Mathematics | Intuitive Topology
M321 | 27930 | C. Livingston
What is "M321: Intuitive Topology" about? (3 cr) M212 or consent of
instructor. One topic we will cover is map coloring. If you take an
outline map of the United States, then by using only four colors, you
can color in the states so that no states that share a border are the
same color. The same is true for any map, a result called the
Four-Color Theorem. We will study this result, seeing how it can be
stated in formal mathematics, and then we will extend it to a
different topological setting, proving that any map that can be drawn
on a "torus," the surface of a donut, can be "seven colored."
Another topic we will study is "Knot Theory." Everyone has dealt with
knots before; we will consider ways in which mathematics can be
rigorously applied to study knotting and linking. We'll also explore
surface theory, for instance exploring the strange world of
"non-orientable" surfaces, starting with paper, tape, and scissors, to
understand Mobius bands better; from there we will prove the
"classification of surfaces."
Why is the course called intuitive? Our goal is to understand some of
the fascinating topics that are encompassed by topology, and to get
there we will sometimes bypass technical foundational results, leaving
those to advanced courses. We will be doing plenty of rigorous
theorem proving; for instance, topological results offer great
examples in which to learn and build skills with proof by induction.
As a text for the course I have selected, "Knots and Surfaces," by
Farmer and Stanford, an inexpensive discovery style book published by
the Math Association of America. We will also use a text I wrote,
"Knot Theory," which I can make available to students at a highly
discounted rate. If you have any questions about the course, please
don't hesitate to drop me an email.