## Course Descriptions

MTH 338: Non-Euclidean Geometry
MTH 434/534: Differential Geometry
MTH 437/537: General Relativity
PH 429/529: Reference Frames

### MTH 338: Non-Euclidean Geometry The world we live in is not Euclidean!

Euclid tried (unsuccessfully!) to formulate a series of postulates for the geometry of a (flat, infinite) piece of paper. His abstract model was supposed to accurately reflect the world around us in that his postulates were to be "self-evident". However, his most famous postulate, the parallel postulate, turns out to be independent of the others. This leaves room for non-Euclidean geometries in which the other postulates are satisfied but which have too many or too few parallel lines.

This course investigates several types of geometry which differ fundamentally from the geometry of a piece of paper. We will discuss what it means to specify a geometry in terms of a set of axioms, and we will illustrate this concept with simple finite geometries. We will then study the classic alternatives to the parallel postulate, namely geometries with too many parallel lines (hyperbolic geometry) or too few parallel lines (elliptic geometry). Considerable time will be spent discussing specific models for these geometries in addition to their general properties.

We will also study the beautiful model of geometry called taxicab geometry, which is based on the notion that the "taxicab" distance between two points is not in a straight line "as a crow flies" but rather along streets "as a taxicab drives". This model has many surprising and "non-Euclidean" features, but actually does satisfy Euclid's postulates, including the parallel postulate. (It does not, however, satisfy the modern, corrected versions of Euclid's postulates and is thus classified as non-Euclidean.)

We will construct and investigate all of these models using computer graphics generated with Mathematica, and we will meet regularly in the Math Learning Center (MLC) Computer Lab. No prior experience with computers will be assumed. This course is also a Writing Intensive Course (WIC), and a 5-7 page paper on a topic in geometry will be required. The prerequisite for MTH 338 is MTH 252 or consent of the instructor. Every effort will be made to make the course self-contained, although prior background in Euclidean geometry and the ability to construct simple proofs would be helpful.

### MTH 434/534: Differential Geometry Differential geometry is vector calculus done right.

This course is a self-contained introduction to the many uses of differential forms. This approach emphasizes geometric content in a coordinate independent way. A good analogy is the use of vectors, rather than their components, to describe a given situation. While we will spend some time developing the necessary mathematical tools, the emphasis will be on applying these tools to concrete examples drawn from the physical sciences.

The formal prerequisites for MTH 434 are MTH 312 & MTH 341. However, the main prerequisite is a certain amount of scientific maturity, rather than background in a particular area. The only specific requirements are a working knowledge of multivariable calculus and linear algebra.