(hold your cursor over any image to learn more)
What should be the goals of a course entitled Advanced Topics in Mathematics that is taught to seniors?Answer:I have developed five main themes for this course. Here they are:
- It should expose students to what mathematics outside of a high school classroom setting really looks like.
- It should give to students something memorable about mathematics while engaging them in a meaningful way and showing them a different face of the diverse mathematical world.
    
- The theory of prime numbers
- Mersenne primes
- Abundant/deficient/perfect numbers
- Proof
- Pythagorean Triples
- Patterns in number theory
- Modulus Arithmetic
- Cryptography
- Bar Codes/2D Matrix Codes/Error Checking
- Assorted topics in number theory
    
- Linear Programming and its applications
- The mathematics of the Global Positioning System
- Structure of the GPS
- Systems of lines/circles/spheres
- Projections
- Satellite signals and technology
- Geography
- The shape of space
- Fermi Problems
/Chaos Theory and Applications
    
- Fractals defined
- Fractal dimension
- Expansion/Removal/Space-Filling fractals
- Fractals in nature
- Using Fractree software
- Chaos Theory
- Sensitive Dependence On Initial Conditions
- Iteration, seeds, orbits, fates
- The logistic equation
- Population Biology
- Chaos in nature
- Chaos in the Double Pendulum System
- Complex numbers/the complex plane
- The Mandelbrot Set
- Using Fractint software
    
- Topological equivalence
- Faces/edges/vertices/EulerŐs Formula
- Traversability
- 3-D topology
- Polyhedra
- Trees/Rooted Trees
- Kruskal's Algorithm
- Knot invariants
- Applications of Knot theory
- Biometrics
At the bottom of each of my web pages I have included suggestions for books on each subject that I have found useful and entertaining. Here are some that are general to mathematics:
Recommended Readings:![]() |