Lecture Notes on Applied Mathematics and Thinking Strategies

Introduction

Speaker: Applied mathematician, visiting for the third time
Background: Formerly worked at Oxford University until 2005
Key message: Focus on understanding the world through mathematics, not just calculations

Ronald Fisher: Pioneer in applied mathematics and statistics
- Background: Arrogant, believed he was smarter than professors
- Contribution: Connected math to reality; worked on significant statistical experiments
- Example: Tea tasting experiment to test if milk first or tea first makes a difference
  - Experiment Methods:
    - Pairwise challenge (less effective)
    - Tray method (more effective; better discrimination)
  - Mathematical Insight: Probabilities and combinatorics used to show tray method is superior
- Legacy: Developed foundational principles in experimental design, still used today
Statistics in Sports:
- Example: Gary Neville's skepticism on measuring player's impact after a goal down
- Measure: Performance of players (e.g., Trent Alexander Arnold) and statistical significance
- Gary Neville Statistic: Used Fisher’s exact test to measure impact accurately
Limitations of Statistics:
- Small Effect Size: Example of grit in success (Angela Duckworth's study, 4% variance)
- Moral and Scientific Issues: Fisher’s support of eugenics and anti-smoking campaigns
- Key Points:
  - Doesn't provide all answers
  - Effect sizes might be small
  - Correlation ≠ causation

Key Figure: Alfred J. Locker, shifted from chemistry to mathematical biology
Model Example: Predator-prey dynamics (rabbits and foxes)
- Unbalanced Equations: Imagining reactions, leading to differential equations
- Interpreting Equations: Rate of change and causative factors
Applications:
- Social Interactions: Clap experiments, social recovery
- Fish Behavior: Modeling fish movements and escape waves
- Football: Marking territory and attacking strategies (e.g., Marcus Rashford's runs)
Limitations: Models often need more than just interactive elements, Lotka’s incomplete models

Margaret Hamilton: Pioneer in software engineering and chaos theory
- Background: Worked with Edward Lorenz on weather prediction models
- Butterfly Effect: Small changes lead to vastly different outcomes
- Demonstration: Exercises and cobweb diagrams
- Practical Application: Reliable software for Apollo moon mission
Key Insights:
- Embrace chaos, but control important aspects
- Balance between order (Yang) and disorder (Yin)

Simulation Example: Cellular automata and simple rules creating complex patterns
Hero: Andrei Kolmogorov
- Definition of Complexity: Length of the shortest description that captures detail and nuance
Implication: Capturing complexity can enhance understanding and modeling in mathematics