaces with [derivative (finance)](https://en.wikipedia.org/wiki/Derivative_(finance)) contexts. - Scott on 2/2 @ 4pm: Now is a pretty interesting time. Is silver volatility driven by speculation or by fundamentals? ### Stochastic Dynamics for Finance and Biology #### Original responses - Study random/stochastic behavior, dynamics, and uncertainty systems - Focus on financial markets and quantitative risk modeling - Explore biological transport models, particle interaction systems, and network dynamics - Work on finance problems involving time evolution, feedback, and randomness (price formation, optimal execution, volatility dynamics) - Use stochastic modeling, Monte Carlo methods, numerical methods for differential equations, optimization, and statistical estimation - Build and test models rather than only theoretical derivations #### Summary of guidance - Use the [random walk](https://en.wikipedia.org/wiki/Random_walk) → [geometric Brownian motion](https://en.wikipedia.org/wiki/Geometric_Brownian_motion) scaffold and its [heat equation](https://en.wikipedia.org/wiki/Heat_equation) link; prototype [Monte Carlo methods](https://en.wikipedia.org/wiki/Monte_Carlo_method) and calibrate on market data while noting parallels to [biological oscillators](https://en.wikipedia.org/wiki/Biological_oscillator); emphasize iterative build-and-test with [numerical methods for DEs](https://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations) and [optimization](https://en.wikipedia.org/wiki/Mathematical_optimization). - Stochasticity - [Stochastic Oscillators](https://drive.google.com/file/d/1MDFDDnMjTjBJTFt8wOUE83nvg5QbCp20/view) - Juan MR Parrondo: Spanish Physicist (So the lectures aren't as rigorious as we might like, but they do get to some important points about building up stochastic ODE. We will study random walks, which underpins this type of thinking.) - [Lesson 6 (1/5). Stochastic differential equations. Part 1](https://youtu.be/vcGpD7nZ3UM?si=hTXH-SJuX-K95A-A), [2of5](https://youtu.be/vRsfjcXBGuI?si=YoR1WHuTr8OZ4UiW), [3of5](https://youtu.be/9zfw_CoPYNE?si=R19zadjbq2UyqkBT), [4of5](https://youtu.be/h1eNpKDOa2c?si=aRax0FO4rz14Drpu), [5of5](https://youtu.be/7J82tcLynaU?si=3LM5Sq2Mnv2GIkx-) - I think that I have notes for the first three. - Finance - [An Introduction to Mathematical Finance with Applications : Understanding and Building Financial Intuition, Petters and Dong (2016)](https://mines.primo.exlibrisgroup.com/permalink/01COLSCHL_INST/1jb8klt/alma997357550602341) - Chapters 1 and 2 give a good introduction to vocabulary and chapter 5 talks about binomial trees and pricing options. - [MCM 2022 PROBLEM C: Trading Strategies](https://www.comap.org/membership/member-resources/item/trading-strategies) - Link to a math modeling contest where you were asked to hold gold and bitcoin and develop a strategy to optimize profit. - [2022 Mathematical Contest In Modeling: Trading Strategies - Solution Attempt](https://engineerweijiezhang.com/portfolio/2022-mcm-trading-strategies/) ### Multi-Factor Finance and Health Crossovers #### Original responses - Study finance contexts including human behavior in markets, inflation rates, and multiple economic factors - Explore health and biological functions (inspired by mathematical biology) - Use regression analysis and run multiple simulations - Prefer data analysis over computation over symbolic approaches #### Summary of guidance - Pair multi-factor [regression analysis](https://en.wikipedia.org/wiki/Regression_analysis)/simulation with the finance cluster; align variables, data sources, and validation plans; compare behavior-informed factor models across datasets. ## Games and strategy analytics ### Route Planning and Risk in Route-Building Games #### Original responses - Deep dive into Ticket to Ride board games to find patterns and graphical representations - Look into other games for analysis - Use data analysis and symbolics with openness to simulations #### Summary of guidance - Ground in [graph theory](https://en.wikipedia.org/wiki/Graph_theory) with the [shortest path problem](https://en.wikipedia.org/wiki/Shortest_path_problem) and [network flow](https://en.wikipedia.org/wiki/Maximum_flow_problem), card-draw [probability](https://en.wikipedia.org/wiki/Probability), and multiplayer [game theory](https://en.wikipedia.org/wiki/Game_theory); pilot [Monte Carlo](https://en.wikipedia.org/wiki/Monte_Carlo_method) strategy evaluation and consider simpler exemplar games like [Hex](https://en.wikipedia.org/wiki/Hex_(board_game)) or [Nim](https://en.wikipedia.org/wiki/Nim) to cement theory. - [Markov Chains and the Game Monopoly](https://www.taylorfrancis.com/chapters/edit/10.1201/9781003092872-17/markov-chains-game-monopoly-j%C3%B6rg-bewersdorff) - [ GPT 5.2 vs Claude Opus 4.5 vs Grok 4.1 vs Gemini 3 Pro - AI Plays Monopoly](https://youtu.be/8agmhNyvmcs?si=vY0MXAi0GzQltSSO) ### Team Matchmaking and Win-Margin Prediction #### Original responses - Create a game match maker that predicts which team wins and by how much - Develop better scoring methods for team performance (currently using points won ratio) - Find analytical methods for scoring functions to minimize manual tweaking - Work on precision issues in best 2/3 game modes #### Summary of guidance - Specify the game/domain; use [Glicko rating system](https://en.wikipedia.org/wiki/Glicko_rating_system)/[TrueSkill](https://en.wikipedia.org/wiki/TrueSkill) baselines, simulate best-of-k series for variance/precision, fit margin models, and compare analytical vs learned scoring with validation on simulations or real data. ## Synchronization, oscillations, chaos, and fluids ### Simple Harmonic Motion to Metronome Synchrony #### Original responses - Study classical mechanics problems involving springs and pendulums for simple harmonic motion - Use computation and simulations for mathematical modeling #### Summary of guidance - Bridge [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion) to weakly coupled metronome synchronization (see [synchronization of coupled oscillators](https://en.wikipedia.org/wiki/Synchronization_of_coupled_oscillators)); identify parameters enabling synchrony, derive simplified coupled-oscillator models, and compare simulations to tabletop experiments. - Understanding of nonlinear oscillators: It turns out that a nonlinear alteration to a mass-spring equation is difficult with calculus related to fundamental problems in mathematics. My advice is: - Understand the derivation of the mass-spring equation. From it derive conservation of energy and relate curves on its Hamiltonian energy surface to phase space trajectories. - Introduce the idea of hard/soft springs, i.e., Duffing equation, and discuss the phase space trajectories from the associated Hamiltonian energy surface. - Using a perturbative approach, show that the nonlinearity forces the amplitude and frequency to couple. - Understand the idea of hysteresis and from this, convince yourself that the Duffing system is capable of hysteresis. - Define a context for nonlinear oscillators and try to port over these ideas to make predictions about the oscillator. - For example, a seasonally driven Stommel's Box Model one might find interesting outcomes in the nonlinear oscillator that is the Atlantic meridional overturning circulation - [Synchronization of metronomes, Pantaleone (2002)](https://www.uaa.alaska.edu/academics/college-of-arts-and-sciences/departments/physics-and-astronomy/_documents/metro.pdf) - [Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales, Goldsztein et al., (2021)](https://pubs.aip.org/aip/cha/article-abstract/31/2/023109/1078905/Synchronization-of-clocks-and-metronomes-A?redirectedFrom=fulltext) ### From Firefly Synchrony to Turbulence and Noise-Induced Order #### Original responses - Study fireflies synchronizing their flashes - Analyze coupled chaotic oscillators that shouldn't sync but sometimes do - Model transition from smooth flow to turbulent flow - Understand why vortices form in turbulent fluids - Study energy transfer from large to small swirls in turbulence - Explore partial synchronization phenomena - Analyze chaos emerging from simple equations - Study how noise can create order instead of destroying it - Compare local vs global interactions in systems - Understand why systems suddenly "snap" from ordered to chaotic behavior - Prefer symbolic over simulation over computational over data analysis approaches #### Summary of guidance - Split efforts into (A) ODE-based synchronization/chaos (e.g., [Kuramoto model](https://en.wikipedia.org/wiki/Kuramoto_model), coupling topologies, [noise](https://en.wikipedia.org/wiki/Noise_(signal_processing))) and (B) select fluid phenomena via minimal vorticity/instability models; map [bifurcations](https://en.wikipedia.org/wiki/Bifurcation_theory) and [phase transitions](https://en.wikipedia.org/wiki/Phase_transition). For fluids see [turbulence](https://en.wikipedia.org/wiki/Turbulence), [vortex](https://en.wikipedia.org/wiki/Vortex), and [energy cascade](https://en.wikipedia.org/wiki/Energy_cascade); for noise-induced order, see [stochastic resonance](https://en.wikipedia.org/wiki/Stochastic_resonance). ## Sound, music, voice, and instrument classification ### Speech and Sound-to-Structure Translation #### Original responses - Study music or sound waves - Model how technology translates human voices and sound waves into understandable formats - Prefer data analysis over computation over simulation #### Summary of guidance - Start with sound/voice fundamentals; build feature pipelines ([spectrogram](https://en.wikipedia.org/wiki/Spectrogram), [MFCC](https://en.wikipedia.org/wiki/Mel-frequency_cepstrum), [formant](https://en.wikipedia.org/wiki/Formant)), test simple ML for [automatic speech recognition](https://en.wikipedia.org/wiki/Automatic_speech_recognition), and leverage statistical patterns of phonemes before scaling complexity. - Understanding acoustics with a PDE background: - [Applied Partial Differential Equations, Logan (2015)](https://mines.primo.exlibrisgroup.com/permalink/01COLSCHL_INST/fmq2oa/cdi_askewsholts_vlebooks_9783319124933) - Chapter 1.5 on vibrations and acoustics derives the wave equation for small deviations to the pressure field, i.e., acoustic waves. (Nonlinear corrections are needed for sonic booms.) - If you are far away from the sound, then it is treated as a point source of a spherical wave, and it can be shown that the amplitude of the wave decays inversely with respect to radius. This is why concerts use arrays of speakers. (See: [Why are speakers in concerts aligned in a vertical array?](https://www.reddit.com/r/askscience/comments/8r1vwg/why_are_speakers_in_concerts_aligned_in_a/)) - In terms of text-to-speech and speech-to-text, there is a lot of Fourier thinking at work, i.e., identify primitive parts through the Fourier transform and then reassemble different primitives by joining parts and inverting the Fourier transform. ### Instrument Timbre Classification via Spatial Acoustics #### Original responses - Work on instrument categorization - Explore classifiers and classification problems - Use data analysis and simulations - Express uncertainty about specific direction but interest in the tools #### Summary of guidance - Use nodal/vibrational modes (see [Chladni figures](https://en.wikipedia.org/wiki/Chladni_figure)) and the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) to connect structure to [timbre](https://en.wikipedia.org/wiki/Timbre); collect multi-microphone recordings to capture spatial inhomogeneity; expand features beyond frequency and compare classifier families. - [Dan Russel Animations](https://www.acs.psu.edu/drussell/demos.html) (Q: How can modal analysis help in instrument classification?) - [Modal Analysis of an Acoustic Folk Guitar](https://www.acs.psu.edu/drussell/guitars/hummingbird.html) - [Modal Analysis of an Electric Guitar](https://www.acs.psu.edu/drussell/guitars/electric.html) - [What are harmonics?](https://www.youtube.com/watch?v=znbfY-tXROk) (Q: Can the data within the Fourier transform for audio help with instrument classification?) - [Standing waves on the "A" string of a guitar: animation and frequency of harmonics.](https://www.youtube.com/hashtag/shorts/shorts)](https://www.youtube.com/shorts/fzo8gcmcds8) ## Climate and environmental data ### Climate Signals: CO2, Ice, and Weather Variability #### Original responses - Study climate, weather patterns, sea ice/glaciers, and CO2 levels in the atmosphere - Work with the Keeling Curve and climate data - Use statistical and data analysis approaches - Experiment with simulations and differential equations (especially PDEs) #### Summary of guidance - Anchor on accessible datasets like the [Keeling Curve](https://en.wikipedia.org/wiki/Keeling_Curve) and [sea ice](https://en.wikipedia.org/wiki/Sea_ice) indices with a focused question (trend/seasonality decomposition, lag/lead); add simple mechanistic pieces as needed and define evaluation metrics early. - [Mathematics and Climate, Kelpler and Engler](https://epubs.siam.org/doi/10.1137/1.9781611972610) (We have copies and/or can get more.) - [Arthur Lakes Library Holding](https://mines.primo.exlibrisgroup.com/permalink/01COLSCHL_INST/1jb8klt/alma996670653502341) -Consider looking through chapters 8(Climate and Stats) and 9 (regression), with the goal of getting to chapter 10 (Mauna Loa CO2 Data), which would give some good data analysis. - [Mathematics and Climate Research Network](https://sites.google.com/view/math-climate/home) - [Climate Mathematics: Theory and Applications](https://climatemathematics.sdsu.edu/toc.html) - [Youtube Channel For Text](https://www.youtube.com/@climatemathematicstheoryan6848/videos) - The topic of PDE with respect to climate is hard. It turns out that fluid mechanics occurring on a shell of a rotating heated sphere is wrought with difficulties. That said, I do have materials that could be useful, but it would require some chatting so that we get the right scope. - Something that might be interesting and accessible is understanding the somewhat recent work [Rossby waves and extreme weather](https://youtu.be/MzW5Isbv2A0?si=bXoWq-lsJd1E8jXp). ## Personal and biological signals ### Quantified-Self and Habit Prediction Across Platforms #### Original responses - Model biological signals - Analyze personal data from Google, Spotify, and other platforms to identify yearly/monthly trends - Use Apple Watch data (sleep, heart rate, steps) for daily scale analysis - Combine multiple datasets to predict habits (e.g., walking/exercise as function of Spotify listening time) - Use statistical modeling and inference techniques - Create fun and interactive data visualizations #### Summary of guidance - Utilize a [quantified self](https://en.wikipedia.org/wiki/Quantified_self)-style personal-signals corpus and literature; define prediction targets and align time scales across sources (e.g., [Apple Watch](https://en.wikipedia.org/wiki/Apple_Watch) exports); preregister metrics and pair interactive visualization with inferential checks. - [The StudentLife study at Dartmouth College](https://studentlife.cs.dartmouth.edu/index.html) (Note: As of 2/2/26, 3pm, the Dartmouth links seem broken. [Here](https://web.archive.org/web/20240818122456/https://studentlife.cs.dartmouth.edu/index.html) is a connection through the [way back machine](https://web.archive.org/) and [here](https://web.archive.org/web/20240814173504/https://studentlife.cs.dartmouth.edu/findings.html) is some of their findings. ) - "In 2013, we initiated the original StudentLife study at Dartmouth College to capture the hidden stress and strain across a single academic term of 48 undergraduate students, using smartphones as human behavioral sensors (see results). In 2016, we conducted a study with 83 students over two terms using smartphones and wearables to investigate the dynamics of depression and anxiety (see results). In 2018, we tracked 200 high school students throughout their four years at Dartmouth College, which included the challenging period of the COVID-19 pandemic (see results)." ## Neuroscience, dreams, and mental health ### Brain Waves, Dreams, and Mental Health Modeling #### Original responses - Study mental illnesses (depression, anxiety, personality disorders) using brain waves or thought patterns - Analyze themes occurring within dreams and predict mental illness in others - Understand where dreams come from, their nature, and duration - Use simulations to model brain behavior and help understand the brain's intricacy #### Summary of guidance - Time-box the exploratory sweep, then lock measurable constructs (e.g., [electroencephalography (EEG)](https://en.wikipedia.org/wiki/Electroencephalography) bands, dream-theme codings, diagnostic proxies); choose tractable subsystems and ensure testability and scope fit. See background: [Dream](https://en.wikipedia.org/wiki/Dream), [Mental disorder](https://en.wikipedia.org/wiki/Mental_disorder). ## Language interaction and geometry in art ### Symmetry, Tessellation, and Aesthetic Structure #### Original responses - Study geometry in art, including symmetry and repeating patterns in designs #### Summary of guidance - Frame with [symmetry group](https://en.wikipedia.org/wiki/Symmetry_group) operations and [tessellation](https://en.wikipedia.org/wiki/Tessellation); define quantifiable pattern descriptors and targets for classification or generation; optionally reference [sacred geometry](https://en.wikipedia.org/wiki/Sacred_geometry) exemplars. ### Language Interaction in Bilingual Communities #### Original responses - Model how different languages interact over time, especially in bilingual populations - Use data analysis and computation for modeling #### Summary of guidance - Scope to household/community dynamics with [bilingualism](https://en.wikipedia.org/wiki/Bilingualism) and [language contact](https://en.wikipedia.org/wiki/Language_contact); specify transition models, observables, and evaluation criteria. ## Pop culture and social media analytics ### K-Pop Fan Economies and Parasocial Dynamics #### Original responses - Model K-pop fan culture, including "Celebrity Worship" tendencies - Study parasocial relationships and extreme investment (financial, time, consumption) - Analyze how K-pop companies market idols (perfect group ratios, diversity, English-speaking influences) - Create predictive models for trends, profitability, comeback timelines, and success factors - Study spatial data of K-pop fans vs Korean/Asian communities - Examine K-pop's relationship with TikTok/social media and globalization - Analyze idol-training systems and success rates - Study revenue sources (music vs social media vs endorsements) - Analyze K-pop survival shows (voting, screen time, editing effects, predictions) - Compare with American pop artists/groups #### Summary of guidance - Apply strict scoping to a data-backed, model-ready question; consider [survival analysis](https://en.wikipedia.org/wiki/Survival_analysis), [sentiment analysis](https://en.wikipedia.org/wiki/Sentiment_analysis), and [spatial analysis](https://en.wikipedia.org/wiki/Spatial_analysis); identify sources and a validation plan early. Background: [K-pop](https://en.wikipedia.org/wiki/K-pop), [Parasocial interaction](https://en.wikipedia.org/wiki/Parasocial_interaction). ## Geology and materials ML ### Predicting Rock Properties from Measurable Features #### Original responses - Study rock properties including hardness, density, color, and fracture patterns - Use neural networks to predict rock properties based on other known properties - Acknowledge inexperience with neural network creation and training #### Summary of guidance - Start with strong baselines ([random forest](https://en.wikipedia.org/wiki/Random_forest), [XGBoost](https://en.wikipedia.org/wiki/XGBoost)) on geological datasets (e.g., [USGS](https://en.wikipedia.org/wiki/United_States_Geological_Survey)); engineer features ([crystallography](https://en.wikipedia.org/wiki/Crystallography), composition, formation conditions), use careful [cross-validation](https://en.wikipedia.org/wiki/Cross-validation_(statistics)), and explore [neural networks](https://en.wikipedia.org/wiki/Artificial_neural_network)/[transfer learning](https://en.wikipedia.org/wiki/Transfer_learning) later. Background properties: [Mohs scale of mineral hardness](https://en.wikipedia.org/wiki/Mohs_scale_of_mineral_hardness), [Fracture (geology)](https://en.wikipedia.org/wiki/Fracture_(geology)). ## Learning and skill acquisition dynamics ### Sinusoidal Growth Patterns in Practice and Learning #### Original responses - Study human psychology of improvement and learning rates over time - Model growth patterns when practicing new skills using sinusoidal functions - Analyze improvement statistics that follow upward-trending sinusoidal patterns - Use data analysis and computation to describe trends, ratios, and periods of growth vs decline #### Summary of guidance - Select a tractable dataset and test [learning curve](https://en.wikipedia.org/wiki/Learning_curve) variants and sinusoid-plus-trend models against alternatives via time-series fits; define phases, periods, and metrics a priori to evaluate model fit and interpretability. - Scott @ 2/2/26: I was thinking about this more and wonder about depth and biology. For example, if we did the rule test, I'm sure that we would see diminishing returns as the biological resources for "focus" become depleted. At the same time, we have the idea of spaced repetition and its benefits for learning. An idea I have rolling around is: - Hypothesis: Individuals who are engaged with problems solved with strong hand-eye coordination skills are better able to adapt in other settings. - Plan: survey to get individual background, test, see if there are correlations.