Lectures

Week 1:

[Jan 18, 2011] Lecture 1 - Mesopotomian and Egyptian Mathematics.

- For a more historical viewpoint, see also Math 629, in particular this chapter.

[Jan 20, 2011] Lecture 2 - Greek Mathematics

- See also Boas&Geller Chapter 6, pages 57-65, for a discussion on what kinds of numbers can be constructed via a straightedge and compass.

Week 2:

[Jan 25, 2011] Lecture 3 - Axiomatic Systems

- See also Boas&Geller Chapter 8, Pages 73-78

[Jan 27, 2011] Lecture 4 - Axiomatic Systems (cont'd)

Goedel's Incompleteness Theorem, Boas & Geller page 78.
- Sketch of a proof via Cantor's diagonalization process.

Week 3:

[Feb 1, 2011] Lecture 5 - Limits, part I. Chapter 9, Boas & Geller.

- Dealing with infinity. Discussion of early concepts of limits (geometrical, philosophical).
- Making limits precise.

[Feb 3, 2011] Lecture 6 - Limits, part II. Chapter 9, Boas & Geller.

- Geometrical Examples
- Recursion
- Iterated Functions
- Dynamical Systems

Week 4:

[Feb 8, 2011] Lecture 7 - Limits, part III. Chapter 9, Boas & Geller.

- Classical epsilon-delta

[Feb 10, 2011] More examples of limits

Week 5:

[Feb 15, 2011] Lecture 8 - Functions, part I. Chapter 10, Boas & Geller.

- Notation and historical comments
- some historical problems and questions

[Feb 17, 2011] Lecture 9 - Functions, part II. Chapter 10, Boas & Geller.

- limits of sequences of functions
- unusual functions
- dimension matters

Week 6:

[Feb 22, 2011] Lecture 10 - Geometry, part I. Chapter 11, Boas & Geller.

- Euclidean geometry,
- Algebraic geometry,
- Analytic Geometry

[Feb 24, 2011] Lecture 11 - Geometry, part II. Chapter 11, Boas & Geller.

- Curvature

Week 7:

[March 1, 2011] Lecture 12 - Geometry, part III. Chapter 11, Boas & Geller.

- Geometer's Sketchpad
- Non Euclidean Applet

[March 3, 2011] Lecture 13 - Applications of Geometry [Summary]

Week 8:

[March 8, 2011] Lecture 14 - Calculus of Variations, part I.

[March 11, 2011] Lecture 15 - Calculus of Variations, part II.

SPRING BREAK - March 14-18, 2011

Week 9:

[March 22, 2011] Lecture 16 - Optimization.

[March 24, 2011] - More applications of Optimzation

Week 10:

[March 29, 2011] Lecture 17 - Complex Arithmetic

See Chapter 12 of Boas & Geller

[March 31, 2011] Lecture 18 - Computer Graphics, using Complex Numbers and Quaternions

Week 11:

[April 5, 2011] Lecture 19 - Survey of Historically important problems. Ideas for projects.

[April 7, 2011] Lecture 20 - More Historically important problems.

Week 12:

[April 12, 2011] Lecture 21 - (Very) recent results concerning the distribution of primes.

[April 14, 2011] Lecture 22 - Proof of Fermat's (Little) Theorem

Week 13:

[April 19, 2011] Lecture 23 - Cryptography, Prime Factorizations, and Public Key encryption

[April 21, 2011] Lecture 24 - Problems and Puzzles involving primes

Week 14: