Modern Models in the Social and Biological Sciences Dan Teague NC School of Science and Mathematics

New Models/New Mathematics 1985 – 2015 – 2045 Why is the content (and intent) of the mathematics curriculum what it is? If the mathematics curriculum is designed to focus on the methods and techniques for the current models of our world, will the curriculum change when those models change?

New Models/New Mathematics 1985 – 2015 – 2045 Why are Latin and Mathematics the only subjects which appear to have been unaffected by developments in the 20 th century? Which good mathematics students are being left out of the math pipeline, and why do they leave?

What was Hot in 1985?

What Happened to Fractals?

Mandelbrot’s Goal “In the whole of science, the whole of mathematics, smoothness was everything. What I did was to open up roughness for investigation.” Fractal Geometry is the true geometry of the natural world.

Air view of Nördlingen, an old walled city on the "Romantische Strasse" (Romantic Road) in Southern Germany.

Alpo Dog Food An organism’s metabolic rate is the rate at which its cells convert nutrients to energy. The organism gives off heat at the same rate as a by product.

Max Rubner’s Suface Hypothesis (1880’s) In order to safely radiate the heat that is generated in the metabolic process through the boundary (skin), the metabolic rate should scale with body mass in the same way as volume and surface area. So, the metabolic rate should scale with body mass to the 2/3 power. Hence, the Alpo bag.

Kleiber’s Law (1937)

Space Filling Curve

Brown, Enquist, West Metabolic Scaling Theory

Life? Hunting the Hidden Dimension

Game of Life Cellular Automata led to Mathematica… then what?

Cellular Automata Peer Pressure Each agent (cell) looks at their neighbors and decides what to do based on what their neighbors have decided to do. This simple idea has led to a new and important area of mathematical modeling known as agent based models.

2-Dimensional Cellular Automata “Standard” Neighborhoods

Schelling Segregation Model David Batten, Discovering Artificial Economies, Westview Press, 2000.

The Cascade Begins

Thomas Schelling (2005 Nobel in Economics)

Phase Transition

Tipping Points and Leadership

I will if you will cascades… Requires diversity of threshold values in the individual agents. Imagine if everyone has a threshold of 4, what would happen. You need at least one person with a threshold of 1, another with a threshold of 2, and several with a threshold of 3.

I will if you will… Revolutions - Moral Mondays Acceptance of Innovations (Smart Phones, Electric Autos) (advantages of owning the first fax machine) Local Influence in TI vs Casio

Local Decisions Create Emergent Behavior Traffic Economy Immune System Consciousness Ant Colony Behavior More is Different

Classical DE Models

Theoretical vs Actual

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Agent-Based Predator-Prey Model

Network as Cellular Automata

Network Model If every vertex connects to every other vertex, then we have the classical Mass Action or Mean Field model.

Fundamental Fact of Life in a Social Network Your friends have more friends than you do.

Theoretical Results (YFHMFTYD)

Actual Results

Network Disease Models

Network Medicine Network of all known human diseases

Network Dynamics Network Structure Dynamics on Networks Dynamics of Networks

If the Models Change, Does the Curriculum Follow?

Wrong Optimization Model Teach as much as you can of subject A because this may be the last math course they take. Teach subject A in such a way that it is followed by B and C.

Modern Models in the Social and Biological Sciences Dan Teague NC School of Science and Mathematics