2017 SYLLABUS
PMU 199H1S LEC 0291 Aha! Mathematical Discovery and Creative Problem Solving
Week; topics from Burger and Starbird's "The Heart of Mathematics" 4th Ed.
COUNTING
1 S2.1 The pigeonhole principle, estimation, and quantitative reasoning:
``Are two non-bald people alive with the same number of body hairs''
2 S2.3, 2.6-2.7 Primes, rationals, irrationals and real numbers
INFINITY
3 S3.1-3.3 `The buddy system:' uncountability of the the irrationals
GEOMETRY AND TOPOLOGY
4 S5.1-5.2 Topological equivaluence; Mobius strips; classifying surfaces
5 S4.5, 6.2 Platonic solids and the Euler characteristic proof
that there are only five. (S5.4 in 3rd Ed.)
ENUMERATIVE GEOMETRY AND COMBINATORICS
6 Cut Plane: "Space can be divided into two regions by a single plane,
four regions by a pair of planes, and eight regions by using three planes.
What is the maximum number of regions that space can be divided into by
using k planes?"
OPTIMIZATION
7 The spider and the ant: "A spider and an ant occupy a 12 x 12 x 24 room.
If the spider is in one corner of the room, where should the ant position
himself to maximize his crawling distance from the spider? The opposite
corner is a an obvious guess, and is the farthest point away as the crow
flies. But a spider is not a crow..."
FRACTALS
8 S7.1-7.5 Iterated maps; Cantor middle thirds set; fractal dimension
Complex numbers; complex dynamics; Julia and Mandelbrot sets
PROBABILITY AND STATISTICS
9 S8.1-8.2 The Monte Haul problem: Let's make a deal
10 S8.4-8.5 Coin tossing experiments, probability and risk, Bayesian inference
11 S2.2 Patterns and proofs
DECISION THEORY AND SOCIAL CHOICE
12 S10.5 Fair-allocation of scarce resources: envy-free divisions