Mathematics 435:01: Geometry
		Professor Zheng-Chao Han

Tentative Schedule for  Fall 2006:

Lecture 1. Introduction; 1.1

Lecture 2. 1.1, 1.2 

Lecture 3. 1.3, rotation matrices.

Lecture 4. 1.3

Lecture 5. Catch up; 2.1, supplement on rotation/reflection matrices

Lecture 6. More on rotation/reflection matrices, 2.2

Lecture 7. 2.3

Lecture 8. 2.4 (not all the proofs, more examples to illustrate the theorems), 2.5

Lecture 9. 2.5, 3.1. 

Lectore 10. 3.2 (add more geometric interpretations)

Lecture 11. 3.3

Lecture 12. Midterm

Lecture 13. 3.4 (emphasize more geometric arguments, such as moving a line to infinity; this chapter
                 should be streamlined to show more geometric aspects, less algebraic aspects; some
		 extra problems towards this goal should be supplemented.)

Lecture 14. 8.1; more on 3.4

Lecture 15. Go over midterm; duality 

Lecture 16. 3.5 (only have time to do 3.5.2, 3.5.3), 7.1

Lecture 17.  7.2

Lecture 18. 7.3.1

Lecture 19. 7.3.3 (do not require the cosine rules); Re-examination of Euclidean geometry 

Lecture 20. Re-examination of Euclidean geometry (start using Euclid's
	    "Elements" Book I and III) : review and discussion of
            basic postulates, discussion of I. 1---12 (mainly I. 4, I.5-6, I.8, I.1).

Lecture 21. Discussion of I. 13---20 (mainly I.16), Congruence theorems: ASA, AAS, ASS, 
            and Hypotenuse-Leg theorem.

Lecture 22. Midterm 2.

Lecture 23. Properties related to the parallel postulate: I.27-30, I.32, Playfair,
           and equidistance properties; Properties
            related to the areas of parallelograms and triangles: I. 33 --- 41.
            Student presentations from Book I.

Lecture 24. Equivalence of several propositions related to parallel lines; Negations
            of these propositions. More student presentations from Book I.

Lecture 25. Pythagaras' theorem. More student presentations on circles from Book III. (Should include 
            some discussion on similar triangles)

Lecture 26. Introduction to  hyperbolic geometry.

Lecture 27. Some basic geometric properties of hyperbolic geometry.

Lecture 28. Review

Comments:  Some geometry topics on my wish list: geometry of maps, algebra of compass/ruler constructions, 
	   incorporation of visualization softwares, some solid geometry(dihedral and trihedral angles,
	   projections)---in addition to its intrinsic value, this part is helpful in the explanation 
	   for the Dandelin spheres in relation to the conic sections. Of course this list could go on
	   with such topics as the role of the algebra of complex numbers in geometry, symmetries in
	   wallpapers, etc, but it seems unrealistic to even touch upon these topics in a one semester
	   course.