Math-Based Research Platform
Pattern
Research investigations, datasets, and interactive tools on how sector returns, audio spectrums, NBA shot selection, tumor growth curves, and human behavior are systems you can model.
Domain Relationship Map
Hover over a node to reveal connections between disciplines
5 Major Research Investigations
Five long-form papers, one per discipline. Each takes a real dataset and concrete question and follows the math through to an answer.
The Information Content of Earnings Call Language
A Sector-Level Sentiment Study of Predictive Signal in Management Discourse
Earnings call transcripts, mandated for U.S. public companies under SEC Regulation FD since 2000, constitute one of the largest structured corpora of deliberate corporate communication in existence, exceeding 80,000 transcripts per year. This paper asks whether the aggregate linguistic tone of management discourse across an entire GICS sector carries measurable predictive signal for subsequent sector-level capital flows and returns. Using the Loughran-McDonald financial sentiment lexicon applied to prepared remarks from S&P 500 constituents over 2018 to 2023, we construct a quarterly sector sentiment index and test its predictive relationship with next-quarter returns, realized volatility, and fund flows. We find that sector tone explains roughly 12 to 18 percent of subsequent quarter returns in Technology and Healthcare, that uncertainty language predicts elevated 30-day realized volatility with an R-squared of about 0.24, and that the signal is strongest in Q4, consistent with year-end reallocation. We situate these findings within the broader literature on text-based asset pricing and examine the practical implications for sector rotation strategies, comparing a time-series momentum overlay against passive buy-and-hold over 2000 to 2023.
Computational Approaches to Hindustani Raga Identification
Spectral Methods, Perceptual Timbre Spaces, and the Limits of Discrete Representation
Hindustani classical music builds melody around the raga, a framework that fixes a set of pitch classes together with the rules for using them: which notes to ascend and descend through, which to dwell on, how to ornament them, and, by long tradition, when in the day they should be heard. A trained ear can tell two ragas apart from these rules alone, even when their underlying notes overlap. This paper asks whether that distinctiveness is also recoverable from the raw audio signal, and which spectral features carry the most discriminative information. Using chroma and timbral features extracted from the Dunya (Compmusic) corpus and the University of Iowa instrument samples, we find that chroma-CQT features achieve about 71 percent accuracy in 10-class raga identification, that ragas sharing a parent scale remain separable, and that timbral features alone perform poorly. We extend the analysis to the perceptual timbre space recovered by multidimensional scaling of spectral features, confirming that computable acoustic features reconstruct the psychoacoustic dimensions of brightness and attack. We close by modeling jazz improvisation as a dynamical system via recurrence quantification analysis, finding that experienced improvisers exhibit higher determinism, evidence that phrase-level coherence is a measurable structural property.
The Three-Point Revolution and the Geometry of Efficient Offense
Expected Value, Shot Selection Optimization, and the Mathematics of Spacing
Between the 2012-13 and 2022-23 NBA seasons, three-point attempts per game increased from roughly 18.4 to 35.1, while mid-range two-point attempts fell by more than 40 percent, among the most dramatic tactical shifts in the history of any major professional sport. This paper argues that the revolution is explicable almost entirely through the lens of expected value, and that its lagged adoption reflects organizational inertia rather than informational gaps. Using NBA shot log data from 2001-02 to 2022-23, we show that expected points per attempt from three first durably exceeded mid-range at the league level in 2010-11, with mass adoption following a two to three year lag consistent with roster construction constraints. We then construct the full expected-value landscape of the shot chart, solve the constrained shot-selection optimization problem under defensive response, and quantify the contribution of offensive spacing to scoring efficiency. We find that the optimal shot mix is approximately 35 percent corner and wing threes, 45 percent restricted area, and 20 percent all else, and that teams in the top quartile of spacing score 4.7 more points per 100 possessions.
Mathematical Models of Growth and Spread
From Tumor Kinetics and Epidemic Dynamics to Predator-Prey Oscillation
Living systems grow and spread according to laws that, while generated by complex biological machinery, are often well approximated by compact differential equations. This paper unifies three quantitative investigations of biological dynamics: the growth law of tumors, the trajectory of epidemics, and the oscillation of predator-prey systems. Using tumor volume data from Benzekry et al. (2014), COVID-19 case counts from the JHU CSSE repository, and the Hudson's Bay Company pelt records, we fit and compare classical ordinary differential equation models. We find that the Gompertz model achieves the lowest AICc in roughly 67 percent of tumor series, that the basic SIR model predicts peak timing within 8 days for three of four countries but systematically overestimates peak height by 40 to 120 percent until corrected by time-varying transmission, and that the Holling Type II predator-prey model produces 73 percent correlation with the Hudson's Bay data versus 61 percent for the classical Lotka-Volterra system. The unifying theme is that simple mechanistic models capture biological dynamics with surprising accuracy, but that their failures are systematic and informative, pointing to the specific biological complexities (angiogenesis, behavioral change, predator satiation) that the simplest models omit.
Calibration, Bias, and Rational Belief Updating in Human Forecasting
Overconfidence, Bayesian Heuristics, and the Optimization of Risk and Reward
Human probability judgment is not a noisy approximation of Bayesian reasoning but is systematically distorted by identifiable cognitive shortcuts. This paper unifies three quantitative investigations of decision under uncertainty: the calibration of probability forecasts, the heuristics that approximate Bayesian belief updating, and the optimization of risk and reward under the mean-variance framework. Using the Good Judgment Project dataset of hundreds of thousands of geopolitical forecasts, base-rate problems from the experimental literature, and portfolio data from 2005 to 2023, we find that median forecasters exhibit substantial overconfidence (events assigned 90 percent probability occur roughly 76 percent of the time) while superforecasters are near-perfectly calibrated (slope of about 0.96), that base-rate neglect occurs in roughly 74 percent of participants when case-specific information is provided but a simple rounding heuristic recovers most of the lost accuracy, and that mean-variance optimization generalizes beyond finance but is consistently undermined by estimation error. The unifying theme is that human decision under uncertainty is systematically suboptimal in predictable ways, and that simple, well-chosen corrections (training, rounding, shrinkage estimation) recover much of the lost performance.
5 Domains Connected By Math
Finance
Markets, earnings, portfolio theory, prediction, and risk. We investigate how mathematical models explain price formation, volatility clustering, and the limits of forecastability in financial time series.
Music
Signal processing, acoustics, Fourier analysis, and harmonic structure. We treat music as a mathematical object — sequences, spectral functions, and dynamical systems that encode cultural meaning in physical form.
Basketball
Shot optimization, efficiency metrics, lineup analysis, and game theory. Basketball has become one of the most data-rich sports, offering a real-time laboratory for optimization and spatial statistics.
Biology
Tumor growth modeling, population dynamics, genetics, and epidemiology. Mathematical biology has moved from metaphor to mechanism — we show how differential equations govern living systems.
Decision Science
Behavioral economics, probability, forecasting, and optimization. We study how humans actually make decisions, where they systematically deviate from rationality, and what mathematics prescribes instead.
20 Investigations
Each investigation starts with a real dataset and a concrete question. The analysis runs through models and visuals to wherever the data leads.
15 Documented Datasets
Every dataset links to a public source you can check yourself. Nothing is invented, and every variable comes with a method and a caveat.
5 Interactive Tools
Calculators that do real math on real data: sector-portfolio backtesting, harmonic decomposition, NBA shot efficiency, growth-curve fitting, and Bayesian updating.