I. Number and Title of Course

Mathematics 620, Modern Geometry: Selected Topics

A. Geometry is an important area of study for graduate students who are high school mathematics teachers. This course would replace a course, Math 621 non-Euclidean Geometry. This course would be part of the breadth requirement for M.S. in Ed. students.

B. The course is offered as a topics course in order to provide flexibility in the course and not to restrict its content to one sub field of Geometry. It would be expected that an instructor would choose a particular area to study.

A. To provide the student with intensive study of one of a range of special topics in Geometry in order to broaden their perceptions of various sub fields of geometry.

B. To encourage the student to participate in the exposition, elaboration, and presentation of special topic of interest in geometry. To encourage the student to undertake independent work in the area.

One or two topics would typically be chosen from among the following list. The list is not meant to be exhaustive.

A. Foundations of Geometry

B. Real Projective Planes

1. Elementary Properties

2. Transformations and Groups of Transformations

3. Desarguesian Planes

4. Pappian Planes

5. Planes over Division Rings and Fields

C. Axiomatic Projective Geometries

D. Linear Geometry

1. Transformations in E2

2. Affine and Euclidean Geometries

3. Projective and non Euclidean Geometries

E. Geometric Algebra

1. Affine and Projective Geometry

2. Sympletic and Orthogonal Geometry

3. The General Linear Group

F. Finite Geometries

1. Basic Conceits

2. Designs

3. Projective and affine Planes

4. Colineations of finite planes

5. Construction of finite Planes

6. Inversive Planes

1. Hyperbolic

a. Parallels with a common perpendicular

b. Parallels without a common perpendicular

c. Horo cyles

d. Triangle Relation

2. Elliptic Geometry

a. Double Elliptic

b. Single Elliptic

Artin, E. An introduction to non-Euclidean Geometry, New York Interscience Publishers, Inc., 1957.

Artzy Rafael. Linear Geometry, Reading, Mass.: Addison-Wesley, 1965

Blumenthal, Leonard M. A Modern View of Geometry, San Francisco: W. H. Freeman and Co., 1961.

Blumenthal, Leonard M. and Menger, Karl. Studies in Geometry, San Francisco: W. H. Freeman and Co., 1971.

Choquet, Gustave. Geometry in a Modern Setting. Boston Houghton-Mifflin Company. 1969

Coxeter, H.S.M.  Introduetion to Geometry, 2nd edition. New York: ?New York: John Wiley and Sons, Inc. 1969.

Coxeter, H.S.M. Projective Geometry, New York: Blaisdell Publishing Company,: l964

Coxeter, H.S.M. Twelve Geometry Essays, Carbondale and Edwardsville: South Illinois University Press. 1968.

Dembowski, P. Finite Geometries. New York. Springer- Verlag, 1969.

Gans, David. An Introduction to non-Euclidean Geometry. New York: Academic Press, 1977.

Grunbaum, Branko.? Gonvex-PoIytopes, New York:? Interscience Publishers, A division of John Weily and Sons, 1967.

Pedoe, Daniel. An Introduction to Projective Geometry. Oxford: Pergamon Press, 1963.

Sadenberg, A. Lectures in Projective Geometry, Princeton, N.J.: D. Van Noshand Company, Inc. 1962.

Stevenson, Frederick AT.? Projective Planes. San Francisco: W. H. Freeman and Co., 1972.

Lectures, discussions, student presentations in class, assigned readings. Oral and written examinations and class presentations.

Acceptance as a graduate student in Mathematics and Math 222 or equivalent.

Three semester hours

This course proposal was examined in accordance with the recommended procedures and was approved by the Graduate Faculty of the Mathematics Department.

Richard A. Wiesen, Chair

Math 620 Modern Geometry: Selected Topics. The topics will be selected from the following list. Foundations, Axiomatic Projective Geometry, Real Projective Geometry, Linear Geometries, Finite Geometries, Non Euclidean Geometries.

A M.A. in Mathematics with special interest in Geometry. There are a number of faculty currently on our staff qualifications exceed these minimum requirements. All have taught undergraduate and graduate geometry courses.

Mabel D. Montgomery, Professor of Mathematics Ph.D. (Mathematics), University of Buffalo, 1953. Guy B. Torchinelli, Associate Professor of Mathematics M.S. (Teaching of Mathematics) University of Illinois, 1958.

Richard A. Wiesen, Professor of Mathematics M.S. (Mathematics) Syracuse University, 1962. Ed.D. (Mathematics Education) State University of New York at Buffalo, 1970.