Math 645: A Survey of Mathematical Problems I
Frank Sottile

• This week begins a four week dive into graph theory, which is a fairly large part of mathematics, particularly discrete mathematics that people do not typically see in an undergraduate program. A graph is a mathematical object that is used to model connectivity. Graphs were at the beginnings of topology, and are ceratinly on the minds of the people who design, build, and study the internet.

  In "Dots and Lines", do pay attention to the Preface (it is included for a reason), and let your mind rest on Russell's paradox; it is both important and subtle. While it was not the first such result in mathematics that indicated limits to our knowledge (look up the story of the parallel postulate), it is the first of several results in mathematics and physics in the 20th century that were counterintuitive and pointed to limitations in what is possible to know. It is also one of several paradoxes in mathematics; each of you will study you will study one such paradox in depth later in the semester for your term paper.

  Newman's reading (I think) reproduces an English translation of the great Leonhard Euler's (pronounced 'Oiler') discussion on the Bridges of Königsberg (a city that was obliterated in WWII and is now called Kaliningrad. That city was the birthplace of both David Hilbert and Hermann Minkowski, two great German mathematicans around 1900. Euler's solution of the Königsberg bridge problem was an important beginning of graph theory, and of topology.

  Tucker's piece uses graph theory to solve a simple (and on its face silly) problem. It is a good illustration of how graphs, while apparently completely unstructured, possess some structure that may be exploited to solve problems, not all of which are so silly.

• Make sure to have read the selection from "Dots and Lines" by Trudeau, number 6 in the course packet. This is foundational for our work on graph theory.
• Read number 7 in the course packet, by Newman. Skip #8, for now.
• Read number 9, by Alan Tucker.
• Chapter 3 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 4.

1. Exercises 3.1, 3.2, and 3.3, all of which are on page 30 of A Survey of Mathematical Problems. (Do give a proof of Exercise 3.3.)
2. Problems 3.1, 3.3, and 3.4. For 3.3, do use 3.2, and appreciate its proof in Dr. Geller's Lecture notes.