Math 510, Introduction to the Foundations of Geometry
(5:35-6:50 PM MW in C-8)


Syllabus: Here is the syllabus.

Blackboard: This class is also on blackboard.

Homework: The homework is due at 7 PM on the due date in the envelope on my office door.

Assigned
 Due
 Read
 Exercises
1/22
Chs. 0-2
1/29
 2/5
 Lecture notes
(copies of these notes are available on my office door for your convenience)
 Prove Theorems 2, 3b-d, and 5.
Hint: On Thm 5, try to construct a similar argument to the proof of Thm 4.
2/5
Lecture notes No new exercises today. New HW will be assigned Wed.
2/7
 2/14
 Lecture notes, Ch. 3
 Prove Theorem 6 and Lemmas 3-5 of the lecture notes.
2/15
 2/21
 Lecture notes, Ch. 3 Finish the proof that our definition of line in Rn satisfies Axiom 3. You need to show k is a subset of l. See around the middle of p. 5 in the lecture notes, where it says "Now let X be any point in k." We proved that C and D are in l so you can give an argument analogous to the one we used in showing that l is a subset of k.
2/21
Lecture notes No new exercises today. Use the time to prepare for the exam. New homework will be assigned on 2/28.
2/28
Lecture notes Still no new exercises. New homework will be assigned on 3/5 once we cover some new material.
3/5
 3/12
 Lecture notes
 Prove that Axiom 9 holds in Rn. Here is a sketch of the proof: Use the fact that Theorem 3(d) holds in Rn (explain why). Use Axiom 7 to list the three possible configurations for A,B,D. Eliminate BAD using proof by contradiction. Suppose BAD is true. You know ABC. Now show that if BAD and ABC then CAD. Look at the proof that Axiom 7 holds in Rn for ideas and consider several cases. Notice that CAD contradicts ADC by Theorem 1.

You don't have to follow the above sketch and are welcome to come up with your own proof. But don't wait until the last moment because it will probably take you some time to develop the intuition needed for the right argument. There is a good chance you can't just sit down and crank out this argument. It may take a few attempts. Don't give up easily. Draw pictures for inspiration.
3/12
 3/21
 Lecture notes Prove Theorems 17 and 18
3/21
 4/4
 Lecture notes, Ch. 3 Prove Theorems 19.(b), (c) and 22
4/4
 4/11
 Lecture notes, Ch. 5
 Prove Theorems 25 and 26
4/11
Lecture notes, Ch. 5
 No new homework. Prepare for exam on 4/16.
4/18
Ch. 6
 No more homework. Work on your presentations.

Here is a link to the GPS Visualizer. This will allow you too see what a shortest path (i.e. an arc of a great circle) looks like on a map. You can even visualize the output using Google Earth if you have it installed. Try a path from Calexico (CXL) to New York (JFK) or to Budapest (BUD) or any other long path you want.

The problem of the fortnight: The Mathematics Department in San Diego posts a new fun problem every two weeks.  You can submit solutions to them by mail, or if you happen to go to SD, in person.  Winners receive prizes.

Exam solutions:
 Useful links: The links below lead to sites with encyclopedias of math terms.  You can use them to find definitions, examples, and some theorems.
Some scholarship opportunities for prospective teachers: