Polynomials, Dynamics, and Choice

Scott Crass is a professor of mathematics at California State University, Long Beach, where he created the Long Beach Project in Geometry and Symmetry. The project’s centerpiece is The Geometry Studio, where students explore math in experimental and perceptual ways. Advised by Peter Doyle, his Ph.D. thesis at UCSD was ‘Solving the Sextic by Iteration: A Complex Dynamical Approach’. His research interests involve blending the algebra and geometry induced by finite group actions on complex spaces, in an effort to discover and study symmetrical structures and associated dynamical systems. A prominent feature of his work involves using maps with symmetry in order to construct elegant algorithms that home in on a polynomial’s roots.

Section I. Polynomials: Symmetries and Solutions. 1. Solving Equations: A Fundamental Problem. 1.1. Polynomial Primer. 1.2. What Numbers Do We Use? 1.3. Roots And Coefficients. 2. What is Symmetry? 2.1. Mirrors and Reflections. 2.2. Mathematical Symmetry. 2.3. Exploring Geometric Symmetry. 2.4. Groups in the Abstract. 2.5. Posing and Solving Problems with Symmetry. 2.6. Structure in the Abstract. 2.7. A look at Higher Dimensions. 2.8. What is Geometry? 2.9. Molecular Symmetry. 2.10. Conservation Laws. 2.11. Thermodynamic Systems. 3. Geometry of Choice: Symmetry’s Cost. 3.1. Spaces Where the Roots Live. 3.2. Shuffling Roots and Solving Equations. 4. Compute First, Then Choose. 4.1. Simplifying a Polynomial. 4.2. Solutions from a Formula and a Choice. 4.3. Reducing a Polynomial’s Symmetry. 4.4. What Goes Wrong. 5. Choose First, Then Compute. 5.1. A Line that becomes a Sphere. 5.2. Symmetrical Structures. 5.3. Fundamentals of Dynamics. 5.4. Dynamical Geometry and Symmetry. 5.5. Solving Equations by Iteration. Section II. Beyond Equation. Chapter 6. Interlude: Modeling Choice. 7. Learning to Choose. 7.1. Making Rational Decisions. 7.2. The Heart Has its Reasons. 7.3. Give Chance a Choice. 8. Choosing to Learn. 8.1. A Crowd Decides. 8.2. When in Doubt, Simulate. 8.3. Give Choice a Chance. 9. Conclusion. 9.1 Symmetry, More or Less. 9.2. Choosing As Metaphor. 9.3 Random Choice is Unavoidable.