Fundamental Problems of Gometry


Math 1040, Spring 2012
Meets: MWF 2:00-2:50 Barus and Holley 158


Notes: Here are some course notes (Last Updated 3/1/2012) that you might find useful. I will try to keep them up to date as we move along.


Reading Assignments:
(Read by Wed., Feb. 1): Euclid's "Elements" (c. 300 BC) can be found in any reasonable library, or various places on the internet (there are links on the Wikipedia page, for example). The Elements consists of 13 ``Books". Read the beginning of Book 1, including the definitions and the axioms, then read several of the propositions to get a feel for it. You will have an assignment related to this reading in Set 1.
(Read by Wed., Feb. 1): Read the "conics" article on the Wikipedia. Reconcile as much of it as possible with what you have learned in class.
(Read by Wed., Feb. 1): Read the first 12 pages of Dieudonne's article on the Historical Development of Algebraic Geometry from the American Mathematical Monthly (Vol. 79 No. 8 (1972) 827-866). Here is a link to it. I am probably infringing on copyrights with this link, but note that you, as a Brown student, could access this freely through JSTOR, or the library. Don't worry too much about understanding the mathematical details, but try to get a feeling for the problems considered by early geometers and the general methods developed to solve these problems over the years.
(Read by Fri., Mar. 2): David Joyce has a nice illustrated version of Euclid's Elements on his website. Read the following propositions and their proofs: III.21, VI.32, IV.5. (Roman numeral is the book number, arabic numberal is the proposition number.) We used propositions roughly like these when discussing the van Yzeren proof of Pascal's Theorem, though VI.32 isn't quite the same as the statement about triangles with pairwise parallel sides that we used, though the proofs are similar.

(References for Axiomatic Geometry): Our notions of "incidence geometry," "projective plane," and the various parallel postulates are fairly standard. The material presented in class is based loosely on the following references (these are readable books written for undergraduates):

R. Hartshorne, Foundations of Projective Geometry, W. A. Benjamin, 1967 (QA471.H28 in the Brown Science Library)
T. E. Faulkner, Projective Geometry, Dover, 2006.
Marvin Greenberg, Euclidean and Non-Euclidean Geometries (Third Edition), W. H. Freeman and Co., New York, 1993.
H. S. M. Coxeter, Non-Euclidean Geometry (6th Edition), MAA, Washington DC, 1998.
D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, 1952.

(We have read some excerpts from the Hilbert/Cohn-Vossen book. Chapter III is most closely related to the things we have done in class. The whole book is fantastic.)
The proof of Hilbert's Theorem (a projective plane is $P^2(F)$ for a skew-field $F$ iff it satisfies Desargues' Theorem) given in class is taken from Hartshorne's book above.

Some more advanced references for projective planes:

Veblen and/or Young, Projective Geometry (in the Brown library they have one by Veblen (call no. QA471.V35) and one by Young (call no. QA471.Y7 c3 in the ANNEX)
Adrian Albert and Reuben Sandler, An Introduction to Finite Projective Planes, New York, Holt, Rinehart and Winston, 1968.
D. Hughes and F. Piper, Projective Planes, Springer-Verlag 1973.
Frederick Stevenson, Projective Planes, San Francisco: W.H. Freeman and Company, 1972.
A. Beutelspacher, Projective planes, pp.107-136 in Handbook of Incidence Geometry, ed. F. Buekenhout, North-Holland, 1995.
Charles Weibel, Survery of non-Desarguesian planes. There is a typo in Weibel's article in the definition of ``near-field."

Some more direct references to the literature:
F. R. Moulton, A simple non-Desarguesian plane geometry, Trans. AMS.
Marshall Hall, Projective planes. Trans. AMS. 54(2) (1943) 229-277.
Clement Lam, The Search for a Finite Projective Plane of Order 10, American Mathematical Monthly 98(4) (1991) 305-318.

I guess the above search turned up empty, because Moorhouse asserts on the link below that there is no finite projective plane of order 10. Moorhouse's page contains computer input data for projective planes of small order:
Projective Planes of Small Order

The construction of projective planes from near-fields (and more general algebraic ``field-like" structures) discussed on April 18 is taken from:
Veblen and Wedderburn, "Non-desarguesian and non-Pascalian geometries," Trans. Amer. Math. Soc. 8(3) (1907) 379-388.

Syllabus

Problem Sets:
Set 1 (Due Friday, Feb. 3)
Set 2 (Due Friday, Feb. 10)
Set 3 (Due Friday, Feb. 17)
Set 4 (Due Friday, Feb. 24)
Set 5 (Due Friday, Mar. 2)
Set 6 (Due Friday, Mar. 9)
Set 7 (Due Friday, Mar. 16)
Set 8 (Due Friday, Mar. 23)
Set 9 (Due Friday, Apr. 6)
Set 10 (Due Monday, Apr. 23)
Set 11 (Due Monday, Apr. 30)


Selected Solutions:
Set 3
Set 4
Set 6

W. D. Gillam